Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations are one of the most important building blocks in Cambridge KS3 mathematics. They appear in algebra, problem solving, and even in geometry and statistics questions. A linear equation is an equation where the unknown variable has a power of 1, such as 3x + 5 = 17 or y = 2x − 1. Being able to solve these equations confidently will help you in Checkpoint tests and in IGCSE work later on.

线性方程是剑桥 KS3 数学中最重要的基础内容之一。它们出现在代数、问题解决，甚至几何和统计题中。线性方程是指未知变量的次数为 1 的方程，例如 3x + 5 = 17 或 y = 2x − 1。能够熟练地解这些方程将帮助你在 Checkpoint 考试以及以后的 IGCSE 学习中更加自信。

1. What Is a Linear Equation? | 什么是线性方程？

A linear equation has variables that are not squared, cubed, or multiplied together. The graph of a linear equation is always a straight line. In KS3, most linear equations have one unknown, usually written as x, y, or n.

线性方程中的变量不会出现平方、立方或彼此相乘的情况。线性方程的图像总是一条直线。在 KS3 阶段，大多数线性方程只有一个未知数，通常用 x、y 或 n 表示。

For example, 2x + 3 = 11 is linear because x has a power of 1. However, x² + 3 = 11 is not linear because x is squared. Similarly, xy + 3 = 11 is not linear because two variables are multiplied together.

例如，2x + 3 = 11 是线性方程，因为 x 的次数为 1。然而，x² + 3 = 11 不是线性方程，因为 x 被平方了。同样，xy + 3 = 11 也不是线性方程，因为两个变量相乘了。

You can recognise a linear equation by checking that the unknown only appears as a single term, for example 4x, −7y, or n/2. There should be no x², x³, 1/x, or √x terms.

你可以通过检查未知数是否只以单项形式出现来识别线性方程，例如 4x、−7y 或 n/2。方程中不应出现 x²、x³、1/x 或 √x 这样的项。

2. The Balance Method | 天平法

An equation is like a balance scale. The left side and the right side are equal, so the scale is level. If you add, subtract, multiply, or divide one side by a number, you must do exactly the same to the other side to keep the equation balanced.

方程就像一架天平。左边和右边相等，所以天平保持水平。如果你对一边进行加、减、乘、除，就必须对另一边做完全相同的运算，才能保持方程平衡。

This is called the balance method. It is the core idea behind solving linear equations. For example, if x + 3 = 10, you can subtract 3 from both sides. Then x + 3 − 3 = 10 − 3, which simplifies to x = 7.

这被称为天平法。它是解线性方程的核心思想。例如，如果 x + 3 = 10，你可以在等式两边同时减去 3。于是 x + 3 − 3 = 10 − 3，化简后得到 x = 7。

The same principle works for multiplication and division. If 4x = 20, divide both sides by 4: 4x ÷ 4 = 20 ÷ 4, so x = 5. Never divide by zero, but you can divide both sides by any non-zero number.

同样的原则也适用于乘法和除法。如果 4x = 20，两边同时除以 4：4x ÷ 4 = 20 ÷ 4，所以 x = 5。永远不要除以零，但你可以在两边同时除以任何非零的数。

3. One-Step Equations | 一步方程

One-step equations only need one operation to isolate the unknown. There are four main types: addition, subtraction, multiplication, and division equations.

一步方程只需要一步运算就能求出未知数。主要有四种类型：加法方程、减法方程、乘法方程和除法方程。

For an addition equation like x + 6 = 15, subtract 6 from both sides.

对于像 x + 6 = 15 这样的加法方程，两边同时减去 6。

x + 6 − 6 = 15 − 6 → x = 9

For a subtraction equation like x − 4 = 12, add 4 to both sides.

对于像 x − 4 = 12 这样的减法方程，两边同时加上 4。

x − 4 + 4 = 12 + 4 → x = 16

For a multiplication equation like 7x = 42, divide both sides by 7.

