Solving Linear Equations | 解线性方程

📚 Solving Linear Equations | 解线性方程

Linear equations are the foundation of algebra. In KS3 mathematics, you will learn how to solve linear equations to find unknown values. This skill is essential for progressing to more complex topics such as simultaneous equations and graphs. In this article, we will explore the methods for solving linear equations step by step, from simple one-step equations to those with variables on both sides. We will also discuss common mistakes and provide practical tips for your Cambridge KS3 exam.

线性方程是代数的基础。在 KS3 数学阶段，你将学习如何解线性方程以求出未知值。这项技能对于后续学习方程组、函数图像等更复杂的主题至关重要。在本文中，我们将逐步介绍解线性方程的方法，从简单的一步方程到两边都含有变量的方程。我们还会探讨常见错误，并为你的剑桥 KS3 考试提供实用建议。

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

A linear equation is an equation where the highest power of the variable (usually x) is 1. For example, 2x + 5 = 11 is a linear equation. The graph of a linear equation is a straight line. The goal when solving a linear equation is to find the value of the variable that makes the equation true.

线性方程是变量的最高次幂为 1 的方程。例如，2x + 5 = 11 就是一个线性方程。线性方程的图像是一条直线。解线性方程的目标是找出使方程成立的变量的值。

The standard form of a linear equation in one variable is ax + b = c, where a, b, and c are constants. Solving the equation means finding the value of x.

一元一次线性方程的标准形式为 ax + b = c，其中 a、b 和 c 是常数。解方程就是求出 x 的值。

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2. Understanding Inverse Operations | 理解逆运算

To isolate the variable and solve an equation, we use inverse operations. An inverse operation reverses the effect of another operation. The table below shows the main pairs of inverse operations.

为了隔离变量、解出方程，我们需要使用逆运算。逆运算可以抵消原运算的效果。下表列出了主要的逆运算配对。

Operation	Inverse Operation
Addition (+)	Subtraction (−)
Subtraction (−)	Addition (+)
Multiplication (×)	Division (÷)
Division (÷)	Multiplication (×)

When you add 5 to a number, you can ‘undo’ it by subtracting 5. Similarly, if you multiply by 3, divide by 3 to get back to the original value. Solving equations relies on applying these inverse operations to both sides of the equation.

当你对一个数加 5 时，可以通过减 5 来“撤销”。同样，如果乘以 3，再除以 3 就能回到原来的值。解方程就是依赖在方程两边同时运用这些逆运算。

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3. Solving One-Step Equations | 解一步方程

One-step equations require only one inverse operation to find the solution. Let us look at a simple addition equation.

一步方程只需要运用一次逆运算就可以求出解。我们来看一个简单的加法方程。

x + 7 = 15

To undo the addition of 7, subtract 7 from both sides:

为了抵消加 7，两边同时减去 7：

x + 7 − 7 = 15 − 7

x = 8

You can check: 8 + 7 = 15, so the solution is correct. Now consider a multiplication equation: 3x = 12. Here the variable is multiplied by 3, so divide both sides by 3.

你可以检验：8 + 7 = 15，因此解是正确的。现在考虑乘法方程：3x = 12。这里变量被乘以 3，所以两边同时除以 3。

3x ÷ 3 = 12 ÷ 3

x = 4

For division equations such as x/4 = 2, multiply both sides by 4 to get x = 8. Remember: whatever you do to one side, you must do to the other.

对于除法方程如 x/4 = 2，两边同时乘以 4 得到 x = 8。记住：你对一边做了什么，对另一边也必须做同样的事情。

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4. Solving Two-Step Equations: Example 1 | 解两步方程：示例 1

Two-step equations involve two operations. We use inverse operations in the reverse order of the usual order of operations (BIDMAS/BODMAS). For the equation 2x + 5 = 11, the variable x is first multiplied by 2 and then 5 is added. To solve, we undo the addition first, then the multiplication.

两步方程包含两种运算。我们需要按照与运算优先级（BIDMAS/BODMAS）相反的顺序来使用逆运算。对于方程 2x + 5 = 11，变量 x 先乘以 2，然后加上 5。解方程时，我们先撤消加法，再撤消乘法。

2x + 5 = 11

Subtract 5 from both sides to remove the constant term:

两边减去 5，去掉常数项：

2x + 5 − 5 = 11 − 5

2x = 6

Now divide both sides by 2:

现在两边除以 2：

2x ÷ 2 = 6 ÷ 2

x = 3

Always check by substituting x = 3 back into the original equation: 2(3) + 5 = 6 + 5 = 11, so it works.

