📚 Solving Linear Equations | 解一元一次方程
Linear equations are one of the most important building blocks in KS3 algebra. They appear in almost every Cambridge Lower Secondary Checkpoint test, and they often show up in word problems, geometry problems, and graphs. This article revises the key methods for solving one-step, two-step, bracket, fraction, and variable-on-both-sides equations, with worked examples and common mistakes to avoid.
一元一次方程是 KS3 代数中最重要的基础内容之一。它们几乎出现在每一份剑桥初中 Checkpoint 考试中，也经常出现在文字题、几何题和图像题里。本文复习解一步方程、两步方程、含括号方程、含分数方程以及两边含未知数方程的关键方法，并配有例题和常见错误提醒。
1. What Is a Linear Equation? | 什么是一元一次方程？
wer of 1. For example, 3x + 2 = 11 is a linear equation because x is not squared or cubed.
一元一次方程是一种代数等式：两个表达式相等，并且未知数只出现一次方（即不平方、不立方）。例如，3x + 2 = 11 是一元一次方程，因为 x 没有平方或立方。
e a, b and c are numbers. The goal is to find the value of x that makes the equation true.
在 KS3 阶段，你通常会看到形如 ax + b = c 的方程，其中 a、b、c 是数字。目标是求出使等式成立的 x 的值。
2. The Balance Method | 天平平衡法
Think of an equation as a balance scale. The left side and the right side must have exactly the same value. Whatever operation you do to one side, you must do exactly the same to the other side.
把方程想象成一个天平。左边和右边的值必须完全相等。无论你对一边做了什么运算，都必须对另一边做完全相同的运算。
For example, to solve x + 5 = 12, subtract 5 from both sides:
例如，解 x + 5 = 12，两边同时减去 5：
x + 5 - 5 = 12 - 5
This simplifies to x = 7.
化简得 x = 7。
3. One-Step Equations | 一步方程
A one-step equation needs only one inverse operation to solve. The inverse of addition is subtraction, the inverse of subtraction is addition, the inverse of multiplication is division, and the inverse of division is multiplication.
一步方程只需要一次逆运算就能解出答案。加法的逆运算是减法，减法的逆运算是加法，乘法的逆运算是除法，除法的逆运算是乘法。
If x + 7 = 15, subtract 7 from both sides: x = 8.
若 x + 7 = 15，两边减去 7：x = 8。
If x - 4 = 10, add 4 to both sides: x = 14.
若 x - 4 = 10，两边加上 4：x = 14。
If 5x = 35, divide both sides by 5: x = 7.
若 5x = 35，两边除以 5：x = 7。
If x ÷ 4 = 9, multiply both sides by 4: x = 36.
若 x ÷ 4 = 9，两边乘以 4：x = 36。
Always use the opposite operation to 'undo' the equation step by step.
一定要用相反的运算逐步“解开”方程。
4. Two-Step Equations | 两步方程
Two-step equations involve two operations. The usual strategy is to undo addition or subtraction first, then undo multiplication or division.
两步方程涉及两种运算。通常的策略是先把加减法逆运算处理掉，再处理乘除法逆运算。
Example: Solve 2x + 3 = 13.
例题：解 2x + 3 = 13。
2x + 3 - 3 = 13 - 3
2x = 10
2x ÷ 2 = 10 ÷ 2
x = 5
Notice that we first subtracted 3, then divided by 2. The order matters because we undo the operations in reverse order of the original expression.
注意我们先减去 3，再除以 2。顺序很重要，因为我们按原表达式相反的运算顺序来解。
5. Equations with Brackets | 含括号的方程
sing the distributive law. For example, 3(x - 2) means 3 × x minus 3 × 2, which is 3x - 6.
当方程含有括号时，先用分配律展开。例如，3(x - 2) 表示 3 × x 减去 3 × 2，得到 3x - 6。
Example: Solve 3(x - 2) = 2x + 5.
例题：解 3(x - 2) = 2x + 5。
3x - 6 = 2x + 5
Now subtract 2x from both sides:
两边同时减去 2x：
3x - 2x - 6 = 2x - 2x + 5
x - 6 = 5
Then add 6 to both sides:
然后两边加上 6：
x = 11
Always expand brackets carefully, especially when a negative sign is outside the bracket.
展开括号时一定要仔细，尤其是括号外有负号时。
6. Equations with Variables on Both Sides | 两边含未知数的方程
When x appears on both sides, collect all x terms on one side and all number terms on the other side. This usually means subtracting the smaller x term from both sides first.
当 x 出现在等式两边时，把所有含 x 的项移到一边，把所有数字项移到另一边。通常先将较小的 x 项从两边减去。
Example: Solve 5x - 4 = 2x + 8.
例题：解 5x - 4 = 2x + 8。
5x - 2x - 4 = 2x - 2x + 8
3x - 4 = 8
Add 4 to both sides:
两边加上 4：
3x = 12
Divide by 3:
除以 3：
x = 4
This method keeps the coefficient of x positive, which reduces sign errors.
这种方法可以让 x 的系数保持为正，从而减少符号错误。
7. Equations with Fractions | 含分数的方程
When an equation has a fraction, multiply both sides by the denominator to clear the fraction. If there is more than one fraction, multiply by the lowest common denominator.
当方程含有分数时，两边乘以分母来去掉分数。如果有多个分数，则乘以最小公分母。
Example: Solve x ÷ 3 + 2 = 5.
例题：解 x ÷ 3 + 2 = 5。
x ÷ 3 + 2 - 2 = 5 - 2
x ÷ 3 = 3
x = 9
Example: Solve (2x + 1) ÷ 5 = 3.
例题：解 (2x + 1) ÷ 5 = 3。
