📚 Solving Linear Equations: Balance Method and Step-by-Step Worked Examples | 线性方程求解：平衡法及分步详解
Linear equations are the foundation of algebra at KS3 Cambridge Mathematics. They appear in almost every topic, from number problems to geometry and graphs. This article explains how to solve one-step, two-step, and multi-step linear equations using the balance method, with clear worked examples and common mistakes to avoid.
线性方程是剑桥 KS3 数学中代数部分的基础。它们在几乎所有主题中都会出现，从数字问题到几何和图像。本文介绍如何使用平衡法解一步、两步和多步线性方程，并给出清晰的例题和常见错误分析。
1. What Is a Linear Equation? | 什么是线性方程？
A linear equation is an equation in which the unknown variable, usually written as x, has an exponent of 1. The graph of a linear equation is always a straight line. For example, 3x + 5 = 20 is a linear equation because x is not squared, cubed, or under a root.
线性方程是指未知数（通常写作 x）的指数为 1 的方程。线性方程的图像总是一条直线。例如，3x + 5 = 20 就是一个线性方程，因为 x 没有平方、立方或开方。
In KS3, you will meet equations such as x + 7 = 12, 4x − 3 = 17, 5x + 2 = 3x + 10, and 2(x + 3) = 14. The goal is always the same: find the value of x that makes the equation true.
在 KS3 阶段，你会遇到诸如 x + 7 = 12、4x − 3 = 17、5x + 2 = 3x + 10 和 2(x + 3) = 14 的方程。目标总是相同的：求出使等式成立的 x 的值。
2. The Balance Method | 平衡法
An equation is like a pair of balanced scales. The left side must equal the right side. Whatever you do to one side, you must also do to the other side to keep the equation balanced.
方程就像一架平衡的天平。左边必须等于右边。无论你对一边做什么操作，也必须对另一边做同样的操作，才能保持方程平衡。
Suppose 3x + 5 = 20. To find x, we first subtract 5 from both sides, then divide both sides by 3. This gives 3x = 15 and then x = 5.
假设 3x + 5 = 20。为了求出 x，我们首先从两边减去 5，然后将两边除以 3。这样得到 3x = 15，然后 x = 5。
Using the balance method helps you show each step clearly. It also reduces mistakes because you treat both sides equally.
使用平衡法有助于清晰地展示每一步，还能减少错误，因为你平等地处理方程的两边。
3. Solving One-Step Equations | 解一步方程
A one-step equation requires only one operation to isolate x. For example, x + 9 = 15 is solved by subtracting 9 from both sides, giving x = 6.
一步方程只需一步运算就能分离出 x。例如，x + 9 = 15 可以通过从两边减去 9 来求解，得到 x = 6。
If the equation is x − 4 = 11, you add 4 to both sides, so x = 15. If the equation is 5x = 35, you divide both sides by 5, so x = 7.
如果方程是 x − 4 = 11，你需要在两边加上 4，所以 x = 15。如果方程是 5x = 35，你需要把两边都除以 5，所以 x = 7。
If the equation is x ÷ 6 = 3, you multiply both sides by 6, so x = 18. Always check that your answer makes the original equation true.
如果方程是 x ÷ 6 = 3，你需要将两边都乘以 6，所以 x = 18。始终检查你的答案是否使原来的等式成立。
4. Solving Two-Step Equations | 解两步方程
A two-step equation needs two operations to isolate x. For example, 2x + 5 = 13. First subtract 5 from both sides: 2x = 8. Then divide both sides by 2: x = 4.
两步方程需要两步运算来分离 x。例如，2x + 5 = 13。首先从两边减去 5：2x = 8。然后将两边除以 2：x = 4。
The order of operations is reversed when solving equations. In the original expression, x was multiplied by 2 and then 5 was added. To solve, we undo the addition first, then undo the multiplication.
解方程时运算顺序是相反的。在原来的表达式中，x 先被乘以 2，然后再加上 5。解方程时，我们先撤销加法，再撤销乘法。
Worked example: Solve 7x − 4 = 24.
例题：解 7x − 4 = 24。
Add 4 to both sides: 7x = 28.
两边同时加 4：7x = 28。
Divide both sides by 7: x = 4.
两边同时除以 7：x = 4。
Check: 7 × 4 − 4 = 28 − 4 = 24, which is correct.
