📚 Solving Linear Equations Step by Step | 循序渐进解一元一次方程
Linear equations are the backbone of KS3 algebra. In this revision guide, we explain how to solve one-variable linear equations using the balance method, inverse operations, and systematic checking. You will learn how to handle brackets, fractions, unknowns on both sides, and word problems.
一元一次方程是 KS3 代数的核心。在本复习指南中，我们将讲解如何用平衡法、逆运算和系统检验来解一元一次方程。你将学会处理括号、分数、等号两边的未知数以及应用题。
1. What Is a Linear Equation? | 什么是一元一次方程？
A linear equation is an equation where the unknown variable appears only to the first power. It can usually be written in the form ax + b = c, where a, b and c are numbers and x is the unknown. For example, 2x + 3 = 11 is a linear equation because the power of x is 1.
一元一次方程是指未知数只出现一次幂的方程。它通常可以写成 ax + b = c 的形式，其中 a、b、c 是数字，x 是未知数。例如，2x + 3 = 11 就是一个一元一次方程，因为 x 的指数是 1。
The word ‘linear’ tells you that the graph of the equation is a straight line. At KS3 level, the main goal is to find the value of x that makes the equation true.
“线性”一词表示方程的图像是一条直线。在 KS3 阶段，主要目标是求出使等式成立的 x 的值。
x + 7 = 15 | 3x − 4 = 8 | 5(x + 2) = 35
2. The Balance Method | 天平平衡法
Think of an equation as a set of balance scales. The left side and the right side must have exactly the same value. If you add, subtract, multiply or divide one side, you must do exactly the same to the other side.
把方程想象成一台天平。左边和右边的值必须完全相等。如果你在一侧进行加、减、乘或除，就必须在另一侧进行完全相同的运算。
For example, to solve x + 5 = 12, subtract 5 from both sides. This keeps the equation balanced. You get x + 5 − 5 = 12 − 5, which simplifies to x = 7.
例如，要解 x + 5 = 12，需要在等式两边同时减去 5。这样方程仍然保持平衡。你得到 x + 5 − 5 = 12 − 5，化简后得到 x = 7。
x + 5 = 12 → x = 12 − 5 → x = 7
This method prevents you from making the common mistake of changing only one side. Always write the operation on both sides, even if you later do the arithmetic mentally.
这种方法可以防止你只改变一侧的常见错误。即使你之后用心算完成计算，也要在两边写下相同的运算。
3. Inverse Operations | 逆运算
To isolate the unknown, use the opposite operation. Addition and subtraction are inverse operations. Multiplication and division are inverse operations. Squaring and taking the square root are also inverse, though at KS3 linear work this appears less often.
为了把未知数单独留在一边，要使用相反的运算。加法和减法互为逆运算。乘法和除法互为逆运算。平方和开平方也是逆运算，不过在 KS3 的一元一次方程中较少出现。
| Operation | 运算 | Inverse Operation | 逆运算 |
| + 5 | − 5 |
| − 3 | + 3 |
| × 4 | ÷ 4 |
| ÷ 2 | × 2 |
If a number is added to x, subtract that number from both sides. If x is multiplied by a number, divide both sides by that number. Using the correct inverse operation is the key to solving equations efficiently.
如果 x 加上一个数，就在两边同时减去这个数。如果 x 乘以一个数，就在两边同时除以这个数。使用正确的逆运算正是高效解方程的关键。
4. Solving Two-Step Equations | 解两步方程
A two-step equation has two operations acting on the unknown. For example, in 3x + 4 = 19, x is first multiplied by 3 and then 4 is added. To solve it, reverse these operations in the opposite order.
两步方程中有两个运算作用于未知数。例如，在 3x + 4 = 19 中，x 先乘以 3，然后再加上 4。要解这个方程，就要按照相反的顺序反向进行这些运算。
First subtract 4 from both sides: 3x = 15. Then divide both sides by 3: x = 5. This is like undoing the last step first, then the first step.
首先在两边同时减去 4：3x = 15。然后在两边同时除以 3：x = 5。这就像先撤销最后一步，再撤销第一步。
3x + 4 = 19 → 3x = 15 → x = 5
Always ask yourself: ‘What was done to x, and in what order?’ Then reverse the order. This prevents confusion with negative numbers and coefficients.
