Solving Linear Equations: Balance, Brackets and Word Problems | 解一元一次方程：天平法、括号与应用题

📚 Solving Linear Equations: Balance, Brackets and Word Problems | 解一元一次方程：天平法、括号与应用题

This article explains the balance method, common problem types and exam-ready strategies.

线性方程是 KS3 代数的核心。在剑桥初中数学中，你需要会建立、求解并检验像 2x + 3 = 11 或 5(x − 2) = 4x + 3 这样的方程。本文将讲解天平法、常见题型以及备考策略。

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

form ax + b = c or ax + b = cx + d, where a, b, c and d are numbers and x is the unknown.

一元一次方程可以写成 ax + b = c 或 ax + b = cx + d 的形式，其中 a、b、c、d 是数，x 是未知数。

The word 'linear' means the variable x is only raised to the power 1. There are no x², x³, 1/x or xy terms.

"线性"意味着变量 x 只有 1 次方，不存在 x²、x³、1/x 或 xy 这样的项。

Recognising this structure helps you choose the right method to solve the equation.

识别这一结构有助于你选择正确的方法来解方程。

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2. The Balance Method | 天平法（等式平衡）

Think of an equation as a balance scale. The left side and the right side have equal weight.

把方程想象成一台天平。左边和右边的重量相等。

To keep the balance, whatever operation you perform on one side must also be performed on the other side.

为了保持平衡，你对一边进行的任何运算，必须对另一边也进行同样的运算。

Use inverse operations: addition undoes subtraction, subtraction undoes addition, multiplication undoes division, and division undoes multiplication.

使用逆运算：加法与减法互逆，减法与加法互逆，乘法与除法互逆，除法与乘法互逆。

x + 5 = 12 ⇒ x = 7

Here we subtract 5 from both sides to isolate x.

这里我们从两边同时减去 5，以分离 x。

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3. Solving One-Step Equations | 解一步方程

When x + 7 = 15, subtract 7 from both sides to get x = 8.

当 x + 7 = 15 时，两边同时减去 7，得到 x = 8。

When 7x = 42, divide both sides by 7 to get x = 6.

当 7x = 42 时，两边同时除以 7，得到 x = 6。

When x / 4 = 9, multiply both sides by 4 to get x = 36.

当 x / 4 = 9 时，两边同时乘以 4，得到 x = 36。

careful with signs: if x − 6 = 3, then x = 9; if 6 − x = 3, then x = 3 after rearranging.

注意符号：如果 x − 6 = 3，则 x = 9；如果 6 − x = 3，移项后 x = 3。

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4. Solving Two-Step Equations | 解两步方程

3 = 11, first undo the addition or subtraction, then undo the multiplication or division.

对于 2x + 3 = 11 这样的两步方程，先处理加法或减法，再处理乘法或除法。

2x + 3 = 11

2x = 8

x = 4

Subtract 3 from both sides, then divide both sides by 2.

两边先减去 3，再两边除以 2。

For 7x − 5 = 30, add 5 to both sides to get 7x = 35, then divide by 7 to get x = 5.

对于 7x − 5 = 30，两边加 5 得到 7x = 35，再除以 7 得到 x = 5。

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5. Equations with Brackets | 含括号的方程

If a bracket is multiplied by a number, expand it first using the distributive law.

如果括号乘以一个数，先用分配律展开。

Example: 3(x + 2) = 21 becomes 3x + 6 = 21, so 3x = 15 and x = 5.

例如：3(x + 2) = 21 变为 3x + 6 = 21，所以 3x = 15，x = 5。

You may also divide both sides by the multiplier first if it is a factor of the other side, but expanding is safer when signs are involved.

如果乘数是另一边的因数，也可以先两边除以该乘数，但涉及符号时展开更稳妥。

For 2(x − 4) = 10, expand to 2x − 8 = 10, then 2x = 18 and x = 9.

对于 2(x − 4) = 10，展开为 2x − 8 = 10，然后 2x = 18，x = 9。

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6. Unknowns on Both Sides | 未知数在等式两边

When x appears on both sides, collect the x terms on one side and the number terms on the other.

当 x 出现在两边时，把含 x 的项移到一边，常数项移到另一边。

5x + 2 = 2x + 14

5x − 2x = 14 − 2

3x = 12 ⇒ x = 4

Subtract 2x from both sides, then subtract 2 from both sides, then divide by 3.

两边减去 2x，再减去 2，然后除以 3。

n subtract 4x from both sides and solve carefully: −9 = −3x + 3, then −12 = −3x, so x = 4.

如果 x 的系数变为负数，例如 4x − 9 = x + 3，可以两边减去 4x 并小心求解：−9 = −3x + 3，然后 −12 = −3x，所以 x = 4。

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7. Equations with Fractions | 含分数的方程

To clear a fraction, multiply every term by the denominator.

为了去分母，将每一项乘以分母。

x / 3 + 2 = 6

x / 3 = 4

x = 12

by 3 after isolating the fraction, or multiply the whole equation by 3 first: x + 6 = 18,

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