Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

In KS3 Cambridge Mathematics, solving linear equations is one of the most important algebra skills. You will meet equations like x + 5 = 12 or 4x − 3 = 9 and need to find the value of the unknown. This article explains the key methods step by step, with clear examples and common exam tips. It follows the Cambridge Lower Secondary topic often labelled Section 6.2: Solving Linear Equations.

在剑桥 KS3 数学中，解一元一次方程是最重要的代数技能之一。你会遇到像 x + 5 = 12 或 4x − 3 = 9 这样的方程，并需要求出未知数的值。本文将逐步解释关键方法，并给出清晰的例子和常见考试提示。本内容对应剑桥初中数学教材中常标记为第 6.2 节的主题：解一元一次方程。

1. What is a Linear Equation? | 什么是一元一次方程？

A linear equation is an equation where the unknown appears only to the power of 1. For example, x + 3 = 10 and 2x − 5 = 7 are linear equations. The word ‘linear’ means that when you draw the graph, you get a straight line.

一元一次方程是指未知数的最高次数为 1 的方程。例如，x + 3 = 10 和 2x − 5 = 7 都是一元一次方程。“线性”一词表示当你画出图像时，会得到一条直线。

In KS3, the unknown is usually represented by a letter such as x, y, a or n. Solving the equation means finding the value of the letter that makes the equation true.

在 KS3 阶段，未知数通常用 x、y、a 或 n 等字母表示。解方程意味着求出使等式成立的字母的值。

The expression on each side of the equals sign must have the same value. This idea is the foundation of every method below.

等号两边的表达式必须具有相同的值。这个思想是下面所有方法的基础。

2. The Balance Method | 天平法

The balance method treats an equation like a set of scales. Whatever you do to one side, you must do to the other side to keep the equation balanced. This is the golden rule of solving equations.

天平法把方程看作一台天平。你对一边做的任何运算，都必须对另一边做同样的运算，才能保持方程平衡。这是解方程的黄金法则。

For example, if you add 3 to the left side, you must add 3 to the right side. If you divide the left side by 2, you must divide the right side by 2.

例如，如果你给左边加 3，就必须给右边也加 3；如果你把左边除以 2，就必须把右边也除以 2。

The table below shows common operations and their inverse operations, which are used to isolate the unknown.

下表列出了常见的运算及其逆运算，这些逆运算用于将未知数单独留在方程一边。

Equation operation	Inverse operation
+ 5	− 5
− 7	+ 7
× 3	÷ 3
÷ 4	× 4

Always apply the inverse operation in the reverse order of operations when you work with multi-step equations.

在解多步方程时，始终按照与运算顺序相反的顺序使用逆运算。

3. Solving by Adding or Subtracting | 通过加法或减法求解

When the unknown has a number added to it or subtracted from it, use the opposite operation to both sides. For example, look at the equation:

当未知数加上或减去一个数时，对两边使用相反的运算。例如，看下面的方程：

x + 7 = 15

To get x on its own, subtract 7 from both sides:

为了得到 x，两边同时减去 7：

x + 7 − 7 = 15 − 7

x = 8

You can check this mentally: 8 + 7 equals 15, so the solution is correct.

你可以在心里检验：8 + 7 等于 15，因此解是正确的。

Similarly, for an equation like y − 4 = 11, add 4 to both sides to find y = 15.

类似地，对于像 y − 4 = 11 这样的方程，两边同时加 4，得到 y = 15。

4. Solving by Multiplying or Dividing | 通过乘法或除法求解

If the unknown is multiplied by a number, divide both sides by that number. For example:

如果未知数乘了一个数，两边同时除以这个数。例如：

3x = 21

Divide both sides by 3:

两边同时除以 3：

3x ÷ 3 = 21 ÷ 3

x = 7

If the unknown is divided by a number, multiply both sides by that number. For example:

如果未知数除以了一个数，两边同时乘以这个数。例如：

x ÷ 4 = 3

Multiply both sides by 4:

两边同时乘以 4：

x ÷ 4 × 4 = 3 × 4

x = 12

Remember that multiplication and division are inverse operations, just like addition and subtraction.

请记住，乘法和除法互为逆运算，就像加法和减法一样。

5. Two-Step Equations | 两步方程

Many KS3 equations require two steps. For example, the equation 2x + 3 = 13 has both a multiplication and an addition. You must undo the addition first, then the multiplication.

许多 KS3 方程需要两步求解。例如，方程 2x + 3 = 13 既有乘法又有加法。你必须先消去加法，再消去乘法。

Start by subtracting 3 from both sides:

首先两边同时减去 3：

2x + 3 − 3 = 13 − 3

2x = 10

Then divide both sides by 2:

然后两边同时除以 2：

2x ÷ 2 = 10 ÷ 2

x = 5

This works because you are reversing the order of operations. In the expression 2x + 3, the multiplication by 2 happens first, then the addition of 3. To solve, reverse the order: subtract 3 first, then divide by 2.

这样做是因为你在反转运算顺序。在表达式 2x + 3 中，先乘以 2，再加上 3。为了解方程，需要反转顺序：先减去 3，再除以 2。

For an equation like 4 − x = 9, be careful: subtracting x is not the same as x being subtracted from 4. You can first add x to both sides to get 4 = 9 + x, then subtract 9 to find x = −5.

