Linear equations are one of the most important algebra skills in the Cambridge Lower Secondary Mathematics course. Exercise 94.2 focuses on using the balance method and inverse operations to find unknown values. This article walks through the key ideas, worked examples and common errors so you can build confidence before attempting the questions.
A linear equation is an equation where the unknown, usually x, only appears to the power of 1. There are no x², x³, or fractions such as 1/x. Examples include x + 5 = 12, 3x − 4 = 11 and 2(x + 3) = 14.
In KS3, most linear equations involve one unknown and can be solved in a small number of steps. Being able to recognise them quickly is the first step to solving them accurately.
Think of an equation as a balance. The left side and right side must remain equal. Whatever operation you perform on one side, you must also perform on the other side. This keeps the equation true while you simplify it.
Multiply or divide both sides by the same non-zero number
例如：方程 x + 5 = 12，两边同时减 5，得到 x = 7。天平法确保每一步都不会破坏等式的平衡。
3. Inverse Operations | 逆运算
Solving an equation usually means undoing the operations that have been applied to x. Addition and subtraction are inverse operations, as are multiplication and division. Use the inverse operation to isolate x on one side of the equation.
解方程通常意味着逆向消除作用在 x 上的运算。加法和减法互为逆运算，乘法和除法也互为逆运算。使用逆运算可以把 x 单独留在等号一边。
Operation
Inverse Operation
+ a
− a
− a
+ a
× a
÷ a
÷ a
× a
For example, if x has been multiplied by 3, divide both sides by 3. If 4 has been added to x, subtract 4 from both sides.
例如，如果 x 被乘以 3，就将两边同时除以 3；如果 x 被加上 4，就将两边同时减去 4。
4. Solving One-Step Equations | 解一步方程
A one-step equation needs only one inverse operation. Example: solve x + 7 = 15. Subtract 7 from both sides to get x = 8.
一步方程只需要进行一次逆运算。例如：解 x + 7 = 15。两边同时减去 7，得到 x = 8。
x + 7 = 15 → x = 8
Another example: solve 4x = 20. Divide both sides by 4 to get x = 5.
另一个例子：解 4x = 20。两边同时除以 4，得到 x = 5。
4x = 20 → x = 5
5. Solving Two-Step Equations | 解两步方程
Two-step equations require two inverse operations. Solve 2x + 3 = 11. First subtract 3 from both sides: 2x = 8. Then divide both sides by 2: x = 4.
Always undo addition or subtraction before multiplication or division. This order follows the reverse of the usual order of operations.
一定要先处理加减，再处理乘除。这个顺序与通常运算顺序的逆过程一致。
6. Dealing with Negative Solutions | 处理负数解
Some equations give negative answers. Example: x + 9 = 4. Subtract 9 from both sides to get x = −5. Negative solutions are just as valid as positive ones.
有些方程会得到负数答案。例如：x + 9 = 4。两边同时减去 9，得到 x = −5。负数解和正数解同样有效。
Example: solve 10 − x = 6. Add x to both sides to get 10 = x + 6, then subtract 6 from both sides to find x = 4.
例如：解 10 − x = 6。两边同时加上 x，得到 10 = x + 6，再两边同时减去 6，得到 x = 4。
10 − x = 6 → 10 = x + 6 → x = 4
7. Equations with Brackets | 带括号的方程
When an equation contains brackets, expand them first. Example: solve 2(x + 3) = 14. Expand to 2x + 6 = 14, then subtract 6 and divide by 2 to get x = 4.
Remember to multiply every term inside the bracket by the number outside. This is the distributive law at work.
记住：括号外的数要乘以括号内的每一项。这实际上就是乘法分配律的应用。
8. Variables on Both Sides | 两边都有未知数的方程
If x appears on both sides, collect all x terms on one side first. Example: solve 5x + 2 = 3x + 10. Subtract 3x from both sides: 2x + 2 = 10. Then subtract 2 and divide by 2 to get x = 4.
如果 x 出现在等号两边，要先把所有含 x 的项移到同一边。例如：解 5x + 2 = 3x + 10。两边同时减去 3x，得到 2x + 2 = 10；然后减去 2，再除以 2，得到 x = 4。
5x + 2 = 3x + 10 → 2x + 2 = 10 → x = 4
Always do the same operation to both sides when collecting like terms. This keeps the equation balanced and prevents sign errors.
合并同类项时，两边必须始终做同样的运算。这样可以保持等式平衡，避免符号错误。
9. Checking Your Solution | 检验你的解
After finding x, substitute it back into the original equation. If the left side equals the right side, your solution is correct. For x = 4 in 2x + 3 = 11: 2(4) + 3 = 8 + 3 = 11, which is true.
求出 x 后，把它代回原方程。如果左边等于右边，说明解是正确的。例如把 x = 4 代入 2x + 3 = 11：2(4) + 3 = 8 + 3 = 11，等式成立。
Checking helps catch arithmetic mistakes before you move on to the next question. It is a quick and reliable way to confirm your work.
检验可以帮助你在做下一题之前发现计算错误。这是一种快速可靠地确认答案是否正确的方法。
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📚 Solving Linear Equations: Balanced Methods for KS3 | 解一元一次方程：KS3 平衡法与技巧
Linear equations are one of the most important building blocks of KS3 Cambridge mathematics. They appear in algebra, number problems, geometry and real-life contexts, so being able to solve them confidently is essential for progress.
A linear equation is an algebraic statement where two expressions are set equal to each other, and the unknown letter, usually x, is only raised to the power 1. For example, 2x + 3 = 11 is a linear equation because the highest power of x is 1.
一元一次方程是将两个代数式用等号连接起来的等式，其中未知数字母通常为 x，并且 x 的最高次数为 1。例如 2x + 3 = 11 是一元一次方程，因为 x 的最高次数是 1。
An equation is different from an expression. An expression such as 3x + 5 does not have an equals sign, while an equation such as 3x + 5 = 20 states that two things are equal.
The solution of a linear equation is the value of x that makes the equation true. Solving means finding that value systematically rather than guessing.
一元一次方程的解就是使等式成立的 x 的值。解方程是指用系统的方法求出这个值，而不是靠猜测。
2x + 3 = 11 ⇒ x = 4
2. The Balance Method | 平衡法
Think of an equation as a balanced set of scales. The left-hand side and the right-hand side must always stay equal, so whatever you do to one side, you must do to the other side.
If you add the same number to both sides, subtract the same number, multiply both sides by the same non-zero number, or divide both sides by the same non-zero number, the equality is preserved.
The balance method is the foundation of all equation solving at KS3 and Cambridge Checkpoint level. It helps you keep the equation true while you rearrange it into the form x = a.
平衡法是 KS3 和剑桥 Checkpoint 阶段解所有方程的基础。它能帮助你在逐步变形为 x = a 的形式时保持等式始终成立。
3. Solving One-Step Equations | 解一步方程
Some equations only need one operation to isolate x. For x + 5 = 12, subtract 5 from both sides: x + 5 – 5 = 12 – 5, so x = 7.
有些方程只需要一步运算就能求出 x。对于 x + 5 = 12，两边同时减去 5：x + 5 – 5 = 12 – 5，所以 x = 7。
For x – 7 = 3, add 7 to both sides: x – 7 + 7 = 3 + 7, giving x = 10.
对于 x – 7 = 3，两边同时加上 7：x – 7 + 7 = 3 + 7，得到 x = 10。
For 3x = 18, divide both sides by 3: 3x ÷ 3 = 18 ÷ 3, so x = 6.
对于 3x = 18，两边同时除以 3：3x ÷ 3 = 18 ÷ 3，所以 x = 6。
For x ÷ 4 = 9, multiply both sides by 4: x ÷ 4 × 4 = 9 × 4, so x = 36.
对于 x ÷ 4 = 9，两边同时乘以 4：x ÷ 4 × 4 = 9 × 4，所以 x = 36。
A quick table can help you remember the inverse operation you need for each type of one-step equation.
下面的表格可以帮助你记住每种一步方程所需的逆运算。
Equation | 方程
Operation to undo | 要消去的运算
Inverse step | 逆运算步骤
x + a = b
add a
subtract a
x – a = b
subtract a
add a
ax = b
multiply by a
divide by a
x ÷ a = b
divide by a
multiply by a
4. Solving Two-Step Equations | 解两步方程
A two-step equation involves two operations, such as 2x + 3 = 11. First undo the addition by subtracting 3 from both sides: 2x = 8. Then undo the multiplication by dividing both sides by 2: x = 4.
两步方程包含两种运算，例如 2x +
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Linear equations are one of the most important topics in the Cambridge KS3 mathematics curriculum. They appear in almost every assessment, from classroom tests to end-of-stage examinations. This article explains the key methods for solving one-variable linear equations, including equations with brackets, equations with the unknown on both sides, and simple fractional equations. It is designed to match the skills tested in exercises such as p89_2 and to build confidence step by step.
A linear equation is an algebraic statement that contains one unknown variable, usually written as x, and an equals sign. The word ‘linear’ means that the highest power of the variable is 1. The standard form can be written as:
In this form, a, b, and c represent known numbers, while x is the unknown value we need to find. Solving a linear equation means finding the single value of x that makes both sides equal. For example, in the equation x + 5 = 12, the solution is x = 7 because 7 + 5 = 12. The solution must keep the equation balanced, just like a pair of scales.
在这个形式中，a、b 和 c 代表已知数字，而 x 是需要求出的未知值。解一元一次方程就是找出使等号两边相等的唯一 x 值。例如，在方程 x + 5 = 12 中，解是 x = 7，因为 7 + 5 = 12。方程的解必须保持等号两边平衡，就像一架天平一样。
2. Inverse Operations | 逆运算
The most powerful tool for solving linear equations is the idea of inverse operations. Addition and subtraction are inverse operations, just as multiplication and division are inverse operations. To isolate the unknown, we perform the opposite operation on both sides of the equation at the same time. This keeps the equation balanced and moves us closer to the value of x.