永远要问自己：“对 x 做了什么运算，顺序是什么？”然后按照相反的顺序反向运算。这可以避免负数和系数带来的混淆。
5. Equations with the Unknown on Both Sides | 未知数在等号两边的方程
Sometimes the variable appears on both sides of the equation, such as 5x + 2 = 2x + 14. The aim is still the same: collect all x terms on one side and all number terms on the other side.
有时变量会出现在等号两边，例如 5x + 2 = 2x + 14。目标仍然相同：把所有含 x 的项移到一边，把所有数字项移到另一边。
A reliable method is to subtract the smaller x term from both sides. Here, subtract 2x from both sides to get 3x + 2 = 14. Then subtract 2 from both sides to get 3x = 12, and finally divide by 3 to get x = 4.
一个可靠的方法是先在两边同时减去较小的含 x 项。这里在两边同时减去 2x，得到 3x + 2 = 14。然后在两边同时减去 2，得到 3x = 12，最后两边同时除以 3，得到 x = 4。
5x + 2 = 2x + 14 → 3x + 2 = 14 → 3x = 12 → x = 4
If the variable term has a negative coefficient, you may prefer to add that term to both sides first. For example, if you have 10 − x = 3, add x to both sides to make the coefficient positive.
如果变量项带有负系数，你可能更愿意先在两边同时加上这个项。例如，如果你遇到 10 − x = 3，可以先把 x 加到两边，使系数变为正数。
6. Equations with Brackets | 含括号的方程
When an equation contains brackets, the standard first step is to expand them. For example, solve 3(x + 2) = 18. Expanding gives 3x + 6 = 18. Then subtract 6 from both sides: 3x = 12. Finally divide by 3: x = 4.
当方程中含有括号时，标准的步骤是先展开括号。例如，解方程 3(x + 2) = 18。展开后得到 3x + 6 = 18。然后在两边同时减去 6：3x = 12。最后两边同时除以 3：x = 4。
Alternatively, if the outside number divides exactly into the right-hand side, you can divide both sides by that number first. In 3(x + 2) = 18, dividing both sides by 3 gives x + 2 = 6, so x = 4. This is usually quicker.
另外，如果括号外面的数能够整除等号右边，你可以先在两边同时除以这个数。在 3(x + 2) = 18 中，两边同时除以 3 得到 x + 2 = 6，所以 x = 4。这样通常更快。
Watch out for negative signs outside brackets. In 2 − (x + 3) = 5, expanding carefully gives 2 − x − 3 = 5, which simplifies to −x − 1 = 5. Add 1 to both sides to get −x = 6, so x = −6.
注意括号外面的负号。在 2 − (x + 3) = 5 中，仔细展开得到 2 − x − 3 = 5，化简后为 −x − 1 = 5。两边同时加 1 得到 −x = 6，所以 x = −6。
7. Equations with Fractions | 含分数的方程
Fractions in equations may look difficult, but they become easier if you clear the denominator. For example, solve x/4 + 1 = 3. Subtract 1 from both sides: x/4 = 2. Then multiply both sides by 4: x = 8.
方程中的分数看起来可能很难，但如果去掉分母，就会变得更容易。例如，解方程 x/4 + 1 = 3。两边同时减去 1：x/4 = 2。然后两边同时乘以 4：x = 8。
When the fraction is a single term, multiply every term by the denominator. For (x + 2)/3 = 5, multiply both sides by 3 to get x + 2 = 15. Then subtract 2 from both sides to get x = 13.
当分数是一个单独的项时，要把每一项都乘以分母。对于 (x + 2)/3 = 5，两边同时乘以 3 得到 x + 2 = 15。然后两边同时减去 2，得到 x = 13。
(x + 2) ÷ 3 = 5 → x + 2 = 15 → x = 13
If there are two fractions, multiply every term by the lowest common denominator. This removes all fractions in one step and leaves a simpler equation to solve.
如果方程中有两个分数，就把每一项都乘以最小公分母。这样可以一步去掉所有分数，留下一个更简单的方程来解。
8. Word Problems and Forming Equations | 应用题与列方程
Many KS3 questions ask you to form an equation from a real-life situation. Identify the unknown quantity and give it a letter, such as x. Then translate the words into mathematical operations step by step.
许多 KS3 题目要求你根据实际情况列方程。先确定未知量，并用一个字母表示，如 x。然后把文字逐步翻译成数学运算。
- English: ‘Three more than a number is eleven’ becomes x + 3 = 11.