对于像 7x = 42 这样的乘法方程，两边同时除以 7。

7x ÷ 7 = 42 ÷ 7 → x = 6

For a division equation like x/3 = 5, multiply both sides by 3.

对于像 x/3 = 5 这样的除法方程，两边同时乘以 3。

(x/3) × 3 = 5 × 3 → x = 15

4. Two-Step Equations | 两步方程

Two-step equations require two operations. A common form is ax + b = c. First undo the addition or subtraction, then undo the multiplication or division.

两步方程需要两步运算。常见形式是 ax + b = c。首先消去加法或减法，然后消去乘法或除法。

Solve 3x + 2 = 14. First subtract 2 from both sides: 3x = 12. Then divide both sides by 3: x = 4.

解方程 3x + 2 = 14。首先两边同时减去 2：3x = 12。然后两边同时除以 3：x = 4。

3x + 2 = 14 → 3x = 12 → x = 4

Solve x/4 − 5 = 3. First add 5 to both sides: x/4 = 8. Then multiply both sides by 4: x = 32.

解方程 x/4 − 5 = 3。首先两边同时加上 5：x/4 = 8。然后两边同时乘以 4：x = 32。

x/4 − 5 = 3 → x/4 = 8 → x = 32

Always perform the inverse operations in the correct order: undo addition or subtraction first, then undo multiplication or division. This is the reverse of the order of operations you use when substituting a value.

一定要按正确的顺序进行逆运算：先消去加法或减法，再消去乘法或除法。这与你代入数值时使用的运算顺序相反。

5. Equations with Brackets | 含括号的方程

When an equation contains brackets, expand them first. Use the distributive law: a(b + c) = ab + ac. Then solve the resulting linear equation using inverse operations.

当方程中含有括号时，首先展开括号。使用分配律：a(b + c) = ab + ac。然后使用逆运算解所得的线性方程。

Solve 5(x + 2) = 35. Expand the left side: 5x + 10 = 35. Subtract 10 from both sides: 5x = 25. Divide by 5: x = 5.

解方程 5(x + 2) = 35。展开左边：5x + 10 = 35。两边同时减去 10：5x = 25。两边同时除以 5：x = 5。

5(x + 2) = 35 → 5x + 10 = 35 → 5x = 25 → x = 5

Solve 2(x − 3) + 4 = 12. First expand: 2x − 6 + 4 = 12. Simplify the left side: 2x − 2 = 12. Add 2 to both sides: 2x = 14. Divide by 2: x = 7.

解方程 2(x − 3) + 4 = 12。首先展开：2x − 6 + 4 = 12。化简左边：2x − 2 = 12。两边同时加上 2：2x = 14。两边同时除以 2：x = 7。

Be careful with negative signs inside brackets. For example, −2(x + 3) means −2 × x + (−2) × 3 = −2x − 6. A common mistake is to write −2x + 6 instead.

注意括号内的负号。例如，−2(x + 3) 表示 −2 × x + (−2) × 3 = −2x − 6。一个常见错误是写成 −2x + 6。

6. Unknowns on Both Sides | 变量在等式两边

Some equations have the unknown on both sides, such as 5x + 2 = 2x + 14. Start by collecting the unknown terms on one side and the number terms on the other side.

有些方程的两边都含有未知数，例如 5x + 2 = 2x + 14。首先把含有未知数的项移到一边，把常数项移到另一边。

Solve 5x + 2 = 2x + 14. Subtract 2x from both sides: 3x + 2 = 14. Then subtract 2 from both sides: 3x = 12. Divide by 3: x = 4.

解方程 5x + 2 = 2x + 14。两边同时减去 2x：3x + 2 = 14。然后两边同时减去 2：3x = 12。两边同时除以 3：x = 4。

5x + 2 = 2x + 14 → 3x + 2 = 14 → 3x = 12 → x = 4

Solve 7x − 3 = 3x + 9. Subtract 3x from both sides: 4x − 3 = 9. Add 3 to both sides: 4x = 12. Divide by 4: x = 3.