解一元一次方程最有力的工具是逆运算的概念。加法和减法互为逆运算，乘法和除法也互为逆运算。为了把未知数单独留在等号一边，我们同时对等号两边进行相反的运算。这样可以保持方程平衡，并使我们更接近 x 的值。
Operation
Inverse operation
中文对照
+ 4
− 4
加 4 的逆运算是减 4
− 7
+ 7
减 7 的逆运算是加 7
× 3
÷ 3
乘 3 的逆运算是除以 3
÷ 5
× 5
除以 5 的逆运算是乘 5
Remember that whatever operation you do to one side of an equation, you must also do to the other side. If you only change one side, the equation is no longer balanced. This rule is often called the golden rule of algebra and it is the reason why every correct solution follows the same logical pattern.
Let us start with a simple addition equation. To solve x + 7 = 15, we need to remove the number 7 that is added to x. The inverse of adding 7 is subtracting 7, so we subtract 7 from both sides:
让我们从一个简单的加法方程开始。要求解 x + 7 = 15，我们需要去掉加在 x 上的数字 7。加 7 的逆运算是减 7，因此我们在等号两边同时减去 7：
x + 7 − 7 = 15 − 7
x = 8
For a multiplication equation such as 3x = 21, the coefficient 3 is multiplied by x. The inverse operation is division by 3, so we divide both sides by 3:
对于乘法方程，例如 3x = 21，系数 3 与 x 相乘。它的逆运算是除以 3，因此我们在等号两边同时除以 3：
3x ÷ 3 = 21 ÷ 3
x = 7
When an equation involves both addition and multiplication, we undo them in the reverse order of the order of operations. BIDMAS tells us that multiplication should be done before addition, so when solving equations we reverse BIDMAS: first deal with addition or subtraction, then deal with multiplication or division.
Brackets are very common in KS3 linear equations. There are two main ways to solve an equation such as 2(x + 3) = 14. The first method is to expand the bracket by multiplying each term inside by 2:
The second method is to divide both sides by the number outside the bracket first. This is often faster when the number outside the bracket is a factor of the number on the right-hand side:
第二种方法是先将两边同时除以括号外的数字。当括号外的数字能整除右边的数时，这种方法通常更快：
2(x + 3) ÷ 2 = 14 ÷ 2
x + 3 = 7
x = 4
Both methods give the same answer. Choose the method that looks quicker for each question, but always write your working clearly so that marks can be awarded for method in exams.
Many KS3 equations have the unknown on both sides of the equals sign, for example 5x + 2 = 2x + 11. The first step is to collect the x terms on one side and the number terms on the other side. To remove the smaller x term, subtract 2x from both sides:
许多 KS3 方程中，未知数出现在等号两边，例如 5x + 2 = 2x + 11。第一步是把含 x 的项移到一边，把数字项移到另一边。为了消去较小的 x 项，两边同时减去 2x：
5x − 2x + 2 = 2x − 2x + 11
3x + 2 = 11
Now the equation is in a simple form. Subtract 2 from both sides and then divide by 3:
现在方程变成了简单的形式。两边同时减 2，然后除以 3：
3x = 9
x = 3
A common mistake is to add x terms to the wrong side. You may move the variable to either side, as long as you perform the same operation on both sides. However, it is usually easier to move the smaller variable term so that the final coefficient of x is positive.
一个常见的错误是把 x 项加到了错误的一边。你可以把变量移到任意一边，只要你对两边进行相同的运算即可。但是，通常移走较小的变量项会更简单，这样最后 x 的系数为正。
6. Fractional Equations | 分数方程
Fractional equations may look difficult, but they follow the same principles. Consider the equation x/4 + 1 = 3. The term x/4 means x divided by 4. To isolate x, first subtract 1 from both sides:
分数方程看起来可能很难，但它们遵循相同的原理。考虑方程 x/4 + 1 = 3。x/4 这一项表示 x 除以 4。为了把 x 单独留在一边，首先两边同时减 1：
x/4 = 2
Now x is being divided by 4. The inverse operation is multiplication by 4, so multiply both sides by 4:
现在 x 被 4 除。它的逆运算是乘以 4，因此两边同时乘以 4：
(x/4) × 4 = 2 × 4
x = 8
For equations with more than one fraction, you can multiply every term by the lowest common denominator to clear the fractions in one step. This often makes the equation much easier to solve. Always check that your final answer does not make any denominator zero.
After finding a value for x, it is important to check that the value is correct. You do this by substituting the value back into the original equation. Each side should simplify to the same number. For example, if you solve 2x + 3 = 13 and get x = 5, check as follows:
求出 x 的值后，检查这个值是否正确非常重要。你可以把这个值代回原方程进行验证。等号两边化简后应该得到相同的数字。例如，如果你解方程 2x + 3 = 13 得到 x = 5，可以这样检查：
2(5) + 3 = 10 + 3 = 13
The left-hand side equals the right-hand side, so the solution is correct. Checking your solution is useful in exams because it can help you catch small arithmetic errors. It also gives you confidence that your answer is accurate before moving on to the next question.
Students often lose marks on linear equations because of a few repeated mistakes. The most common errors are listed below, together with the correct approach:
学生在解线性方程时经常因为一些重复出现的错误而失分。以下是最常见的错误以及正确的做法：
Forgetting to do the same operation on both sides. Always apply every step to both the left-hand side and the right-hand side.
忘记对等号两边进行相同的运算。每一步都必须同时应用于左边和右边。
Incorrect sign when moving a term. For example, changing 5 to −5 without a proper inverse step. Use inverse operations carefully.
移项时符号错误。例如，在没有进行正确逆运算的情况下把 5 变成 −5。请小心使用逆运算。
Expanding brackets incorrectly, such as writing 2(x + 3) = 2x + 5. Multiply every term inside the bracket by the factor outside.
Dividing at the wrong time. Remember to undo addition and subtraction before multiplication and division unless using the shortcut of dividing through a bracket.
除法的时机错误。除非使用先除以括号外因数的简便方法，否则要先逆算加减，再逆算乘除。
Reading the question carefully and writing every step on a new line will help you avoid these mistakes. A tidy layout also makes it easier for examiners to give you method marks.
仔细读题并把每一步写在新的一行可以帮助你避免这些错误。整洁的书写格式也便于考官给你方法分。
9. Word Problems | 文字题
Linear equations often appear inside word problems. The key skill is to translate the English sentence into an algebraic equation before solving it. Let the unknown number be x, then build the equation step by step. For example, the sentence ‘three times a number plus five equals twenty’ becomes:
Subtract 5 from both sides and divide by 3 to find the number:
两边同时减 5，再除以 3，即可求出这个数：
3x = 15
x = 5
Other common word problems involve ages, perimeter, cost, or consecutive numbers. In every case, identify the unknown quantity, write an equation that describes the relationship, then solve it using the same inverse-operation methods. Always answer the question with a sentence, not just a number.
To become confident with linear equations, regular practice is essential. Start with basic one-step equations, then move to two-step equations, equations with brackets, and equations with variables on both sides. As you improve, try mixed exercises under timed conditions so that you can work accurately and efficiently.
Show every step of working, even for easy questions.
即使题目简单，也要写出每一步计算过程。
Keep the equals signs in a vertical line.
让等号在每一行中垂直对齐。
Check your answer by substitution.
用代回原方程的方法检查答案。
If time remains, go back to the hardest questions.
如果还有时间，回头检查最难的题目。
Linear equations are a core part of the Cambridge KS3 mathematics syllabus. By mastering the inverse operations, understanding brackets, and practising word problems, you will build a strong foundation for more advanced algebra in later stages.
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📚 Solving Linear Equations: KS3 Cambridge Guide | 解一元一次方程：KS3 剑桥指南
A linear equation is an equation in which the unknown variable, usually written as x or y, is not raised to any power other than 1. These equations are central to the KS3 Cambridge Mathematics curriculum and appear in a wide range of problems, from basic arithmetic puzzles to geometry and real-life contexts.
一元一次方程是指未知数（通常写作 x 或 y）的指数仅为 1 的方程。这类方程是剑桥 KS3 数学课程的核心内容，出现在从基础算术谜题到几何和现实生活情境的各类问题中。
In this guide we will develop a clear method for solving linear equations. You will learn how to use inverse operations, keep an equation balanced, remove brackets, clear fractions, and check your final answer.
A linear equation states that two expressions are equal, and the unknown has no exponent greater than 1. For example, x + 5 = 12, 3n − 2 = 7, and 2(y + 4) = 18 are all linear equations.
The goal is to find the value of the unknown that makes the statement true. In many KS3 questions, you are asked to solve an equation and show each step clearly.
目标是求出使等式成立的未知数的值。在许多 KS3 题目中，你都需要解方程并清楚地展示每一步。
2. The Balance Method | 平衡法
The balance method is a visual way to understand equations. Think of the equals sign as the pivot of a balance: whatever you do to one side, you must also do to the other side, otherwise the balance is destroyed.
For example, if you add 3 to the left side of x + 5 = 12, you must add 3 to the right side as well. This rule is the foundation of all algebraic equation solving.
例如，如果你在 x + 5 = 12 的左边加 3，那么右边也必须加 3。这条规则是所有代数方程求解的基础。
3. Solving by Adding and Subtracting | 通过加减解方程
To solve an equation like x + 7 = 15, we need to remove the number that is added to x. The inverse of adding 7 is subtracting 7, so we subtract 7 from both sides.