始终把 x = 3 代入原方程检验：2(3) + 5 = 6 + 5 = 11，成立。

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5. Solving Two-Step Equations: Example 2 | 解两步方程：示例 2

Some two-step equations involve subtraction or a negative coefficient. Consider (x/3) − 4 = 1. Here x is divided by 3 and then 4 is subtracted. Undo the subtraction first by adding 4, then undo the division by multiplying by 3.

有些两步方程涉及减法或负系数。考虑 (x/3) − 4 = 1。这里 x 先除以 3，再减去 4。先通过加 4 撤消减法，再乘以 3 撤消除法。

x/3 − 4 = 1

x/3 − 4 + 4 = 1 + 4

x/3 = 5

x/3 × 3 = 5 × 3

x = 15

Another common type is 5 − 2x = 1. In such cases, avoid the mistake of adding 2x incorrectly as a first step. Instead, subtract 5 from both sides to isolate the term with x.

另一种常见类型是 5 − 2x = 1。这种情况下，要避免错误地将第一步设为加 2x。正确的做法是先两边减去 5，把含 x 的项分离出来。

5 − 2x − 5 = 1 − 5

−2x = −4

x = (−4) ÷ (−2) = 2

Dividing a negative by a negative gives a positive result.

负数除以负数得到正数。

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6. Equations with Variables on Both Sides | 两边都有变量的方程

When variables appear on both sides of the equation, we first collect all variable terms on one side and constants on the other. For example, 3x + 2 = 2x + 5.

当方程两边都出现变量时，我们需要先把所有含变量的项移到一边，常数移到另一边。例如，3x + 2 = 2x + 5。

3x + 2 = 2x + 5

Subtract 2x from both sides to bring the variables together:

两边同时减去 2x，使变量项归到一边：

3x − 2x + 2 = 2x − 2x + 5

x + 2 = 5

Then subtract 2 from both sides to solve for x:

然后两边减去 2，解出 x：

x = 5 − 2

x = 3

You can also move the constant 2 first, but moving the variable term is usually more efficient. Always aim to have a positive x term where possible.

你也可以先移动常数 2，但通常先移动变量项更高效。尽量让 x 的系数为正。

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7. Using the Balance Method | 使用天平法

The balance method is a visual way of understanding equation solving. Imagine a scale where both sides must always be balanced. Whatever change you make to one side, you must make the same change to the other side to keep the scale balanced. This is why we ‘do the same to both sides’.

天平法是一种直观理解解方程过程的方法。想象一个天平，两边必须始终保持平衡。你对一边做了什么变化，就必须对另一边做相同的变化，以维持天平平衡。这就是为什么我们要“两边做相同操作”。

For the equation x + 3 = 8, think of the left pan holding an x and 3 weights, and the right pan holding 8 weights. To find the weight of x, remove 3 from both pans; the scale stays balanced and x = 5.

对于方程 x + 3 = 8，想象左盘放着一个 x 和 3 个砝码，右盘放着 8 个砝码。要找出 x 的重量，从两边各拿走 3 个砝码；天平仍然平衡，且 x = 5。

This method helps when solving more complex equations because it reinforces the rule that operations must be applied equally to both sides. It also highlights why we cannot simply move a number without compensating.

这个方法在解更复杂的方程时很有帮助，因为它强调了必须对方程两边同等施行的规则。它也突出了为什么不能在不做补偿的情况下随意移动数字。

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8. Common Mistakes and How to Avoid Them | 常见错误及如何避免

Many students lose marks due to avoidable mistakes. Here are the most frequent ones and how to prevent them.

许多学生丢失分数是因为一些可以避免的错误。以下是最常见的错误及其预防方法。

Mistake 1: Forgetting to apply an operation to both sides. For example, solving 2x = 10 only on the left gives 2x/2 = 10, which is wrong. Always write the operation on both sides.

错误 1：忘记对两边施加同样的操作。例如，解 2x = 10 时只在左边除以 2，写成 2x/2 = 10，这是错误的。一定要在两边都写上运算。

Mistake 2: Incorrect order of inverse operations. In 2x + 3 = 7, dividing by 2 before subtracting 3 leads to an incorrect result. Remember to undo the addition/subtraction before multiplication/division.

错误 2：逆运算的顺序不对。在 2x + 3 = 7 中，先除以 2 再减去 3 会得到错误答案。记住先撤消加减再撤消乘除。

Mistake 3: Mishandling negative signs. For −x = 4, the solution is x = −4, not x = 4. Multiply by −1 if necessary to make x positive.

错误 3：处理负号不当。对于 −x = 4，解为 x = −4，而不是 x = 4。如有必要，两边乘以 −1 使 x 变正。

Mistake 4: Not checking the solution. Always substitute your answer back into the original equation to confirm it works.