解方程 7x − 3 = 3x + 9。两边同时减去 3x：4x − 3 = 9。两边同时加上 3：4x = 12。两边同时除以 4：x = 3。

You can collect the unknown on either side, but many students prefer to keep the coefficient of x positive. If you end up with a negative coefficient, multiply both sides by −1 at the end.

你可以把未知数移到任意一边，但许多学生更喜欢让 x 的系数保持为正。如果最后 x 的系数为负，可以在最后两边同时乘以 −1。

7. Equations with Fractions | 含分数的方程

Equations with fractions can look difficult, but they become easier if you clear the fractions by multiplying every term by the common denominator. Alternatively, use inverse operations step by step.

含分数的方程看起来可能很难，但如果把每一项都乘以公分母，消去分数，就会变得更容易。或者也可以一步一步地使用逆运算。

Solve x/2 + 3 = 7. Subtract 3 from both sides: x/2 = 4. Multiply both sides by 2: x = 8.

解方程 x/2 + 3 = 7。两边同时减去 3：x/2 = 4。两边同时乘以 2：x = 8。

Solve (2x)/3 + 1 = 5. Subtract 1 from both sides: (2x)/3 = 4. Multiply both sides by 3: 2x = 12. Divide by 2: x = 6.

解方程 (2x)/3 + 1 = 5。两边同时减去 1：(2x)/3 = 4。两边同时乘以 3：2x = 12。两边同时除以 2：x = 6。

If the denominator is part of a bracket, such as (x + 2)/3 = 4, multiply both sides by the denominator first. This gives x + 2 = 12, so x = 10. Then substitute back to check: (10 + 2)/3 = 12/3 = 4, which is correct.

如果分母是括号的一部分，例如 (x + 2)/3 = 4，首先将两边同时乘以分母。得到 x + 2 = 12，所以 x = 10。然后代回检验：(10 + 2)/3 = 12/3 = 4，结果正确。

8. Checking Your Solution | 检验答案

After solving an equation, always check your answer by substituting it back into the original equation. Both sides should give the same value.

解完方程后，一定要把答案代回原方程进行检验。两边应得到相同的值。

Solve 4x − 1 = 15. Add 1 to both sides: 4x = 16. Divide by 4: x = 4. Check by substituting x = 4 into the left side: 4 × 4 − 1 = 16 − 1 = 15. The right side is 15, so the solution is correct.

解方程 4x − 1 = 15。两边同时加上 1：4x = 16。两边同时除以 4：x = 4。将 x = 4 代入左边检验：4 × 4 − 1 = 16 − 1 = 15。右边是 15，所以答案正确。

If the two sides do not match, re-check your algebraic steps. Look for arithmetic errors, sign mistakes, or a missed operation. Checking is especially useful in exams because it helps you catch careless errors before moving on.

如果两边不相等，请重新检查你的代数步骤。寻找算术错误、符号错误或遗漏的运算。在考试中检验尤其有用，因为它能帮助你在继续之前发现粗心错误。

9. Word Problems and Forming Equations | 应用题与列方程

In KS3, you will often need to form an equation from a written problem. Read the problem carefully, choose a letter for the unknown, and translate the words into mathematical language.

在 KS3 中，你经常需要根据文字题列出方程。仔细阅读题目，选择一个字母表示未知数，并把文字翻译成数学语言。

Example: I think of a number, multiply it by 5, then subtract 7. The answer is 23. Let the number be n. Then 5n − 7 = 23. Add 7 to both sides: 5n = 30. Divide by 5: n = 6.

例题：我想一个数，把它乘以 5，然后减去 7。答案是 23。设这个数为 n。则 5n − 7 = 23。两边同时加上 7：5n = 30。两边同时除以 5：n = 6。

Another example: A rectangle has length 2x + 1 and width x. Its perimeter is 26. Form the equation 2(2x + 1 + x) = 26. Simplify the bracket: 2(3x + 1) = 26. Expand: 6x + 2 = 26. Subtract 2: 6x = 24. Divide by 6: x = 4. So the width is 4 and the length is 9.