要解 x + 7 = 15 这样的方程，我们需要去掉加在 x 上的数。加 7 的逆运算是减 7，所以我们在两边同时减去 7。
x + 7 − 7 = 15 − 7
This leaves x = 8. We have isolated the unknown, and the equation is solved. Similarly, for x − 4 = 10, we add 4 to both sides because addition is the inverse of subtraction.
这样就得到 x = 8。我们已经将未知数分离出来，方程解出来了。同样地，对于 x − 4 = 10，我们在两边同时加 4，因为加法是减法的逆运算。
Remember: the operation must be done to both sides, not just one side, and you should write each new line clearly in your working.
记住：运算是两边都要做的，不是只做一边，并且你应该在解题过程中把每一行都写清楚。
4. Solving by Multiplying and Dividing | 通过乘除解方程
If the unknown is multiplied by a number, we use division to undo it. For example, in 4x = 28, the x is multiplied by 4, so we divide both sides by 4.
This gives x = 7. If the unknown is divided by a number, such as x/5 = 3, we multiply both sides by 5, because multiplication is the inverse of division.
这样得到 x = 7。如果未知数除以一个数，例如 x/5 = 3，我们就在两边同时乘以 5，因为乘法是除法的逆运算。
In Cambridge KS3 questions, you will often need to combine these inverse operations when the equation has more than one step.
在剑桥 KS3 题目中，当方程不止一步时，你通常需要结合这些逆运算。
5. Two-Step Equations | 两步方程
A two-step equation has two operations acting on the unknown. For example, in 2x + 3 = 11, the x is first multiplied by 2 and then 3 is added. To solve it, we reverse the operations in the correct order.
A common mistake is to divide by 2 before subtracting 3; that makes the working more difficult and can lead to errors. Always undo the addition or subtraction first.
一个常见的错误是在减 3 之前先除以 2；这会使解题更困难并可能导致错误。一定要先处理加法或减法。
6. Equations with Brackets | 含括号的方程
When an equation contains brackets, you should usually expand them first using the distributive law. For example, 3(x + 4) = 27 becomes 3x + 12 = 27.
Alternatively, you could divide both sides by 3 first, giving x + 4 = 9, and then subtract 4. Both methods are correct when applied carefully.
或者，你也可以先在两边同时除以 3，得到 x + 4 = 9，然后再减去 4。只要认真运用，这两种方法都是正确的。
7. Equations with Fractions | 含分数的方程
Equations with fractions can be simplified by clearing the denominator. For example, in x/3 + 2 = 6, we first subtract 2 from both sides to get x/3 = 4.
Then multiply both sides by 3 to find x = 12. If the equation is (2x)/5 = 4, multiply both sides by 5, giving 2x = 20, then divide by 2 to get x = 10.
然后两边同时乘以 3，得到 x = 12。如果方程是 (2x)/5 = 4，两边同时乘以 5，得到 2x = 20，再除以 2，得到 x = 10。
Always check whether the fraction bar applies to the whole expression. Writing clear brackets can help you avoid mistakes.
始终检查分数线是否适用于整个表达式。写清括号可以帮助你避免错误。
8. Forming Equations from Word Problems | 从文字题建立方程
Word problems require you to translate a situation into an equation. For example, “I think of a number, double it, add 6, and the result is 20.” Let the unknown number be n, so the equation is 2n + 6 = 20.
Solve it by subtracting 6 and then dividing by 2: n = 7. This shows that the number you thought of was 7.
通过先减 6 再除以 2 来解这个方程：n = 7。这说明你最初想的数是 7。
For geometry problems, such as finding a missing side of a triangle when the perimeter is given, you can form an equation by adding all the side expressions and setting them equal to the known perimeter.
After solving an equation, you should always check your answer by substituting it back into the original equation. For example, if you solved 5(x − 2) = 20 and got x = 6, substitute 6 into the left side: 5(6 − 2) = 5 × 4 = 20.
Since the left side equals the right side, the answer is correct. Checking is especially important in KS3 tests because it can catch careless arithmetic errors.
由于左边等于右边，答案是正确的。检验在 KS3 考试中尤其重要，因为它能发现粗心的计算错误。
Make checking part of your routine, and write “Check: 20 = 20” or a short note to show your reasoning.
把检验变成一种习惯，并写出 “检验：20 = 20” 或简短注释来说明你的推理。
10. Common Mistakes and How to Avoid Them | 常见错误及如何避免
One common mistake is forgetting to do the same operation on both sides of the equation. Another is subtracting a number before dealing with multiplication in a two-step equation. Also, when expanding brackets, remember to multiply every term inside the bracket.
By practising these steps and checking your work, you will build accuracy and confidence for the Cambridge KS3 mathematics assessments.
通过练习这些步骤并检查你的解答，你将为剑桥 KS3 数学评估建立准确度和信心。
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Pythagoras’ theorem is a cornerstone of Cambridge KS3 mathematics. It gives a direct connection between the three sides of a right-angled triangle, allowing us to calculate an unknown length when two sides are known. The theorem appears in geometry, coordinate work and real-life measurement problems.
Pythagoras’ theorem states that in a right-angled triangle, the area of the square drawn on the hypotenuse is equal to the sum of the areas of the squares drawn on the other two sides. The hypotenuse is the side opposite the right angle and is always the longest side.
Here a and b represent the two shorter sides, often called legs, and c represents the hypotenuse.
这里 a 和 b 表示两条较短的边，通常称为直角边，c 表示斜边。
2. Right-Angled Triangles and Labelling | 直角三角形与标记
Before applying the theorem, always identify the right angle and the hypotenuse. The hypotenuse is opposite the right angle. The two remaining sides are perpendicular to each other and can be labelled in either order as a and b.
在应用定理前，首先要确定直角和斜边。斜边位于直角对面。其余两条边互相垂直，可以任意标记为 a 和 b。
In a triangle with vertices A, B and C, if angle C is 90°, then side AB is the hypotenuse. The other sides AC and BC are the legs.
在顶点为 A、B、C 的三角形中，如果角 C 为 90°，那么边 AB 就是斜边。其他两边 AC 和 BC 是直角边。
3. The Formula in Detail | 公式详解
The formula a² + b² = c² means ‘a squared plus b squared equals c squared’. Squaring a number means multiplying it by itself. For example, 5² = 5 × 5 = 25.
公式 a² + b² = c² 表示 “a 的平方加 b 的平方等于 c 的平方”。平方是指一个数乘以它自身。例如 5² = 5 × 5 = 25。
To find an unknown side, we often need the inverse operation of squaring, which is taking the square root. The square root of x is written √x. For instance, √25 = 5 because 5² = 25.
If you are finding the hypotenuse, use the formula above. If you are finding one of the shorter sides, rearrange the formula first.
如果求斜边，使用上面的公式。如果求直角边，则先对公式进行变形。
4. Finding the Hypotenuse | 求斜边
Suppose a right-angled triangle has legs of 3 cm and 4 cm. To find the hypotenuse, substitute a = 3 and b = 4 into the formula.
假设一个直角三角形的两条直角边分别为 3 cm 和 4 cm。要求斜边，将 a = 3、b = 4 代入公式。
c² = 3² + 4² = 9 + 16 = 25
Then take the square root: c = √25 = 5 cm. The hypotenuse is 5 cm long.
然后开平方根：c = √25 = 5 cm。斜边长为 5 cm。
Always include the correct unit in your final answer. If the sides were in metres, the hypotenuse would be in metres too.
最终答案要写明正确单位。如果
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📚 KS3 Algebra Essentials: Collecting Like Terms and Simplifying Expressions | KS3 代数核心：合并同类项与化简表达式
Algebra can feel like a new language, but at KS3 the most important skill is learning how to simplify expressions by collecting like terms. This topic appears throughout the Cambridge Checkpoint tests and is the foundation for solving equations, rearranging formulae and working with graphs.
A term is a single number, a letter, or a number and letter multiplied together, such as 5, x, or 7y. In the expression 3a + 2b − 4, the terms are 3a, 2b and −4. The numbers 3 and 2 are called coefficients, while the letters a and b are called variables.
项是一个单独的数、字母，或数与字母相乘的结果，例如 5、x、7y。在表达式 3a + 2b − 4 中，各项为 3a、2b 和 −4。数字 3 和 2 称为系数，字母 a 和 b 则称为变量。
Understanding the parts of an expression is the first step to simplifying it. Every term has a coefficient part and a variable part. In the term 6x², the coefficient is 6, the variable is x, and the exponent is 2.
Like terms have exactly the same variables and exponents. For example, 2x and 5x are like terms, while 2x and 3y are unlike terms. Similarly, 4a² and −7a² are like, but 4a² and 4a are not because the exponents differ.
Sorting terms into groups is a useful skill. In the expression 5p + 3q − 2p + 8q, the p terms are 5p and −2p, while the q terms are 3q and 8q. This grouping allows you to simplify the expression quickly.
3. The Golden Rule: Only Combine Like Terms | 黄金法则：只能合并同类项
You can only add or subtract like terms. Combine their coefficients and keep the variable part unchanged. For example, 4p + 3p = (4+3)p = 7p. If terms are unlike, leave them as separate parts in the final expression.
A common expression to simplify is 5x + 2y − 2x + 3y. First collect the x terms: 5x − 2x = 3x. Then collect the y terms: 2y + 3y = 5y. The simplified form is 3x + 5y.
This method works for longer expressions too. For 10m + 4n + 3m − n − 2, collect the m terms to get 13m, collect the n terms to get 3n, and leave the constant −2. The final answer is 13m + 3n − 2.