对于像 4 − x = 9 这样的方程，要特别注意：减去 x 不等于 x 减去 4。你可以先在两边加 x，得到 4 = 9 + x，然后两边减去 9，得出 x = −5。

6. Equations with Brackets | 含括号的方程

When an equation contains brackets, you can expand them first. For example:

当方程含有括号时，你可以先展开括号。例如：

3(x + 2) = 21

Expand the left side to get 3x + 6 = 21. Then subtract 6 from both sides and divide by 3.

展开左边得到 3x + 6 = 21。然后两边减去 6，再除以 3。

3x + 6 − 6 = 21 − 6

3x = 15

x = 5

An alternative is to divide both sides by 3 first, giving x + 2 = 7, then subtract 2. Both methods give the same answer; choose the one you find easier.

另一种方法是两边先除以 3，得到 x + 2 = 7，然后减去 2。两种方法得到同样的答案；选择你觉得更简单的一种即可。

If the brackets have a negative sign in front, such as −2(x − 4) = 10, be careful with signs. Multiply each term inside the bracket by −2 to get −2x + 8 = 10, then subtract 8 and divide by −2 to find x = −1.

如果括号前面有负号，例如 −2(x − 4) = 10，要特别注意符号。将括号内每一项乘以 −2，得到 −2x + 8 = 10，然后两边减去 8，再除以 −2，得出 x = −1。

7. Equations with Fractions | 含分数的方程

Equations can include fractions, such as x/3 + 2 = 6. The key is to remove the fraction by multiplying both sides by the denominator at the right time.

方程中可以含有分数，例如 x/3 + 2 = 6。关键是在合适的时候通过在两边乘以分母来去掉分数。

First subtract 2 from both sides:

首先两边同时减去 2：

x/3 + 2 − 2 = 6 − 2

x/3 = 4

Then multiply both sides by 3:

然后两边同时乘以 3：

x/3 × 3 = 4 × 3

x = 12

For an equation like (x + 2)/3 = 4, the fraction line means the whole expression x + 2 is divided by 3. Multiply both sides by 3 first to get x + 2 = 12, then subtract 2 to find x = 10.

对于像 (x + 2)/3 = 4 这样的方程，分数线表示整个表达式 x + 2 除以 3。先两边乘以 3，得到 x + 2 = 12，然后减去 2，得出 x = 10。

If there are two fractions, multiply both sides by the common denominator. For example, in x/2 + x/3 = 5, multiply every term by 6 to get 3x + 2x = 30, then combine to 5x = 30, so x = 6.

如果有两个分数，两边乘以公分母。例如，在 x/2 + x/3 = 5 中，每一项乘以 6，得到 3x + 2x = 30，然后合并为 5x = 30，所以 x = 6。

8. Equations with Unknowns on Both Sides | 未知数在方程两边

Sometimes the unknown appears on both sides of the equals sign. For example:

有时未知数会出现在等号的两边。例如：

5x + 2 = 2x + 11

You need to collect the variable terms on one side and the number terms on the other. Subtract 2x from both sides:

你需要把含未知数的项移到一边，把常数项移到另一边。两边同时减去 2x：

5x − 2x + 2 = 2x − 2x + 11

3x + 2 = 11

Then subtract 2 from both sides and divide by 3:

然后两边减去 2，再除以 3：

3x = 9

x = 3

It does not matter which side you collect the variable on, but it is often easier to subtract the smaller coefficient from the larger one to keep the variable positive.

把未知数移到哪一边都可以，但通常用较大的系数减去较小的系数，使未知数的系数保持为正，这样更容易计算。

9. Checking Your Answer | 检验答案

After solving an equation, always check your solution by substituting it back into the original equation. This will catch mistakes with signs, fractions and brackets.

解完方程后，一定要把解代回原方程进行检验。这可以发现符号、分数和括号方面的错误。

For example, if your solution to 5x + 2 = 2x + 11 is x = 3, check the left side and right side separately:

例如，如果你解 5x + 2 = 2x + 11 得到 x = 3，分别检验左边和右边：

LHS = 5 × 3 + 2 = 15 + 2 = 17

RHS = 2 × 3 + 11 = 6 + 11 = 17

Since LHS = RHS, the solution is correct. In an exam, you can write a short check with the original equation.

由于左边等于右边，所以解是正确的。在考试中，你可以用原方程写出简短的检验过程。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

One very common mistake is applying an operation to only one side of the equation. Remember that every step must keep the equation balanced.

一个非常常见的错误是只对等式的一边进行运算。请记住，每一步都必须保持方程平衡。

Another common error is forgetting to reverse the sign when moving a term. For example, in x + 5 = 12, you must subtract 5, not add 5. In 3x = 18, you must divide by 3, not multiply.

另一个常见错误是移项时忘记改变符号。例如，在 x + 5 = 12 中，你必须减去 5，而不是加 5。在 3x = 18 中，你必须除以 3，而不是乘以 3。

Also watch out for negative signs when expanding brackets or dividing by a negative coefficient. Work step by step and write each line clearly.

此外，在展开括号或除以负系数时要注意负号。一步一步计算，并清楚地写出每一行。

Exam tips:

考试技巧：

Always show your working, even if you can see the answer quickly.

即使你能很快看出答案，也要写出计算过程。
Do one operation at a time and write the new equation each time.

每次只做一步运算，并写出新的方程。
Check your answer by substitution before moving to the next question.

在进入下一题之前，用代入法检验答案。
Leave fractions in their simplest form where appropriate.

在合适的情况下，将分数化为最简形式。

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Solving Linear Equations | 解一元一次方程