📚 Solving Two-Step Linear Equations | 解两步一元一次方程
In Cambridge KS3 Mathematics, solving equations is a core algebra skill that builds the foundation for more advanced topics such as inequalities, simultaneous equations, and functions. A two-step equation involves exactly two operations applied to an unknown value, such as multiplication followed by addition. This article explains how to solve these equations reliably by reversing the operations and keeping both sides balanced.
在剑桥 KS3 数学中，解方程是一项核心代数技能，为不等式、联立方程和函数等更高级主题打下基础。两步方程是指对未知数进行了两次运算，例如先乘后加。本文将讲解如何通过逆向运算并保持等式两边平衡，可靠地解出这类方程。
1. What Is a Two-Step Equation? | 什么是两步方程？
r example, 2x + 7 = 19 means x is first multiplied by 2 and then 7 is added to the result.
两步一元一次方程是指未知数先乘以一个数，再加上或减去另一个数。例如，2x + 7 = 19 表示 x 先乘以 2，然后在结果上加 7。
d by −3 and then 5 is added. Recognising the two operations is the first step in solving the equation.
其他形式包括 x/4 − 3 = 7，即 x 先除以 4，再减去 3；还有 −3y + 5 = 11，即 y 先乘以 −3，再加 5。识别出这两次运算是解方程的第一步。
2. The Golden Rule: Keep the Equation Balanced | 黄金法则：保持方程平衡
An equation is like a balanced seesaw: whatever you do to one side, you must also do to the other side. If you subtract 5 from the left, you must subtract 5 from the right. If you divide the left side by 3, you must divide the right side by 3 too.
方程就像一个平衡的跷跷板：你对一边做什么，就必须对另一边做同样的操作。如果你从左边减去 5，就必须从右边也减去 5。如果你把左边除以 3，右边也必须除以 3。
This principle allows you to change the form of an equation without changing its solution. The goal is to isolate the unknown on one side of the equals sign by undoing the two operations in reverse order.
这条法则允许你在不改变方程解的情况下改变方程的形式。目标是通过按相反顺序抵消两次运算，将未知数单独留在等号的一边。
3. Worked Example 1: 3x + 5 = 20 | 例题1：3x + 5 = 20
multiplied by 3 and then 5 has been added. To undo these operations, we work backwards: first subtract 5, then divide by 3.
解方程 3x + 5 = 20。未知数 x 先乘以 3，然后加上了 5。为了抵消这些运算，我们逆向操作：先减去 5，再除以 3。
3x + 5 − 5 = 20 − 5 → 3x = 15
3x ÷ 3 = 15 ÷ 3 → x = 5
Check: 3 × 5 + 5 = 15 + 5 = 20, so x = 5 is correct.
检验：3 × 5 + 5 = 15 + 5 = 20，因此 x = 5 是正确的。
4. Worked Example 2: x/4 − 3 = 7 | 例题2：x/4 − 3 = 7
d by 4 and then 3 has been subtracted. To solve, first add 3 to both sides, then multiply both sides by 4.
解方程 x/4 − 3 = 7。这里 x 先除以 4，然后减去了 3。为了求解，先给两边都加上 3，然后两边都乘以 4。
x/4 − 3 + 3 = 7 + 3 → x/4 = 10
x/4 × 4 = 10 × 4 → x = 40
Check: 40/4 − 3 = 10 − 3 = 7, so x = 40 is correct.
检验：40/4 − 3 = 10 − 3 = 7，因此 x = 40 是正确的。
5. Dealing with Negative Coefficients | 处理负系数
ision step must be performed by the same negative number. For example, solve −2x + 6 = 14.
当 x 的系数是负数时，最后一步除法必须除以同一个负数。例如，解方程 −2x + 6 = 14。
First subtract 6 from both sides, then divide both sides by −2.
先给两边都减去 6，然后两边都除以 −2。
−2x + 6 − 6 = 14 − 6 → −2x = 8
−2x ÷ (−2) = 8 ÷ (−2) → x = −4
ve number by a negative number gives a negative result. Check: −2 × (−4) + 6 = 8 + 6 = 14.
请记住，正数除以负数得到负数。检验：−2 × (−4) + 6 = 8 + 6 = 14。
6. Equations with Brackets First | 先处理含括号的方程
an either expand the bracket first or divide both sides by the number outside the bracket.
有些两步方程包含括号。例如，2(x + 3) = 18。你可以先展开括号，也可以两边同时除以括号外的数。
Method 1: divide both sides by 2 first.
方法1：先两边同时除以 2。
2(x + 3) ÷ 2 = 18 ÷ 2 → x + 3 = 9
x + 3 − 3 = 9 − 3 → x = 6
Method 2: expand first.
方法2：先展开括号。
2(x + 3) = 2x + 6, so 2x + 6 = 18
2x + 6 − 6 = 18 − 6 → 2x = 12 → x = 6
Both methods give the same answer. Choose the method that feels easier for each question.