这个方法也适用于更长的表达式。对于 10m + 4n + 3m − n − 2，合并 m 项得到 13m，合并 n 项得到 3n，常数 −2 保持不变。最终答案为 13m + 3n − 2。
6a + 3b − 2a + 5b = 4a + 8b
5. Handling Powers and Indices | 处理幂与指数
When simplifying, notice the difference between x² and x. They are not like terms. For example, 3x² + 2x cannot be combined further. Only terms with the same base and the same exponent can be added, such as 5x² + 2x² = 7x².
Remember that x is the same as x¹, so x and x² have different exponents. When terms include higher powers, group them separately: x³ terms with x³ terms, x² terms with x² terms, and x terms with x terms.
请记住，x 等同于 x¹，所以 x 和 x² 的指数不同。当项中含有更高次幂时，要分别分组：x³ 项与 x³ 项合并，x² 项与 x² 项合并，x 项与 x 项合并。
5x² + 2x² = 7x²
6. Worked Example 2: Expressions with x² and x | 示例 2：含 x² 与 x 的表达式
Simplify 4x² + 3x − x² + 5x. Group the squared terms: 4x² − x² = 3x². Group the linear terms: 3x + 5x = 8x. The result is 3x² + 8x.
This example shows why you must not combine x² and x into x³ or any other power. The coefficient and exponent of each group stay separate until the final expression is built.
这个例子说明了为什么不能把 x² 和 x 合并成 x³ 或任何其他幂。各组的系数和指数保持独立，直到写出最终表达式。
4x² + 3x − x² + 5x = 3x² + 8x
7. Using Algebra in Perimeter Problems | 用代数解决周长问题
Simplified expressions often appear in geometry. Suppose a rectangle has width w and length l. Its perimeter is w + l + w + l, which simplifies to 2w + 2l. If a shape has sides 2x, 3x, x and 4, the perimeter is 2x + 3x + x + 4 = 6x + 4.
化简表达式经常出现在几何中。假设一个长方形的宽为 w，长为 l，其周长为 w + l + w + l，化简为 2w + 2l。如果一个图形的边长为 2x、3x、x 和 4，那么周长为 2x + 3x + x + 4 = 6x + 4。
Perimeter questions test your ability to write a sum and then collect like terms. Always add all side lengths first, then combine any terms that have the same variable and exponent.
周长题考查你写出加法式并合并同类项的能力。一定要先把所有边长相加，然后合并变量和指数相同的项。
Perimeter = 2x + 3x + x + 4 = 6x + 4
8. Simplifying Before Substituting Values | 先化简再求值
When a question asks you to evaluate an expression, simplify first. For example, evaluate 2x + 3x + 4 when x = 5. Combine 2x + 3x = 5x, giving 5x + 4. Substitute x = 5: 5(5) + 4 = 25 + 4 = 29.
Substituting without simplifying often creates more work and more chances for error. A simplified expression keeps the arithmetic shorter and clearer, especially when negative numbers or fractions are involved.
9. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Watch out for these frequent errors when simplifying algebraic expressions. Each mistake involves trying to combine terms that are not like terms, or changing the exponent or sign incorrectly.
Always check the sign in front of each term before grouping. A minus sign belongs to the term that follows it, so −2a must be carried into the calculation as a negative coefficient.
分组前一定要检查每一项前面的符号。减号属于它后面的项，因此 −2a 在计算中必须作为负系数来对待。
10. Quick Practice Questions | 快速练习
Try these simplification questions before checking the answers. Write the expression in its simplest form.
在查看答案之前，请先尝试以下化简练习。把每个表达式写成最简形式。
a) 7a + 2a
b) 12b − 5b
c) 4x + 3y + 2x + y
d) 9m² + 2m − 3m² + 5m
e) 6n + 4 − 2n + 7
f) Perimeter of a triangle with sides x, 2x, and 3x + 1
Compare your working with the simplified forms below. Make sure you grouped only like terms and kept the signs correct.
将你的解题过程与下面的化简结果进行比较。确保只合并同类项，
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📚 Fractions, Decimals and Percentages for Cambridge KS3 Mathematics | 剑桥KS3数学：分数、小数与百分比
Fractions, decimals and percentages are three different ways of representing parts of a whole. In the Cambridge KS3 Mathematics curriculum, learners are expected to move fluently between these forms, perform calculations with them, and apply them to real-life problems such as discounts, interest, and data comparison. This revision guide breaks the topic into core skills that frequently appear in Checkpoint tests and end-of-unit assessments.
A fraction is written as a/b, where a is the numerator and b is the denominator. The denominator tells you how many equal parts the whole has been divided into, and the numerator tells you how many of those parts are being considered. For example, 3/4 means three out of four equal parts.
分数写作 a/b，其中 a 是分子，b 是分母。分母表示整体被分成多少个相等的部分，分子表示考虑其中的多少份。例如，3/4 表示四等份中的三份。
Fractions can represent values less than 1, equal to 1, or greater than 1. A proper fraction has a numerator smaller than its denominator, such as 2/5. An improper fraction has a numerator larger than or equal to its denominator, such as 7/4. A mixed number combines a whole number with a proper fraction, such as 1 3/4.
Equivalent fractions have the same value but different numerators and denominators. You can create equivalent fractions by multiplying or dividing both the numerator and denominator by the same non-zero whole number. For example, 1/2 = 2/4 = 3/6 = 50/100.
To simplify a fraction, divide the numerator and denominator by their highest common factor (HCF). For instance, 18/24 simplifies to 3/4 because the HCF of 18 and 24 is 6, and 18 ÷ 6 = 3, 24 ÷ 6 = 4. A fraction is in its simplest form when the numerator and denominator have no common factor other than 1.
To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient becomes the whole number, the remainder becomes the numerator, and the denominator stays the same. For example, 17/5 = 3 2/5 because 17 ÷ 5 = 3 remainder 2.
To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place the result over the original denominator. For example, 2 3/8 = (2 × 8 + 3)/8 = 19/8.
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In Cambridge KS3 Mathematics, simplifying algebraic expressions is one of the most important skills you will build before studying equations, formulae and graphs. It allows you to rewrite a long or messy expression in a shorter, clearer form without changing its value. This article explains what terms are, how to recognise like terms, how to collect them correctly, and how to avoid common errors.
An algebraic term is a number, a variable, or a number and variable multiplied together, such as 3x, −5y, 7, or a². The number part of a term is called the coefficient. In the term 4x, the coefficient is 4, and the variable is x.
When we write terms in an expression, they are usually separated by addition or subtraction signs. For example, in 3x + 5y − 2, there are three terms: 3x, 5y and −2.
Like terms are terms that have exactly the same variable part raised to the same power. Only like terms can be added or subtracted directly. For example, 3x and 7x are like terms, while 3x and 3x² are not like terms because the powers of x are different.
同类项是指变量的部分完全相同，并且变量的指数也相同的项。只有同类项才能直接相加或相减。例如，3x 和 7x 是同类项，而 3x 和 3x² 不是同类项，因为 x 的指数不同。
Here are some examples of like and unlike terms: 2a and 5a are like; 4xy and −7xy are like; 6m² and 9m² are like; but 5p and 5q are not like, and 8n and 8n³ are not like.
Collecting like terms means adding or subtracting the coefficients while keeping the variable part exactly the same. This is because 3x + 5x means three x’s plus five x’s, which makes eight x’s, so 3x + 5x = 8x.
When all coefficients are positive, collecting like terms is straightforward. For example, simplify 6a + 2a + 3a. Since all three terms are like terms, add the coefficients: 6 + 2 + 3 = 11, so the answer is 11a.
You can also collect like terms that contain exponents, as long as the exponent is the same. For instance, 2x² + 4x² = 6x², but 2x² + 4x cannot be combined because the powers of x are different.
When negative signs are involved, it is helpful to think of the sign as belonging to the coefficient. Simplify 9k − 3k. The coefficient of the first term is 9, and the coefficient of the second term is −3. Adding them gives 9 + (−3) = 6, so the result is 6k.
If the coefficients are both negative, add them as negative numbers. For example, simplify −4m − 5m. The coefficients are −4 and −5, so their sum is −9, giving −9m.
A constant is a term with no variable, such as 3, −7 or 12. Constants are like terms with each other, but they cannot be combined with terms that contain variables. For example, in 4x + 3 + 2x + 5, the variable terms 4x and 2x combine to 6x, and the constants 3 and 5 combine to 8.
This is a common structure in Cambridge KS3 questions: group the x-terms together, group the constants together, and then write the simplified expression with the variable term first.
这是剑桥初中数学题中常见的结构：先把含 x 的项放在一起，再把常数项放在一起，然后将化简后的代数式写成变量项在前、常数项在后的形式。
7. Expressions with Several Variables | 含多个变量的代数式
When an expression contains more than one variable, collect each type of variable separately. For example, simplify 3a + 2b + 5a + 4b. The a-terms are 3a and 5a, which give 8a. The b-terms are 2b and 4b, which give 6b. The final expression is 8a + 6b.
If terms have mixed variables such as xy, they must have exactly the same combination of variables to be like terms. For example, 4xy and 9xy are like terms, but 4xy and 4x are not.
Some expressions include brackets, such as 2(x + 3) + 4x. Before collecting like terms, you must expand the bracket. Use the distributive law: multiply the term outside the bracket by each term inside. Here, 2(x + 3) becomes 2x + 6.
9. Common Errors and How to Avoid Them | 常见错误及避免方法
One common mistake is adding unlike terms. For example, 3x + 4y is sometimes written incorrectly as 7xy. This is wrong because x and y are different variables, so the terms cannot be combined.
一个常见错误是把非同类项相加。例如，有人会把 3x + 4y 错误地写成 7xy。这是错误的，因为 x 和 y 是不同的变量，所以这些项不能合并。
Another mistake is changing the power of a variable. For example, 2x² + 3x² equals 5x², not 5x⁴. Only the coefficients are added; the exponent stays the same.
