Solving Linear Equations and Inequalities | 解一次方程与不等式

📚 Solving Linear Equations and Inequalities | 解一次方程与不等式

This revision guide supports the algebra skills often found in Cambridge KS3 worksheets, including the practice sheet p316_2.pdf. It focuses on solving linear equations and inequalities step by step. Linear equations appear throughout the Cambridge Lower Secondary Mathematics curriculum, especially in Stage 8 and Stage 9, and they form the foundation for more advanced work in IGCSE and A Level.

本复习指南对应剑桥 KS3 常见的代数技能，包括练习页 p316_2.pdf 中的内容，重点逐步讲解解一次方程和不等式的方法。一次方程贯穿剑桥初中数学课程，尤其在 Stage 8 和 Stage 9 中频繁出现，并为 IGCSE 和 A Level 的更深入学习打下基础。

Equations are like balanced scales: whatever you do to one side, you must do to the other. Once you master this principle, you can solve nearly any linear equation confidently. The key is to work systematically, line by line, and always check your answer.

方程就像平衡的天平：你对一边做什么，就必须对另一边做同样的事情。一旦掌握了这个原则，你就能自信地解决几乎所有的一次方程。关键是要有条理地逐行计算，并始终检验答案。

1. What Is a Linear Equation? | 什么是一次方程？

A linear equation is an equation in which the unknown, usually written as x, appears only to the power of 1. This means there are no terms like x², x³, or 1/x. For example, 2x + 5 = 13 is a linear equation because x is not squared or cubed.

一次方程是指未知数（通常写作 x）只以一次幂出现的方程。这意味着方程中没有 x²、x³ 或 1/x 这样的项。例如，2x + 5 = 13 就是一次方程，因为 x 没有平方或立方。

2x + 5 = 13

The goal is always to find the value of the unknown that makes the equation true. In KS3, most linear equations have exactly one solution, but you should also be aware that some equations can have no solution or infinitely many solutions.

我们的目标始终是求出使方程成立的未知数的值。在 KS3 阶段，大多数一次方程恰好有一个解，但你也应该知道，有些方程可能没有解或有无穷多个解。

A linear equation can often be represented as a straight line on a graph, and the solution of the equation is the x-coordinate where the line crosses a given y-value. This link between algebra and graphs is important later in Cambridge mathematics.

一次方程通常可以在图像上表示为一条直线，方程的解就是直线与某个 y 值相交时的 x 坐标。代数与图像之间的这种联系在以后的剑桥数学学习中非常重要。

2. Inverse Operations and One-Step Equations | 逆运算与一步方程

One-step equations need only one inverse operation to solve. Addition and subtraction are inverse operations, and multiplication and division are inverse operations. To isolate x, apply the inverse operation to both sides of the equation.

一步方程只需要一次逆运算即可求解。加法和减法互为逆运算，乘法和除法互为逆运算。为了把 x 单独留在一边，需要对方程两边同时进行逆运算。

Operation	Inverse operation	中文
+ 5	− 5	加 5 的逆运算是减 5
− 3	+ 3	减 3 的逆运算是加 3
× 4	÷ 4	乘 4 的逆运算是除以 4
÷ 7	× 7	除以 7 的逆运算是乘 7

Example 1: Solve x + 8 = 15. Subtract 8 from both sides: x = 15 − 8, so x = 7.

例 1：解方程 x + 8 = 15。两边同时减去 8：x = 15 − 8，所以 x = 7。

Example 2: Solve 6x = 54. Divide both sides by 6: x = 54 ÷ 6, so x = 9.

例 2：解方程 6x = 54。两边同时除以 6：x = 54 ÷ 6，所以 x = 9。

Example 3: Solve x/4 = 9. Multiply both sides by 4: x = 9 × 4, so x = 36.

例 3：解方程 x/4 = 9。两边同时乘以 4：x = 9 × 4，所以 x = 36。

Always apply the same operation to both sides of the equation. This keeps the equation balanced and prevents many common errors.