错误 4：不检验解。始终把答案代回原方程，验证是否成立。

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9. Checking Your Solution | 检验你的解

Checking your solution is a quick way to confirm accuracy. After finding x = 3 for the equation 2x + 5 = 11, substitute 3 into the left-hand side: 2(3) + 5 = 6 + 5 = 11, which matches the right-hand side. If the two sides are equal, your solution is correct.

检验你的解是快速确认准确性的方法。在解出 2x + 5 = 11 得到 x = 3 之后，把 3 代入左边：2(3) + 5 = 6 + 5 = 11，与右边相等。如果两边相等，你的解就是正确的。

Checking is especially useful for catching sign errors or arithmetic mistakes. When working under time pressure in exams, it can save you from losing marks on a question you actually know how to do.

检验对于发现符号错误或算术错误尤其有用。在考试时间紧张的情况下，它可以避免你在一道本会做的题目上丢分。

If the check fails, go back and re-trace your steps. Often the mistake is a simple slip, such as adding instead of subtracting or incorrectly combining like terms.

如果检验不成立，就回头重新梳理步骤。错误往往只是简单的笔误，比如该减却加了，或者合并同类项有误。

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10. Solving Equations with Fractions | 解含有分数的方程

Fractions can appear in linear equations in two main ways: the variable is multiplied by a fraction, or fractions appear as parts of terms. For (2/3)x = 6, multiply both sides by the reciprocal 3/2 to isolate x.

分数在方程中出现主要有两种形式：变量乘以一个分数，或者项中包含分数。对于 (2/3)x = 6，两边同乘以倒数 3/2 即可隔离 x。

x = 6 × (3/2) = 9

When the equation has a denominator like (x/4) + 3 = 7, we can clear the fraction by multiplying every term by the denominator 4: 4 × (x/4) + 4 × 3 = 4 × 7, giving x + 12 = 28, so x = 16. This method eliminates fractions early.

当方程有分母如 (x/4) + 3 = 7 时，我们可以通过给每一项乘以分母 4 来去分母：4 × (x/4) + 4 × 3 = 4 × 7，得到 x + 12 = 28，因此 x = 16。这个方法可以及早消去分数。

If a fraction equation has variable terms on both sides, multiply through by the least common denominator (LCD) to simplify. Always be careful to multiply every term, including the constant terms.

如果分数方程两边都含变量项，可以乘以最小公分母（LCD）来简化。务必注意每一项都要乘，包括常数项。

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11. Real-Life Applications | 实际应用

Linear equations are not just abstract exercises. They can model real situations. For example, if apples cost £0.80 each and you buy a certain number, plus a £1.50 bag, and spend a total of £5.10, we can write the equation 0.80n + 1.50 = 5.10. Solving gives n = 4.5, which indicates 4 apples if we require integer quantities, but the algebraic model shows the relationship.

线性方程并非只是抽象的练习，它们可以模拟真实情境。例如，苹果每个 0.80 英镑，你买了若干个，外加一个 1.50 英镑的袋子，共花了 5.10 英镑，我们可以写出方程 0.80n + 1.50 = 5.10。解出 n = 4.5，如果要求整数就是 4 个苹果，但代数模型显示了数量关系。

Another example: A father is twice as old as his son. In 5 years, the sum of their ages will be 65. Let the son’s age be x, then the father’s age is 2x. Equation: (x+5) + (2x+5) = 65 → 3x + 10 = 65 → x = 18.33? Actually let’s make it nicer: (x+5)+(2x+5)=65 ⇒ 3x+10=65 ⇒ 3x=55 ⇒ x ≈ 18.3, but with a better problem: If the sum is 68, then x=19. This type of word problem helps you to translate words into equations.

另一个例子：父亲的年龄是儿子的两倍。5 年后他们的年龄之和为 65 岁。设儿子年龄为 x，则父亲年龄为 2x。方程：(x+5) + (2x+5) = 65 → 3x + 10 = 65 → x ≈ 18.3。我们可以调整总和数使得 x 为整数，例如和为 68 时，x=19。这类文字题帮助你学会把语言转化为方程。

When working with word problems, define your variable clearly, build the equation step by step, and check the solution in the context of the problem.

解答文字题时，要清晰地定义变量，逐步建立方程，并在问题情境中检验解。

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12. Practice Tips for KS3 Exams | KS3 考试练习技巧

To excel in solving linear equations, regular practice is key. Start with one-step equations and gradually move to two-step and equations with fractions. Write every step clearly, even if you can do it mentally, because showing working can earn you marks even if the final answer is slightly wrong.

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