中文：“一个数加三等于十一”可以写成 x + 3 = 11。 - English: ‘Five times a number decreased by 2 is 18’ becomes 5x − 2 = 18.
中文：“一个数的五倍减 2 等于 18”可以写成 5x − 2 = 18。 - English: ‘The sum of three consecutive numbers is 42’ becomes x + (x + 1) + (x + 2) = 42.
中文：“三个连续数的和是 42”可以写成 x + (x + 1) + (x + 2) = 42。
After forming the equation, solve it using the methods above. Then check that your answer makes sense in the original situation. This is especially important in word problems, because an answer may be mathematically correct but not realistic.
列出方程后，用上面的方法求解。然后检查答案在原情境中是否合理。这在应用题中尤其重要，因为一个答案可能在数学上正确，但在实际中并不现实。
9. Checking Your Solution | 检验答案
Always substitute your solution back into the original equation to check it. If both sides give the same value, your answer is correct. This takes only a few seconds and catches many small errors.
总是要把你的解代回原方程进行检验。如果两边得到相同的值，说明答案正确。这只需几秒钟，却能避免许多小错误。
For example, after solving 2x − 3 = 9, you get x = 6. Substitute 6 into the left side: 2 × 6 − 3 = 12 − 3 = 9. Since the left side equals the right side, the solution is correct.
例如，解出 2x − 3 = 9 后得到 x = 6。把 6 代入左边：2 × 6 − 3 = 12 − 3 = 9。因为左边等于右边，所以这个解是正确的。
2(6) − 3 = 9 → 12 − 3 = 9 → 9 = 9 ✓
In a test, this checking step can help you decide whether to move on or rework the problem. Make it a habit even when the question does not ask you to check.
在考试中，这个检验步骤可以帮助你判断是继续下一题还是重新计算。即使题目没有要求检验，也要养成这种习惯。
10. Common Mistakes to Avoid | 常见错误与避免方法
One common mistake is applying an operation to only one side of the equation. Remember that an equation is a balance. If you subtract 4 from one side, you must subtract 4 from the other side as well.
一个常见错误是只对等式的一边进行运算。请记住，方程就像天平。如果你在一侧减 4，就必须在另一侧也减 4。
Another mistake is expanding brackets incorrectly, especially when a negative sign is outside. Write each term inside the bracket and change signs carefully. For example, 2 − (x + 3) is not 2 − x + 3; it is 2 − x − 3.
另一个错误是括号展开不正确，尤其是括号外面有负号时。要把括号内的每一项都写出来，并仔细变号。例如，2 − (x + 3) 不是 2 − x + 3，而是 2 − x − 3。
- Do not forget to reverse the operation order in two-step equations.
不要忘记在两步方程中反向进行运算。 - Do not divide only part of a side by the coefficient.
不要只用系数去除一侧的部分项。 - Do not leave fractions mixed with whole numbers carelessly.
不要把分数与整数随意混在一起而不处理。
Reading the question carefully and writing every step neatly will reduce most errors. Show your working, because marks are often awarded for method even if the final answer is wrong.
仔细读题并工整书写每一步，可以减少大多数错误。要展示解题过程，因为即使最终答案错误，方法通常也能得到分数。
11. Practice Questions and Quick Tips | 练习题与速记技巧
Try these quick questions before you finish. Solve each equation and then check your answer by substitution.
在结束之前，请尝试以下快速练习。解出每个方程，然后通过代入检验答案。
- Solve: x + 9 = 17 Answer: x = 8
解： x + 9 = 17 答案： x = 8 - Solve: 4x − 3 = 13 Answer: x = 4
解： 4x − 3 = 13 答案： x = 4 - Solve: 7x + 1 = 3x + 17 Answer: x = 4
解： 7x + 1 = 3x + 17 答案： x = 4 - Solve: 2(x − 5) = 12 Answer: x = 11
解： 2(x − 5) = 12 答案： x = 11 - Solve: x/3 + 2 = 6 Answer: x = 12
解： x/3 + 2 = 6 答案： x = 12
Quick tip: always do the opposite of the last operation first. Another quick tip: if the unknown has a negative sign, change it to positive early. These habits will help you solve linear equations confidently and accurately.
速记技巧：总是先做与最后一步运算相反的运算。另一个速记技巧：如果未知数带负号，尽早把它变为正号。这些习惯将帮助你自信而准确地解一元一次方程。
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