This table shows that a constant and a variable term cannot be combined, and that the exponent must not change when collecting like terms.
上表说明，常数项和含变量的项不能合并，而且合并同类项时指数不能改变。
10. Worked Example and Practice | 例题与练习
Worked example: Simplify 5x + 3y − 2x + 7y + 4. Identify the x-terms, y-terms and constants. The x-terms are 5x and −2x, giving 3x. The y-terms are 3y and 7y, giving 10y. The constant is 4. Therefore, the simplified expression is 3x + 10y + 4.
例题：化简 5x + 3y − 2x + 7y + 4。先确定含有 x 的项、含有 y 的项和常数项。含有 x 的项是 5x 和 −2x，合并得到 3x。含有 y 的项是 3y 和 7y，合并得到 10y。常数项是 4。因此，化简后的代数式为 3x + 10y + 4。
5x + 3y − 2x + 7y + 4 = 3x + 10y + 4
Now try this practice question: Simplify 7a + 4b − 3a + 2b + 6. Collect the a-terms first: 7a − 3a = 4a. Then collect the b-terms: 4b + 2b = 6b. Finally, add the constant 6. The answer is 4a + 6b + 6.
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📚 Fractions, Decimals and Percentages for KS3 Cambridge Mathematics | KS3 剑桥数学：分数、小数与百分数
In KS3 Cambridge Mathematics, fractions, decimals and percentages are three connected ways of describing parts of a whole. This revision guide covers the key methods you need to convert, compare, add, subtract, multiply and divide with confidence, and shows how these skills appear in Checkpoint-style questions.
A percentage is a fraction out of 100. The symbol % means ‘per 100’, so 25% = 25/100.
百分数是分母为 100 的分数。符号 % 表示每一百，所以 25% = 25/100。
Form
Meaning
Example
Fraction
Parts out of equal parts
3/4
Decimal
Place value using powers of ten
0.75
Percentage
Parts out of 100
75%
2. Equivalent Fractions and Simplifying | 等值分数与化简
Equivalent fractions have the same value but use different numerators and denominators. Multiply or divide the top and bottom by the same non-zero number.
等值分数数值相同，但分子和分母不同。将分子和分母同时乘以或除以同一个非零数，可得到等值分数。
a/b = (a×k)/(b×k), k ≠ 0
To simplify a fraction, divide the numerator and denominator by their highest common factor (HCF). For example, 12/18 = 2/3 because HCF(12,18) = 6.
Recurring decimals can be written with dot notation, but at KS3 you are usually asked to round or compare them. A recurring decimal such as 0.333… can be written as 0.3 with a dot above the 3.
4. Converting Decimals to Percentages and Back | 小数与百分数的相互转换
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📚 Mastering Fractions, Decimals and Percentages for Cambridge KS3 Maths | 掌握剑桥KS3数学中的分数、小数与百分数
In the Cambridge KS3 Mathematics curriculum, fractions, decimals and percentages are not isolated topics. They form a connected number system that appears in almost every problem-solving context, from sharing quantities to calculating discounts. This article brings together the key skills you need to move fluently between these three representations and apply them with confidence. We will look at equivalent fractions, conversions, ordering, arithmetic, percentage change and common exam-style questions.
Equivalent fractions have the same value even though they look different. You can create an equivalent fraction by multiplying or dividing both the numerator and the denominator by the same non-zero whole number. For example, 1/2, 2/4, 3/6 and 12/24 are all equal because each one represents the same shaded part of a whole.
等值分数虽然看起来不同，但数值相同。你可以将分子和分母同时乘以或除以同一个非零整数，从而得到等
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📚 Mastering Linear Equations for KS3 Cambridge Maths | 掌握剑桥 KS3 数学一元一次方程
Linear equations are one of the most important building blocks in the KS3 Cambridge Mathematics course. They appear in topics such as number patterns, functions, geometry problems and word problems. A strong grasp of solving linear equations gives you the tools to handle harder algebra later, including simultaneous equations and quadratic equations.
A linear equation in one variable is an equation where the unknown, usually written as x, is raised only to the power of 1. This means there are no terms such as x², x³ or 1/x. The graph of a linear equation is always a straight line, which is where the word ‘linear’ comes from.
一元一次方程是指只含有一个未知数（通常用 x 表示），并且未知数的次数仅为 1 的方程。这意味着方程中不会出现 x²、x³ 或 1/x 这样的项。一元一次方程的图像总是直线，所以被称为线性方程。
Ax + B = C
The standard form is Ax + B = C, where A is not zero. For example, 3x + 2 = 11 and x/4 – 5 = 1 are both linear equations.
标准形式为 Ax + B = C，其中 A 不等于零。例如，3x + 2 = 11 和 x/4 – 5 = 1 都是一元一次方程。
When you solve a linear equation, you are finding the value of x that makes the equation true. The solution is the number that balances both sides of the equation exactly.
解一元一次方程，就是找到使等式成立的 x 的值。这个解就是使方程左右两边完全平衡的数。
Exam tip: In Cambridge KS3 tests, you may be asked to identify whether an equation is linear. Always check that the highest power of x is 1.
考试提示：在剑桥 KS3 测试中，可能会要求判断一个方程是否为线性方程。务必检查 x 的最高次数是否为 1。
2. The Balance Method | 天平法
Think of an equation as a balance scale. The left side of the equals sign and the right side must always have the same value. You can only keep the scale balanced if you do the same operation to both sides.
This idea is called the balance method. If you add, subtract, multiply or divide one side by a number, you must do exactly the same to the other side.
这种思想称为天平法。如果你
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Linear equations are the backbone of algebra at KS3. In this Cambridge KS3 p54_2 style revision guide, you will learn how to solve one-step, two-step, bracket and variable-on-both-sides equations using balance and inverse operations. The unknown, usually written as x, only appears to the power of 1, so every equation can be solved step by step.
A linear equation is an equation where the unknown, usually written as x, has degree 1. This means you will only see terms like x, 2x, −3x or x/4, and never x², x³ or 1/x.
The general form is ax + b = c with a ≠ 0. The solution is the value of x that makes the equation true when substituted back into the original equation.
一般形式为 ax + b = c，其中 a ≠ 0。解就是代入原方程后能使方程成立的 x 值。
ax + b = c, a ≠ 0
2. Balance Principle and Inverse Operations | 等式平衡原理与逆运算
An equation is balanced like a set of scales. If you add, subtract, multiply or divide one side, you must do exactly the same to the other side, otherwise the equation becomes false.
The key is to use inverse operations. Addition and subtraction are inverse operations; multiplication and division are inverse operations. Your aim is to isolate x step by step on one side of the equals sign.
关键是使用逆运算。加法与减法互为逆运算；乘法与除法互为逆运算。你的目标是逐步把 x 单独留在等号一边。
The table below shows common inverse pairs you will use when solving equations.
下表展示了你在解方程时常用的逆运算配对。
Operation 运算
Inverse Operation 逆运算
+ 5
− 5
− 7
+ 7
× 3
÷ 3
÷ 4
× 4
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📚 Solving Linear Equations for Cambridge KS3 | 剑桥KS3线性方程解法精讲
Linear equations are one of the most important building blocks in Cambridge KS3 mathematics. They appear in nearly every topic, from number problems to geometry, and they are tested regularly in classwork, homework and end-of-stage assessments. This article explains how to solve linear equations step by step, using clear examples and exam-style tips.
A linear equation is an equation where the unknown, usually written as x, is raised only to the power of 1. This means there are no x² terms, no x³ terms, and no variables inside square roots or denominators if the equation is in standard linear form.
线性方程是指未知数（通常用 x 表示）的指数仅为 1 的方程。这意味着方程中没有 x² 项、没有 x³ 项，并且在标准线性形式下变量不会出现在平方根或分母中。
The general form of a linear equation in one variable is ax + b = c, where a, b and c are numbers. For example, 2x + 3 = 11 is linear, but x² + 2 = 6 is not linear.
An equation is like a balance scale. Whatever you do to one side, you must do to the other side. This rule is the golden rule of algebra and it keeps the equation true.
If you add, subtract, multiply or divide one side by a number, you must do exactly the same to the other side. For example, if x = 5, then x + 3 = 8 only if you add 3 to both sides.
To solve a one-step equation, you perform the inverse operation to isolate the variable. If the equation says x + 4 = 9, subtract 4 from both sides.
要解一步方程，你需要执行逆运算来分离变量。如果方程是 x + 4 = 9，就从两边都减去 4。
x + 4 = 9 x + 4 − 4 = 9 − 4 x = 5
If the equation says 3x = 12, divide both sides by 3 to find x = 4.
如果方程是 3x = 12，把两边都除以 3，得到 x = 4。
3x ÷ 3 = 12 ÷ 3 ⇒ x = 4
4. Solving Two-Step Equations | 解两步方程
A two-step equation involves two operations. For example, 2x + 3 = 11 has multiplication by 2 and addition of 3. You must undo these operations in reverse order: first subtract 3, then divide by 2.
Always reverse the order of operations: addition or subtraction first, then multiplication or division. This is a common exam technique.
始终按运算顺序的逆序操作：先做加法或减法的逆运算，再做乘法或除法的逆运算。这是一项常见的考试技巧。
5. Equations with Brackets | 含括号的方程
When an equation contains brackets, expand them first. Use the distributive law: multiply each term inside the brackets by the number outside.
当方程含有括号时，要先展开括号。使用分配律：将括号内的每一项乘以括号外的数。
3(x + 2) = 18 3x + 6 = 18 3x = 12 x = 4
After expanding, the equation becomes a two-step or one-step equation, which you already know how to solve.