检验：7 × 4 − 4 = 28 − 4 = 24，结果正确。
5. Equations with Unknowns on Both Sides | 未知数在等号两边的方程
When an equation has x on both sides, such as 5x + 2 = 3x + 10, you should first collect the x terms on one side. Subtract 3x from both sides: 2x + 2 = 10.
当方程两边都有 x 时，例如 5x + 2 = 3x + 10，你应该先把含 x 的项移到一边。从两边减去 3x：2x + 2 = 10。
Then subtract 2 from both sides: 2x = 8.
然后从两边减去 2：2x = 8。
Finally divide both sides by 2: x = 4.
最后把两边除以 2：x = 4。
Check: Left side = 5 × 4 + 2 = 22. Right side = 3 × 4 + 10 = 22. Both sides match, so the solution is correct.
检验：左边 = 5 × 4 + 2 = 22。右边 = 3 × 4 + 10 = 22。两边相等，因此解是正确的。
6. Equations with Brackets | 含括号的方程
If an equation contains brackets, expand them first. For example, 2(x + 3) = 14 becomes 2x + 6 = 14 after expansion.
如果方程中含有括号，应首先展开括号。例如，2(x + 3) = 14 展开后变为 2x + 6 = 14。
Then subtract 6 from both sides: 2x = 8. Divide both sides by 2: x = 4.
然后从两边减去 6：2x = 8。两边除以 2：x = 4。
Worked example: Solve 3(2x − 1) = 27.
例题：解 3(2x − 1) = 27。
Expand the brackets: 6x − 3 = 27.
展开括号：6x − 3 = 27。
Add 3 to both sides: 6x = 30.
两边同时加 3：6x = 30。
Divide both sides by 6: x = 5.
两边同时除以 6：x = 5。
Check: 3(2 × 5 − 1) = 3(10 − 1) = 3 × 9 = 27, which is correct.
检验：3(2 × 5 − 1) = 3(10 − 1) = 3 × 9 = 27，结果正确。
7. Equations Involving Fractions | 含分数的方程
When an equation contains a fraction, multiply every term by the denominator to clear it. For example, x/4 + 3 = 7. Multiply both sides by 4: x + 12 = 28.
当方程中包含分数时，将所有项乘以分母以去掉分母。例如，x/4 + 3 = 7。将两边乘以 4：x + 12 = 28。
Then subtract 12 from both sides: x = 16.
然后从两边减去 12：x = 16。
Check: 16/4 + 3 = 4 + 3 = 7, which is correct.
检验：16/4 + 3 = 4 + 3 = 7，结果正确。
If the equation is (x + 2)/3 = 5, multiply both sides by 3 first: x + 2 = 15. Then subtract 2 from both sides: x = 13.
如果方程是 (x + 2)/3 = 5，先将两边乘以 3：x + 2 = 15。然后从两边减去 2：x = 13。
This method works for any equation with a single fraction on one side. If there are fractions on both sides, find the lowest common denominator first.
此方法适用于一边只有一个分数的任何方程。如果两边都有分数，应首先找到最小公分母。
8. Checking Your Answer by Substitution | 用代入法检验答案
After solving an equation, always substitute your answer back into the original equation to check it works. This is called substitution.
解完方程后，一定要把你的答案代回原方程检验是否正确。这叫做代入法。
For example, if you solved 6x + 2 = 20 and found x = 3, substitute x = 3: 6 × 3 + 2 = 18 + 2 = 20. Since both sides equal 20, the answer is correct.
例如，如果你解方程 6x + 2 = 20 得到 x = 3，代入 x = 3：6 × 3 + 2 = 18 + 2 = 20。因为两边都等于 20，所以答案是正确的。
If the left side does not equal the right side after substitution, you have made a mistake. Go back through each step and check your arithmetic and your operations.
如果代入后左边不等于右边，说明你出错了。回到每一步，检查你的算术和运算。
Checking answers is especially important in exams because it can help you catch small sign errors or division mistakes.
检查答案在考试中尤其重要，因为它可以帮助你发现小的符号错误或除法错误。
9. Common Mistakes to Avoid | 常见错误及避免方法
One common mistake is forgetting to do the same operation on both sides. For example, in 2x + 5 = 15, some students subtract 5 only from the left side. This is incorrect.