两种方法得到相同的答案。每题选择你觉得更简单的方法即可。
7. Equations with Simple Fractions | 含简单分数的方程
number, treat that division as the last operation to undo. For example, solve x/3 + 2 = 5.
当未知数除以一个数时，把这个除法当作最后一步反向运算。例如，解方程 x/3 + 2 = 5。
First subtract 2, then multiply both sides by 3.
先减去 2，然后两边都乘以 3。
x/3 + 2 − 2 = 5 − 2 → x/3 = 3
x/3 × 3 = 3 × 3 → x = 9
s multiplying by a fraction, such as 2/3 x = 8, multiply both sides by the reciprocal 3/2.
如果方程涉及乘以一个分数，例如 2/3 x = 8，则两边同时乘以倒数 3/2。
2/3 x × 3/2 = 8 × 3/2 → x = 12
8. Checking Your Solution | 检验你的解
Always substitute your answer back into the original equation to check it works. This habit catches small sign or arithmetic errors before they cost marks.
一定要把你的答案代回原方程进行检验。这个习惯能在你丢分之前发现小的符号或计算错误。
For 3x + 5 = 20 with x = 5: 3 × 5 + 5 = 15 + 5 = 20. True.
对于 3x + 5 = 20，当 x = 5 时：3 × 5 + 5 = 15 + 5 = 20，成立。
For x/4 − 3 = 7 with x = 40: 40/4 − 3 = 10 − 3 = 7. True.
对于 x/4 − 3 = 7，当 x = 40 时：40/4 − 3 = 10 − 3 = 7，成立。
Checking is especially important when negative numbers or fractions are involved, because sign errors are common there.
当涉及负数或分数时，检验尤其重要，因为符号错误在这些地方很常见。
9. Common Mistakes to Avoid | 常见错误与避免
- Forgetting to do the same operation on both sides.
- Adding before dividing, or subtracting before multiplying, in the wrong order.
- Dividing by a positive number when the coefficient is negative.
- Forgetting to check the final answer in the original equation.
一个常见错误是忘记对等式两边做同样的运算。另一个错误是逆序运算时搞错顺序，例如错误的加减或乘除顺序。还有负系数时除以正数，以及忘记把最终答案代回原方程检验。
To avoid these mistakes, write each step on a new line and show the operation being performed on both sides. Do not try to do too much in your head at KS3 level.
为了避免这些错误，每一步都在新的一行写出，并标明你对两边做了什么运算。在 KS3 阶段不要过多心算。
10. Practice Questions | 练习题
Try these Cambridge KS3 style questions. Solve each equation and check your answer.
尝试以下剑桥 KS3 风格练习题。解出每个方程并检验答案。
- 4x + 3 = 23
- x/5 − 2 = 6
- −3x + 7 = 1
- 3(x − 4) = 15
- 2/5 x = 10
Answers: x = 5, x = 40, x = 2, x = 9, x = 25.
答案：x = 5，x = 40，x = 2，x = 9，x = 25。
When practising, write full working for every question. The process shown is just as important as the final answer in Cambridge assessments.
练习时，每一题都要写出完整过程。在剑桥测评中，展示的过程与最终答案同样重要。
11. Summary and Key Points | 小结与关键点
To solve a two-step linear equation, identify the two operations applied to the unknown, then undo them in reverse order. Always keep the equation balanced by performing the same operation on both sides.
要解两步一元一次方程，先识别对未知数进行的两次运算，然后按相反顺序抵消它们。始终对两边做相同的运算，保持方程平衡。
Key steps: undo addition or subtraction first, then undo multiplication or division. If there are brackets, expand or divide first. If the coefficient is negative, divide by that negative number. Finally, substitute your answer to check.
关键步骤：先抵消加减法，再抵消乘除法。如果有括号，先展开或先除以括号外的数。如果系数为负，则除以那个负数。最后，代入答案进行检验。
Mastering this topic will make harder algebra topics much easier in Years 8 and 9.
掌握这个主题将使你在 8 年级和 9 年级学习更难的代数内容时轻松很多。
12. How This Links to Cambridge KS3 | 与剑桥 KS3 的联系
In the Cambridge Lower Secondary Mathematics framework, solving linear equations appears in the Algebra strand across Stages 7 to 9. Learners are expected to construct and solve equations in real-life contexts and to use systematic methods with correct notation.
在剑桥初中数学课程框架中，解一元一次方程出现在 7 至 9 阶段的代数板块。学习者应能在实际情境中建立并求解方程，并使用规范的符号和系统方法。
Questions often combine algebra with shapes, angles, or word problems, such as finding an unknown angle in a triangle when the other two are given as expressions like 2x + 10 and x − 5. The skills in this article directly support those tasks.
考试题目通常将代数与图形、角度或文字题结合，例如已知三角形两个角的表达式为 2x + 10 和 x − 5，求未知角。本文中的技能直接支持此类题目。
Build your confidence by practising a mix of equations daily, and you will be well prepared for the Cambridge Checkpoint test.
每天练习不同类型的方程，建立信心，你就能为剑桥 Checkpoint 考试做好充分准备。
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