展开后，方程就变成一步或两步方程，你已经知道如何求解了。
6. Equations with Variables on Both Sides | 变量在方程两边的方程
Sometimes the variable appears on both sides of the equation, such as 5x + 2 = 3x + 10. The first step is to collect like terms by subtracting the smaller variable term from both sides.
Then solve the resulting two-step equation as normal. Always check that you have collected the variable terms on one side only.
然后像平常一样解得到的两步方程。始终检查你是否已经将变量项集中到了一边。
7. Equations with Fractions | 含分数的方程
If a linear equation contains fractions, the quickest method is to multiply every term by the lowest common denominator (LCD). This clears the fractions.
如果线性方程含有分数，最快的方法是将每一项都乘以最小公分母（LCD）。这样可以清除分数。
x/2 + 3 = 7 2(x/2) + 2(3) = 2(7) x + 6 = 14 x = 8
Alternatively, you can treat the fraction as a division and use the inverse operation, but clearing fractions is usually cleaner and faster in exams.
或者，你可以把分数看作除法并使用逆运算，但在考试中，清除分数通常更简洁、更快捷。
8. Checking Your Solution | 检验你的解
Always substitute your answer back into the original equation to check that the left side equals the right side. This takes a few seconds and can catch careless mistakes.
始终将你的答案代回原方程，检查左边是否等于右边。这只需几秒钟，却能发现粗心错误。
For 2x + 3 = 11, if x = 4, then 2(4) + 3 = 8 + 3 = 11, which is correct.
One frequent mistake is forgetting to do the same operation on both sides. For example, writing x + 4 = 9, then x = 9 − 4 is correct, but writing x + 4 = 9, then x = 13 is wrong because the 4 was added instead of subtracted.
一个常见错误是忘记对两边做相同的运算。例如，x + 4 = 9，正确的下一步是 x = 9 − 4，但如果写成 x = 13 就错了，因为把 4 加到了右边而不是减去。
Another common error is mishandling negative signs. In the equation 3 − x = 5, subtracting 3 gives −x = 2, so x = −2, not x = 2.
另一个常见错误是处理负号不当。在方程 3 − x = 5 中，两边减 3 得到 −x = 2，所以 x = −2，而不是 x = 2。
Also, when expanding brackets, students sometimes forget to multiply all terms inside the bracket. For 2(x + 3), the expansion is 2x + 6, not 2x + 3.
Try these questions without looking at the solutions. They follow the style of Cambridge KS3 assessments.
尝试在不看答案的情况下完成以下题目。它们遵循剑桥 KS3 评估的风格。
Solve x + 9 = 15.
Solve 4x − 5 = 19.
Solve 3(x − 2) = 12.
Solve 7x + 1 = 4x + 13.
Solve x/3 + 2 = 5.
The answers are x = 6, x = 6, x = 6, x = 4, and x = 9. Check each one by substitution.
答案分别是 x = 6、x = 6、x = 6、x = 4 和 x = 9。请通过代入检验每一个答案。
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📚 Mastering Fractions, Decimals and Percentages for Cambridge KS3 | 剑桥KS3数学：掌握分数、小数与百分数
Fractions, decimals and percentages are three connected ways of describing a part of a whole. In the Cambridge KS3 mathematics curriculum, you are expected to convert between these forms, compare them confidently and use them in realistic problems such as discounts, data and measurement.
A fraction, a decimal and a percentage can all represent exactly the same value. For example, one half can be written as 1/2, 0.5 or 50%. These are called equivalent forms because they describe the same part of a whole.
A fraction has a numerator above a denominator, such as 3/4. A decimal uses place value columns after the point, including tenths, hundredths and thousandths. A percentage is simply a number of parts out of 100.
Understanding this equivalence helps you choose the most useful form for a problem. For example, multiplying by a decimal is often easier than multiplying by a fraction, while a percentage is easier to interpret in a sales sign.
To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/8 means 3 ÷ 8, which gives 0.375. This method works for any fraction.
When the denominator is 10, 100 or 1000, you can write the numerator directly in the correct decimal place. For example, 7/100 = 0.07 and 23/1000 = 0.023.
Some fractions produce recurring decimals. For example, 1/3 = 0.333… and 2/9 = 0.222… . In Cambridge KS3 questions, a recurring decimal is often shown with a dot above the repeating digit or with three dots after the number.
For decimals greater than 1, the percentage is greater than 100. For example, 1.25 = 125% and 2.8 = 280%. This often appears in percentage increase questions.
When converting a decimal such as 0.6 to a percentage, you can think of it as
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This article reviews the core fraction and decimal skills that appear in Cambridge KS3 Mathematics, including the types of questions often found on page 50, exercise 2 of practice booklets. The focus is on clear methods, common errors, and exam-style problem solving.
A fraction represents part of a whole and is written as a/b, where a is the numerator and b is the denominator. The denominator tells you how many equal parts the whole is divided into, and the numerator tells you how many of those parts are taken.
分数表示整体的一部分，写作 a/b，其中 a 是分子，b 是分母。分母表示整体被分成多少等份，分子表示取了多少份。
A decimal uses a decimal point to show tenths, hundredths, thousandths and so on. For example, 0.7 means seven tenths, and 0.25 means twenty-five hundredths, which can also be written as 25/100.
Equivalent fractions have the same value but different numerators and denominators. You can create an equivalent fraction by multiplying or dividing both the numerator and denominator by the same non-zero number.
To simplify a fraction, divide the numerator and denominator by their highest common factor (HCF). For example, 18/24 simplifies to 3/4 because the HCF of 18 and 24 is 6.
To add or subtract fractions, they must have the same denominator. If they do not, find the lowest common multiple (LCM) of the denominators and convert each fraction to an equivalent fraction with that common denominator.
When subtracting, use the same method but subtract the numerators. Always simplify your final answer if possible.
减法使用相同的方法，但需要将分子相减。如果可能，最后一定要化简答案。
4. Multiplying Fractions | 分数的乘法
Multiplying fractions is straightforward: multiply the numerators together and multiply the denominators together. You do not need a common denominator.
分数相乘非常简单：分子相乘，分母相乘。不需要公分母。
For example, 2/3 × 3/5 = (2 × 3)/(3 × 5) = 6/15, which simplifies to 2/5. You can also cancel common factors before multiplying to make the numbers smaller.
If a mixed number is involved, such as 1 1/2, convert it to an improper fraction first: 1 1/2 = 3/2, then multiply as usual.
如果涉及带分数，如 1 1/2，先把它转换为假分数：1 1/2 = 3/2，然后按常规方法相乘。
5. Dividing Fractions | 分数的除法
To divide by a fraction, multiply by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and denominator. For example, the reciprocal of 2/5 is 5/2.
For the calculation 3/4 ÷ 2/5, first flip the second fraction and change the division sign to multiplication: 3/4 × 5/2 = 15/8. This can be written as the mixed number 1 7/8.
Remember that dividing by a number smaller than 1 gives an answer larger than the original number, which is a useful check.
请记住，除以一个小于 1 的数，得到的答案会比原数大，这是一个有用的检验方法。
6. Converting Between Fractions and Decimals | 分数与小数之间的转换
A fraction can be converted to a decimal by dividing the numerator by the denominator. For example, 3/8 means 3 ÷ 8, which gives 0.375.
分数可以通过用分子除以分母转换为小数。例如，3/8 表示 3 ÷ 8，得到 0.375。
Some fractions give terminating decimals, such as 1/4 = 0.25 and 7/20 = 0.35. Others give recurring decimals, such as 1/3 = 0.333… and 2/11 = 0.181818…
When adding or subtracting decimals, align the decimal points and then add or subtract each column. For example, 3.45 + 2.7 should be written with 2.7 as 2.70, giving 6.15.
When multiplying decimals, ignore the decimal points initially, multiply the numbers, and then put the decimal point back so that the answer has the same total number of decimal places as the two original numbers combined.
8. Fractions, Decimals and Percentages | 分数、小数与百分数
Fractions, decimals and percentages are three ways of expressing the same proportion. Converting between them is a key KS3 skill.
分数、小数和百分数是表示同一个比例的三种方式。在它们之间进行转换是 KS3 阶段的一项关键技能。
50% means 50 out of 100, so 50% = 50/100 = 1/2 = 0.5.
50% 表示 100 份中的 50 份，因此 50% = 50/100 = 1/2 = 0.5。
To convert a decimal to a percentage, multiply by 100. So 0.75 = 75%.
将小数转换为百分数，乘以 100。因此 0.75 = 75%。
To convert a fraction to a percentage, first convert it to a decimal and then multiply by 100.
将分数转换为百分数，先转换为小数，然后乘以 100。
Fraction
Decimal
Percentage
1/2
0.5
50%
1/4
0.25
25%
3/5
0.6
60%
7/10
0.7
70%
9. Mixed Operations and Order of Operations | 混合运算与运算顺序
When a calculation involves a mixture of fractions, decimals and operations, you must follow the order of operations: Brackets, Indices, Division and Multiplication (from left to right), Addition and Subtraction (from left to right). This is often remembered as BIDMAS or BODMAS.
Fractions and decimals appear in many everyday situations, including money, measurement, cooking and sharing. Being able to switch between forms helps you make quick estimates and accurate calculations.
For example, if a recipe requires 3/4 cup of milk and you want to make half the recipe, you need 1/2 × 3/4 = 3/8 cup. A discount of 25% on a £60 item gives a saving of £15, because 0.25 × 60 = 15.
When solving word problems, write down the information in mathematical form before calculating. This helps avoid misreading the question and makes your working clear.
解决应用题时，先以数学形式写出已知信息，然后再计算。这有助于避免读错题目，并使你的解题过程清晰。
11. Common Errors and Checking Strategies | 常见错误与检查策略
Forgetting to find a common denominator before adding fractions. Always check that the denominators are the same.