一定要对方程两边同时进行相同的运算。这样方程才能保持平衡，也能避免许多常见错误。

3. Solving Two-Step Equations | 解两步方程

Two-step equations involve two operations, such as multiplication and addition. The safest method is to undo addition or subtraction first, then undo multiplication or division. This order is the reverse of the usual order of operations.

两步方程包含两种运算，例如乘法和加法。最稳妥的方法是先消去加法或减法，再消去乘法或除法。这个顺序与通常的运算顺序相反。

Solve 4x + 7 = 31.

解方程 4x + 7 = 31。

4x + 7 − 7 = 31 − 7 → 4x = 24

4x ÷ 4 = 24 ÷ 4 → x = 6

You should always check your solution by substituting x = 6 back into the original equation: 4(6) + 7 = 24 + 7 = 31, which is correct.

你应该把 x = 6 代回原方程进行检验：4(6) + 7 = 24 + 7 = 31，结果正确。

If the equation has subtraction, undo it with addition first. For example, solve 5x − 9 = 16. Add 9 to both sides to get 5x = 25, then divide by 5 to get x = 5.

如果方程中有减法，先用加法消去它。例如，解方程 5x − 9 = 16。两边同时加 9 得到 5x = 25，再除以 5 得到 x = 5。

4. Equations with Brackets | 含有括号的方程

When brackets appear in an equation, expand them first using the distributive law. Multiply each term inside the bracket by the number or term outside. Then solve the resulting equation as usual.

当方程中出现括号时，先用分配律展开括号。将括号内的每一项都乘以括号外的数或项，然后按常规方法解得到的方程。

Solve 3(x + 4) = 27.

解方程 3(x + 4) = 27。

3x + 12 = 27

3x = 15 → x = 5

Pay special attention when a negative number is outside the bracket. For example, −2(x − 3) = 8 expands to −2x + 6 = 8, not −2x − 6 = 8. The negative sign multiplies every term inside the bracket.

当括号外面是负数时要格外小心。例如，−2(x − 3) = 8 展开后是 −2x + 6 = 8，而不是 −2x − 6 = 8。负号要乘以括号内的每一项。

After expanding, always check your signs before moving on. Many marks are lost because of small sign errors in bracket expansion.

展开后一定要先检查符号再继续计算。很多失分都是因为括号展开时的小符号错误造成的。

5. Equations with Unknowns on Both Sides | 未知数在等号两边的方程

If the unknown appears on both sides of the equation, collect the x terms on one side first. You can subtract the smaller x term from both sides so that the coefficient of x remains positive.

如果未知数出现在等号两边，先把含 x 的项移到同一边。你可以从两边同时减去较小的 x 项，使 x 的系数保持为正。

Solve 7x + 2 = 3x + 22.

解方程 7x + 2 = 3x + 22。

7x − 3x + 2 = 3x − 3x + 22 → 4x + 2 = 22

4x = 20 → x = 5

When you move a term to the other side of the equals sign, its sign changes. This is because you are adding or subtracting the same amount from both sides, not because of a mysterious rule.

当你把一项移到等号另一边时，它的符号会改变。这是因为你对两边同时加或减同一个量，而不是因为什么神秘规则。

If both x terms have negative coefficients, you can choose to move either term, but moving the less negative one often keeps the working simpler. Always aim to keep x positive when possible.

如果两个 x 项都是负数，你可以选择移动任意一项，但移动较小负数的项通常会让计算更简单。只要可能，尽量让 x 保持为正。

6. Fractions in Linear Equations | 一次方程中的分数

When an equation contains fractions, multiply every term on both sides by the lowest common denominator, or LCD. This clears the fractions and gives a simpler equation with whole numbers.

当方程中含有分数时，将方程两边每一项都乘以最小公分母，即 LCD。这样可以消去分数，得到一个只含整数的更简单方程。

Solve x/2 + 3/4 = 1.

解方程 x/2 + 3/4 = 1。

The LCD of 2 and 4 is 4. Multiply every term by 4: 2x + 3 = 4. Then 2x = 1, so x = 1/2.

2 和 4 的最小公分母是 4。将每一项乘以 4：2x + 3 = 4。然后 2x = 1，所以 x = 1/2。

Another common type is a single fraction equal to

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