另一个例子：一个矩形的长为 2x + 1，宽为 x。它的周长是 26。列出方程 2(2x + 1 + x) = 26。化简括号：2(3x + 1) = 26。展开：6x + 2 = 26。减去 2：6x = 24。除以 6：x = 4。所以宽是 4，长是 9。

10. Common Mistakes | 常见错误

Many errors in solving equations come from small slips in signs, order of operations, or not applying an operation to every term. Being aware of these can help you avoid them.

解方程中的许多错误来自符号、运算顺序的小失误，或者没有把运算应用到每一项。了解这些错误可以帮助你避免它们。

Mistake 1: Only applying an operation to one side. For x + 5 = 12, you must subtract 5 from both sides, not just the left side. Correct: x = 12 − 5 = 7.

错误 1：只对一边进行运算。对于 x + 5 = 12，你必须两边同时减去 5，而不是只减左边。正确答案：x = 12 − 5 = 7。

Mistake 2: Losing a negative sign. If you have −x = 6, the solution is x = −6, not x = 6. Multiply both sides by −1 to change the sign.

错误 2：丢掉负号。如果得到 −x = 6，解应该是 x = −6，而不是 x = 6。两边同时乘以 −1 来改变符号。

Mistake 3: Expanding brackets incorrectly. Remember that 3(x + 4) = 3x + 12, not 3x + 4. Also −2(x − 5) = −2x + 10, not −2x − 10.

错误 3：括号展开错误。记住 3(x + 4) = 3x + 12，而不是 3x + 4。另外 −2(x − 5) = −2x + 10，而不是 −2x − 10。

Mistake 4: Dividing incorrectly. If 2x = 8, then x = 8 ÷ 2 = 4. Students sometimes subtract instead of dividing. Always use the inverse of the operation attached to x.

错误 4：除法错误。如果 2x = 8，那么 x = 8 ÷ 2 = 4。学生有时会做减法而不是除法。一定要使用与 x 相连的运算的逆运算。

11. Practice Questions with Worked Answers | 练习题与解答

Try these questions before looking at the answers. Cover the answer column and check your work step by step.

在查看答案之前，先尝试做这些题目。遮住答案栏，一步一步检查你的解答。

Question \| 题目	Worked Answer \| 解答
x + 9 = 20	x = 11
4x = 28	x = 7
2x + 5 = 19	2x = 14, x = 7
6(x − 3) = 24	6x − 18 = 24, 6x = 42, x = 7
9x − 4 = 2x + 10	7x = 14, x = 2
x/5 + 2 = 6	x/5 = 4, x = 20

For each question, write out every line of working. This helps you spot mistakes and shows the examiner that you understand the method, not just the final answer.

对于每一道题，写出每一步过程。这能帮助你发现错误，并向考官展示你理解方法，而不仅仅是最终答案。

12. Summary Checklist | 总结清单

Use this checklist to review the key skills for solving linear equations at KS3.

使用这份清单来复习 KS3 解一元一次方程的关键技能。

I can recognise a linear equation and identify the unknown. | 我能识别线性方程并找出未知数。
I can use the balance method to keep both sides equal. | 我能使用天平法保持两边相等。
I can solve one-step equations using inverse operations. | 我能使用逆运算解一步方程。
I can solve two-step equations in the correct order. | 我能按正确顺序解两步方程。
I can expand brackets and solve equations that contain them. | 我能展开括号并解含有括号的方程。
I can collect unknowns on one side and numbers on the other. | 我能把未知数移到一边，把数字移到另一边。
I can solve equations with fractions. | 我能解含有分数的方程。
I can check my answer by substitution. | 我能通过代入检验答案。
I can form and solve equations from word problems. | 我能根据文字题列出并解方程。

If you can tick every box, you are ready for Cambridge KS3 algebra assessments and Checkpoint-style questions on linear equations.

如果你能勾选每一项，你就已经准备好应对剑桥 KS3 代数评估和 Checkpoint 风格的线性方程题目了。

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Solving Linear Equations | 解一元一次方程