分数相加前忘记通分。一定要检查分母是否相同。
Misplacing the decimal point in multiplication or division. Estimate the answer first to see if the result is reasonable.
乘法或除法中小数点位置放错。先估算答案，看结果是否合理。
Not simplifying the final fraction. Always look for a common factor in the numerator and denominator.
最后没有化简分数。一定要检查分子和分母是否有公因数。
Confusing the reciprocal when dividing fractions. Flip only the second fraction, not the first.
分数除法中混淆倒数。只翻转第二个分数，不要翻转第一个。
A useful checking strategy is to substitute your answer back into the original problem or to solve the problem using a different method. For example, check 1/3 + 1/4 = 7/12 by estimating: 1/3 is about 0.33 and 1/4 is 0.25, so the sum should be about 0.58, and 7/12 is about 0.58.
In Cambridge KS3 exams, fraction and decimal questions often combine two or more skills. You may be asked to simplify an expression, convert between forms, or solve a word problem involving money or measurement.
Always show your working clearly, even if the question only asks for the final answer. Marks are often awarded for method, so a clear step-by-step solution can earn partial credit even if you make a small arithmetic error.
Practice with a mix of pure calculations and word problems. Set out your work in a logical order and leave space for checking. If time allows, verify your final answer by estimating or working backwards.
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📚 KS3 Cambridge Mathematics: Solving Linear Equations Step by Step | 剑桥初中数学：逐步解一元一次方程
Linear equations are one of the most important building blocks in KS3 Cambridge mathematics. This guide covers the key skills you need to solve one-step, two-step and multi-step equations with confidence.
A linear equation in one unknown is a statement that two expressions are equal, such as 3x + 5 = 17. The unknown is usually written as a letter, often x, and its highest power is 1.
In Cambridge KS3, you will see equations like x + 4 = 9, 5x = 35, 2x − 3 = 11, and 4(x + 2) = 28. The goal is always to find the value of the unknown that makes the equation true.
Think of an equation as a pair of balanced scales. Whatever you do to one side, you must do exactly the same to the other side to keep the scales balanced.
The two most common inverse operations are adding/subtracting the same number and multiplying/dividing by the same non-zero number.
两种最常见的逆运算是：同加或同减同一个数，以及同乘或同除以同一个非零数。
If x + 3 = 10, subtract 3 from both sides: x = 7.
如果 x + 3 = 10，两边同时减 3：x = 7。
If 2x = 14, divide both sides by 2: x = 7.
如果 2x = 14，两边同时除以 2：x = 7。
3. Solving One-Step Equations | 解一步方程
A one-step equation needs only one inverse operation to isolate the unknown. For example, x + 6 = 13 is solved by subtracting 6 from both sides.
一步方程只需要一次逆运算就能把未知数单独求出来。例如 x + 6 = 13，两边同时减去 6 即可求解。
For x − 4 = 9, add 4 to both sides to get x = 13. For 5x = 45, divide both sides by 5 to get x = 9.
对于 x − 4 = 9，两边同时加 4，得到 x = 13。对于 5x = 45，两边同时除以 5，得到 x = 9。
x + 6 = 13 → x = 13 − 6 → x = 7
x + 6 = 13 → x = 13 − 6 → x = 7
4. Solving Two-Step Equations | 解两步方程
A two-step equation involves two operations. For example, 2x + 3 = 15 has a multiplication and an addition. First subtract 3 from both sides, then divide both sides by 2.
The order matters: always undo addition or subtraction before undoing multiplication or division.
顺序很重要：一定要先处理加法或减法，再处理乘法或除法。
2x + 3 = 15 → 2x = 12 → x = 6
2x + 3 = 15 → 2x = 12 → x = 6
5. Unknown on Both Sides | 方程两边都有未知数
When the unknown appears on both sides, collect all the unknown terms on one side and all the number terms on the other side. For example, 5x − 2 = 3x + 8.
Then subtract 12 from both sides to get 3x = 15, and divide by 3 to find x = 5.
然后两边同时减 12，得到 3x = 15，再除以 3，求出 x = 5。
3(x + 4) = 27 → 3x + 12 = 27 → 3x = 15 → x = 5
3(x + 4) = 27 → 3x + 12 = 27 → 3x = 15 → x = 5
7. Equations with Fractions | 含分数的方程
To solve equations with fractions, multiply every term by the denominator or the lowest common multiple to clear the fractions. For example, x ÷ 4 = 3 can be written as x/4 = 3.
解含分数的方程时，把每一项都乘以分母或最小公倍数，以消去分数。例如 x ÷ 4 = 3 可以写作 x/4 = 3。
Multiply both sides by 4 to get x = 12. For (x + 2) ÷ 3 = 5, multiply both sides by 3 to get x + 2 = 15, then x = 13.
两边同时乘以 4，得到 x = 12。对于 (x + 2) ÷ 3 = 5，两边同时乘以 3，得到 x + 2 = 15，然后 x = 13。
x ÷ 4 = 3 → x = 12
x ÷ 4 = 3 → x = 12
8. Word Problems into Equations | 文字题转化为方程
Many exam questions describe a real-life situation. Read the problem carefully, let the unknown be x, and build an equation using the information given.
许多考试题目会描述一个实际情境。仔细读题，设未知数为 x，并根据已知信息建立方程。
Example: ‘I think of a number, multiply it by 4 and add 7. The result is 31.’ Let the number be x, so 4x + 7 = 31.
If the check fails, go back and look for an arithmetic or balancing error.
如果检验不成立，就返回去检查是否有计算或平衡操作的错误。
10. Common Mistakes | 常见错误
One common mistake is forgetting to apply an operation to both sides. Another is performing the wrong inverse operation, such as adding instead of subtracting.
一个常见错误是忘记对两边同时进行同一种操作。另一个错误是使用了错误的逆运算，例如该减却加了。
Also be careful with negative numbers. For 3x − 8 = 4, you must add 8 to both sides, giving 3x = 12, then x = 4.
Do not move terms without changing signs correctly.
移项时不要忘记正确变号。
Do not divide only part of one side.
不要只除以某一边的一部分。
11. Exam-Style Practice | 考试风格练习
Try these KS3 Cambridge style questions. Solve each equation and check your answer.
试试以下剑桥初中风格的题目。解出每个方程并检验答案。
4x + 9 = 33
7x − 5 = 2x + 20
2(x − 3) = 18
x ÷ 5 + 2 = 7
The solutions are x = 6, x = 5, x = 12 and x = 25. Use the balancing method to show your working clearly.
答案分别是 x = 6、x = 5、x = 12 和 x = 25。使用平衡法清晰展示你的解题步骤。
12. Summary | 小结
To solve linear equations successfully, identify the operations involved, undo them in reverse order using inverse operations, and always keep both sides balanced.
要成功解一元一次方程，先识别涉及的运算，再用逆运算按相反顺序逐步化简，并始终保持两边平衡。
Equation type | 方程类型
Strategy | 策略
One-step | 一步方程
Use one inverse operation | 使用一次逆运算
Two-step | 两步方程
Undo addition/subtraction first | 先处理加减法
Unknown on both sides | 两边有未知数
Collect like terms | 合并同类项
With brackets | 含括号
Expand first | 先展开括号
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📚 Cambridge KS3 Maths: Solving Linear Equations | 剑桥KS3数学：解一元一次方程
This revision guide supports worksheet p42_2.pdf and focuses on solving linear equations, a core skill in the Cambridge Lower Secondary Mathematics curriculum. You will learn how to keep equations balanced, undo operations in the correct order, and check your solutions.
A linear equation is an equation in which the unknown appears only to the first power. It has no squares, cubes, or unknown variables in denominators.
线性方程是未知数只出现一次方的方程。它没有平方、立方，也没有未知数出现在分母中。
Linear equations can usually be written as ax + b = c or ax + b = cx + d, where a, b, c and d are numbers. They are called ‘linear’ because their graphs are straight lines.
线性方程通常可以写成 ax + b = c 或 ax + b = cx + d 的形式，其中 a、b、c、d 是数字。它们被称为“线性”是因为它们的图像是直线。
Examples include x + 5 = 12, 2x − 3 = 9, and 4(x − 1) = 12.
例子包括 x + 5 = 12、2x − 3 = 9 和 4(x − 1) = 12。
2. The Balance Method | 天平法
The balance method is the key idea behind solving any equation. Think of an equation as a balanced set of scales: the left side must equal the right side.
天平法是解任何方程的关键思想。把方程想象成一架平衡的天平：左边必须等于右边。
Add or subtract the same amount from both sides — 在两边加上或减去相同的量。
Multiply or divide both sides by the same non-zero amount — 将两边同时乘以或除以相同的非零量。
As long as you do the same operation to both sides, the equation stays balanced. This allows you to isolate the unknown step by step.
只要你对两边执行相同的运算，方程就会保持平衡。这使你能够一步一步地分离未知数。
3. Solving One-Step Equations | 解一步方程
A one-step equation needs only one inverse operation to solve it. Undo addition with subtraction, subtraction with addition, multiplication with division, and division with multiplication.
一步方程只需一次逆运算即可求解。用减法抵消加法，用加法抵消减法，用除法抵消乘法，用乘法抵消除法。
Example 1: Solve x + 7 = 15.
例1：解方程 x + 7 = 15。
x + 7 − 7 = 15 − 7 → x = 8
Example 2: Solve 5x = 35.
例2：解方程 5x = 35。
5x ÷ 5 = 35 ÷ 5 → x = 7
Example 3: Solve x ÷ 6 = 4.
例3：解方程 x ÷ 6 = 4。
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This revision article covers the key skills from a typical Cambridge KS3 mathematics exercise on fractions, decimals and percentages. You will learn how to switch between these three forms, calculate with fractions, solve percentage problems, and avoid common errors. The examples are designed to match the style of questions you may meet in a Page 41, Exercise 2 worksheet.