一个常见错误是忘记在两边做同样的运算。例如，在 2x + 5 = 15 中，有些学生只从左边减去 5。这是不正确的。
Another mistake is reversing the wrong operation. When solving x − 3 = 8, you should add 3 to both sides, not subtract 3. The correct answer is x = 11.
另一个错误是用错了相反的运算。解 x − 3 = 8 时，应该在两边加 3，而不是减 3。正确的答案是 x = 11。
Some students also forget to apply the operation to the entire side. In 3(x + 2) = 18, you must expand first: 3x + 6 = 18, not 3x + 2 = 18.
还有一些学生忘记对整个一边进行运算。在 3(x + 2) = 18 中，你必须先展开：3x + 6 = 18，而不是 3x + 2 = 18。
A final common mistake is losing a negative sign. In −2x = 10, dividing both sides by −2 gives x = −5, not x = 5. Watch the signs carefully.
最后一个常见错误是丢失负号。在 −2x = 10 中，两边除以 −2 得到 x = −5，而不是 x = 5。要仔细注意符号。
10. Worked Examples from Typical KS3 Questions | KS3 典型题型例题
Example 1: Solve 4x + 7 = 31.
例题 1：解 4x + 7 = 31。
Subtract 7 from both sides: 4x = 24.
从两边减去 7：4x = 24。
Divide both sides by 4: x = 6.
两边除以 4：x = 6。
Example 2: Solve 9x − 3 = 5x + 21.
例题 2：解 9x − 3 = 5x + 21。
Subtract 5x from both sides: 4x − 3 = 21.
从两边减去 5x：4x − 3 = 21。
Add 3 to both sides: 4x = 24.
两边加 3：4x = 24。
Divide both sides by 4: x = 6.
两边除以 4：x = 6。
Example 3: Solve 5(x − 2) = 35.
例题 3：解 5(x − 2) = 35。
Expand the brackets: 5x − 10 = 35.
展开括号：5x − 10 = 35。
Add 10 to both sides: 5x = 45.
两边加 10：5x = 45。
Divide both sides by 5: x = 9.
两边除以 5：x = 9。
Example 4: Solve x/5 + 4 = 10.
例题 4：解 x/5 + 4 = 10。
Subtract 4 from both sides: x/5 = 6.
从两边减去 4：x/5 = 6。
Multiply both sides by 5: x = 30.
两边乘以 5：x = 30。
11. Practice Questions and Answers | 练习题与答案
Try these questions without looking at the answers first.
先不要看答案，试做以下练习题。
Question 1: Solve x + 12 = 25.
问题 1：解 x + 12 = 25。
Question 2: Solve 6x − 4 = 20.
问题 2：解 6x − 4 = 20。
Question 3: Solve 7x + 5 = 3x + 29.
问题 3：解 7x + 5 = 3x + 29。
Question 4: Solve 4(x + 3) = 32.
问题 4：解 4(x + 3) = 32。
Question 5: Solve x/3 + 5 = 9.
问题 5：解 x/3 + 5 = 9。
Answers: Q1 x = 13, Q2 x = 4, Q3 x = 6, Q4 x = 5, Q5 x = 12.
答案：第1题 x = 13，第2题 x = 4，第3题 x = 6，第4题 x = 5，第5题 x = 12。
12. Summary and Exam Tips | 总结与考试技巧
To solve linear equations confidently, always use the balance method. Undo operations in the reverse order of the original expression, and show every step clearly.
要自信地解线性方程，始终使用平衡法。按照原表达式相反的运算顺序逐步撤销，并清晰地展示每一步。
In exams, write your working in a logical line-by-line format. This makes it easier for the examiner to award method marks, even if you make a small arithmetic error.
在考试中，将你的解题过程按逻辑逐行书写。这样即使你出现小的算术错误，阅卷老师也更容易给出方法分。
Always check your answer by substituting it back into the original equation. If both sides are equal, your answer is correct. If not, find and fix the error.
一定要把你的答案代回原方程进行检验。如果两边相等，你的答案就是正确的。如果不相等，找出并改正错误。
Practice regularly with varied equations, including those with fractions, brackets, and unknowns on both sides. This builds confidence and speed for your Cambridge KS3 assessments.
通过不同类型的方程进行定期练习，包括含分数、括号和未知数在等号两边的方程。这将为你的剑桥 KS3 评估增强信心和速度。
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