A fraction represents a part of a whole. The top number is the numerator, and the bottom number is the denominator. Equivalent fractions have the same value but different numerators and denominators, such as 1/2, 2/4, and 3/6.
To simplify a fraction, divide both the numerator and the denominator by their highest common factor, or HCF. For 12/18, the HCF of 12 and 18 is 6, so 12/18 = 2/3.
To find a fraction of an amount, divide by the denominator and multiply by the numerator. For example, 3/5 of 40 is found by 40 ÷ 5 = 8, then 8 × 3 = 24.
This method works because the denominator tells you how many equal parts the whole is split into, and the numerator tells you how many of those parts are needed.
这种方法成立是因为分母表示整体被分成多少等份，分子表示需要取其中的几份。
2/3 of 60 = 40 | 60 的 2/3 = 40
5/8 of 32 = 20 | 32 的 5/8 = 20
7/10 of 90 = 63 | 90 的 7/10 = 63
3. Converting Between Fractions, Decimals and Percentages | 分数、小数和百分数的转换
The three forms are connected. A fraction can be written as a decimal by dividing the numerator by the denominator. A decimal becomes a percentage by multiplying by 100.
这三种形式相互关联。分数可以通过分子除以分母化成小数；小数乘以 100 就变成百分数。
Fraction | 分数
Decimal | 小数
Percentage | 百分数
1/2
0.5
50%
1/4
0.25
25%
3/4
0.75
75%
1/10
0.1
10%
2/5
0.4
40%
3/8
0.375
37.5%
To convert a percentage back to a decimal, divide by 100. To write a decimal as a fraction, use place value, such as 0.35 = 35/100 = 7/20.
To add or subtract fractions, they must have the same denominator. If they do not, find the lowest common multiple, or LCM, of the denominators and write equivalent fractions.
分数加减时，它们必须有相同的分母。如果分母不同，求分母的最小公倍数，即 LCM，并写成等值分数。
Example: 1/4 + 1/6. The LCM of 4 and 6 is 12. So 1/4 = 3/12 and 1/6 = 2/12. The sum is 5/12.
Increase by 15% means multiply by 1.15 | 增加 15% 表示乘以 1.15
Decrease by 15% means multiply by 0.85 |
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This guide covers the core skills in KS3 Cambridge mathematics that appear on p39.2, focusing on fractions, decimals and percentages. You will learn how to convert between the three forms, compare values, and apply percentage calculations in real contexts.
A fraction represents a part of a whole. The numerator is the top number and shows how many equal parts are taken; the denominator is the bottom number and shows the total number of equal parts. For example, in ¾, 3 is the numerator and 4 is the denominator.
There are three main types of fraction: a proper fraction has a numerator smaller than the denominator, an improper fraction has a numerator larger than or equal to the denominator, and a mixed number combines a whole number with a proper fraction.
Equivalent fractions are written differently but have the same value. You can multiply or divide both the numerator and the denominator by the same non-zero number without changing the value of the fraction. To simplify a fraction, divide both parts by their highest common factor.
For example, 18/24 simplifies to ¾ because both 18 and 24 can be divided by 6. Always check whether the numerator and denominator have any common factor before deciding that a fraction is in its simplest form.
3. Converting Between Fractions and Decimals | 分数与小数的互化
To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/8 = 3 ÷ 8 = 0.375. If the division does not terminate, the decimal will repeat, such as 1/3 = 0.333…
To convert a terminating decimal to a fraction, write the decimal as a fraction over a power of 10 and then simplify. For example, 0.45 = 45/100 = 9/20.
4. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分数的互化
Percentage means ‘out of 100’. To convert a fraction to a percentage, first change it to a decimal, then multiply by 100%. To convert a percentage to a fraction, write the percentage over 100 and simplify if possible.
These common conversions should be memorised because they appear frequently in KS3 tests and Checkpoint papers.
这些常见转换应该熟记，因为它们在 KS3 测验和 Checkpoint 试卷中经常出现。
5. Ordering and Comparing | 排序与比较
When ordering fractions, decimals and percentages, change all values into the same form. Converting to decimals is usually the fastest method. Then compare the digits from left to right, starting with the largest place value.
For example, to order ⅖, 0.45 and 38%, first convert each value: ⅖ = 0.4, 0.45 = 0.45, 38% = 0.38. The order from smallest to largest is 38% < ⅖ < 0.45.
A useful check is to imagine a number line from 0 to 1. Fractions and decimals close to 1 are larger than those closer to 0.
一种有效的检查方法是想象一条从 0 到 1 的数轴。接近 1 的分数和小数比接近 0 的更大。
6. Adding and Subtracting Fractions | 分数加减
To add or subtract fractions, first find a common denominator. Rewrite each fraction with that denominator, then add or subtract the numerators and keep the denominator the same. For mixed numbers, convert them to improper fractions first.
Do not add denominators together. This is one of the most common mistakes in KS3 fraction work.
不要将分母相加。这是 KS3 分数学习中最常见的错误之一。
7. Multiplying and Dividing Fractions | 分数乘除
To multiply fractions, multiply the numerators together and the denominators together. Simplify before multiplying if possible to keep numbers small. To divide by a fraction, multiply by its reciprocal.
To find a percentage of an amount, write the percentage as a fraction over 100 and multiply by the amount. This skill is often tested with money, distance, time and other real-life quantities.
To increase an amount by a percentage, multiply by (100 + p)/100. To decrease an amount by a percentage, multiply by (100 – p)/100. The multiplier must match the direction of the change.
10. Common Misconceptions and Exam Tips | 常见误区与考试技巧
Common mistakes include adding denominators when adding fractions, forgetting to simplify, and confusing 0.05 with 5%. Another frequent error is using 0.5 to represent 5%, when 5% is actually 0.05.
In exams, show clear working at each step. Write the common denominator when adding or subtracting fractions, and use estimation to check that percentage answers are reasonable. For example, 40% of 90 should be less than 90 but more than 30.
When a question gives a mixture of fractions, decimals and percentages, convert everything to decimals first. This reduces errors and makes comparison straightforward.
Finally, always re-read the question to check whether it asks for an increase, a decrease, or a simple fraction-of-an-amount calculation. One word can change the multiplier entirely.
最后，一定要重新读题，确认题目要求的是增加、减少还是简单求一个数的几分之几。一个词就可能完全改变乘
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📚 Solving Linear Equations for KS3 | KS3 数学：解一元线性方程
Linear equations are one of the most important building blocks in KS3 mathematics. They appear in algebra, word problems, graphs and even later GCSE work. This article explains how to solve linear equations step by step using the balancing method, with plenty of examples tailored to the Cambridge KS3 syllabus.
A linear equation is a statement that two expressions are equal, and the unknown usually appears only to the power of 1. For example, x + 5 = 12 and 3x – 4 = 2x + 7 are both linear equations because x is not squared or cubed.
线性方程是表示两个表达式相等的语句，未知数通常只出现一次方。例如 x + 5 = 12 和 3x – 4 = 2x + 7 都是线性方程，因为 x 没有平方或立方。
The word ‘linear’ tells us that if we drew the equation as a graph, it would form a straight line. In KS3 we focus on finding the value of the unknown that makes the equation true.
Think of an equation as a set of balance scales. Both sides must always have the same value. Whatever operation we do to one side, we must also do to the other side.
This idea is the key to all equation solving. If we add, subtract, multiply or divide one side by a number, we must do exactly the same to the other side.
这个思想是解所有方程的关键。如果我们对一边加、减、乘、除一个数，就必须对另一边做完全相同的操作。
Forward operation
Inverse operation
Addition (+)
Subtraction (-)
Subtraction (-)
Addition (+)
Multiplication (×)
Division (÷)
Division (÷)
Multiplication (×)
The table above shows the inverse operations you will use constantly. Always choose the inverse of the operation that is attached to the unknown.
上表展示了你将反复使用的逆运算。始终选择与未知数相连的运算的逆运算。
3. Solving One-Step Equations | 解一步方程
In a one-step equation, only one operation is needed to isolate x. For example, x + 7 = 15 can be solved by subtracting 7 from both sides:
在一步方程中，只需要一步运算就能求出 x。例如 x + 7 = 15，可以两边同时减去 7：
x + 7 – 7 = 15 – 7 → x = 8
Similarly, if the equation is 4x = 36, we divide both sides by 4:
类似地，如果方程是 4x = 36，我们把两边同时除以 4：
4x ÷ 4 = 36 ÷ 4 → x = 9
The operation you choose must be the inverse, or opposite, of the operation attached to x. Addition is undone by subtraction, and multiplication is undone by division.
所选的运算必须是与 x 相连的运算的逆运算，也就是相反运算。加法用减法来抵消，乘法用除法来抵消。
For a division equation like x ÷ 3 = 6, multiply both sides by 3 to get x = 18. For a subtraction equation like x – 9 = 21, add 9 to both sides to get x = 30.
对于 x ÷ 3 = 6 这样的除法方程，两边同时乘以 3 得到 x = 18。对于 x – 9 = 21 这样的减法方程，两边同时加 9 得到 x = 30。
4. Solving Two-Step Equations | 解两步方程
Two-step equations involve two operations, such as multiplication and addition. Solve them by reversing the order of operations. For 2x + 5 = 17, first subtract 5 from both sides, then divide by 2:
Remember: we always undo addition or subtraction before undoing multiplication or division. This is because we normally follow BIDMAS in forward order, so we reverse it when solving.
Another example is 5x – 4 = 31. First add 4 to both sides to get 5x = 35, then divide both sides by 5 to get x = 7.
另一个例子是 5x –
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