Solving Linear Equations: Balancing and Inverse Operations | 解一元一次方程：等式平衡与逆运算

📚 Solving Linear Equations: Balancing and Inverse Operations | 解一元一次方程：等式平衡与逆运算

Linear equations are the foundation of algebra at KS3. In the Cambridge Lower Secondary Mathematics curriculum, you need to solve equations such as 3x + 5 = 20 by performing the same operation on both sides of the equals sign. This article explains the balancing method, inverse operations, working with brackets and fractions, and checking your final answer carefully.

一元一次方程是 KS3 代数的基础。在剑桥初中数学课程中，你需要解像 3x + 5 = 20 这样的方程，方法是在等号两边同时进行相同的运算。本文讲解等式平衡法、逆运算、去括号和分数方程的处理，以及如何仔细检验最终答案。

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

A linear equation in one variable is an equation that can be written in the form ax + b = c, where a, b and c are constants and a is not zero. The highest power of the unknown x is 1, so the graph of the related expression is a straight line.

一元一次方程是含有一个未知数的方程，可以写成 ax + b = c 的形式，其中 a、b 和 c 是常数，且 a 不等于零。未知数 x 的最高次数是 1，因此相关表达式的图像是一条直线。

For example, 2x + 7 = 19, 5(x – 3) = 40 and 4x – 1 = 2x + 9 are all linear equations. Solving a linear equation means finding the value of x that makes the left-hand side exactly equal to the right-hand side.

例如，2x + 7 = 19、5(x – 3) = 40 和 4x – 1 = 2x + 9 都是一元一次方程。解一元一次方程就是要找到使左边恰好等于右边的 x 值。

2. The Balancing Principle | 等式平衡原理

An equation is like a balance scale. To keep the two sides equal, you must perform exactly the same operation on both sides. If you add 3 to one side, you must add 3 to the other side. If you divide one side by 5, you must divide the other side by 5 as well.

方程就像一个天平。为了保持两边相等，你必须对两边同时进行完全相同的运算。如果一边加 3，另一边也必须加 3。如果一边除以 5，另一边也必须除以 5。

x + 5 = 12

To solve the equation above, subtract 5 from both sides: x + 5 – 5 = 12 – 5, so x = 7. The balance is maintained because the same amount was removed from each side.

要解上面的方程，两边同时减去 5：x + 5 – 5 = 12 – 5，因此 x = 7。由于两边减去了相同的量，平衡得以保持。

3. Inverse Operations | 逆运算

Inverse operations undo each other. Addition and subtraction are inverse operations, and multiplication and division are inverse operations. When you solve an equation, you use inverse operations to isolate the unknown x.

逆运算可以相互抵消。加法和减法互为逆运算，乘法和除法互为逆运算。解方程时，你使用逆运算来单独求出未知数 x。

For a two-step equation such as 2x + 3 = 11, the operations acting on x are multiplication by 2 and then addition of 3. To undo them, you reverse the order: first subtract 3, then divide by 2.

对于 2x + 3 = 11 这样的两步方程，作用于 x 的运算是先乘以 2，再加 3。要撤销这些运算，你需要反向进行：先减去 3，再除以 2。

2x + 3 = 11

2x = 8

x = 4

Always remember to reverse the order of operations when applying inverse operations: start with the last operation and work backwards.

使用逆运算时，一定要记住反向操作顺序：从最后一步运算开始，逆序回推。

4. Solving One-Step Equations | 解一步方程

One-step equations require only one inverse operation. For example, if x – 4 = 9, add 4 to both sides to get x = 13. If 3x = 18, divide both sides by 3 to get x = 6.

一步方程只需要一步逆运算。例如，如果 x – 4 = 9，两边加 4 得到 x = 13。如果 3x = 18，两边除以 3 得到 x = 6。

If the equation is x/7 = 5, multiply both sides by 7: x = 35. The key is to choose the operation that cancels the number on the same side as x.

如果方程是 x/7 = 5，两边乘以 7：x = 35。关键是选择能消去 x 同侧数字的运算。

Here are more examples:

更多示例如下：

x + 9 = 21 → x = 12

y – 6 = 15 → y = 21

4a = 32 → a = 8

Even with one-step equations, it is good practice to check your solution by substituting the value back into the original equation.

即使是一步方程，也建议将解代回原方程进行检验，这是一个好习惯。

5. Solving Two-Step Equations | 解两步方程

A two-step equation contains two operations acting on the unknown. For example, 5x + 3 = 28 involves multiplication by 5 and addition of 3. To solve it, undo the addition first, then the multiplication.

两步方程包含作用于未知数的两种运算。例如，5x + 3 = 28 涉及乘以 5 和加上 3。解方程时，先撤销加法，再撤销乘法。

5x + 3 = 28

5x = 25

x = 5

Step 1: subtract 3 from both sides. Step 2: divide both sides by 5. You should always write each new line directly below the previous one and keep the equals signs aligned.

第一步：两边减去 3。第二步：两边除以 5。你应该把每一行新式子都写在上一行正下方，并保持等号对齐。

Another example is 7x – 4 = 24. First add 4 to both sides: 7x = 28. Then divide both sides by 7: x = 4. This clear layout helps you avoid sign errors.

另一个例子是 7x – 4 = 24。先两边加 4：7x = 28。然后两边除以 7：x = 4。清晰的书写布局有助于避免符号错误。

6. Equations with Brackets | 带括号的方程

When an equation contains brackets, you usually expand them first using the distributive law. For example, 3(x + 2) = 21 becomes 3x + 6 = 21 after expanding.

当方程中含有括号时，通常先使用分配律展开括号。例如，3(x + 2) = 21 展开后变成 3x + 6 = 21。

3(x + 2) = 21

3x + 6 = 21

3x = 15

x = 5

After expanding, solve the resulting two-step equation as usual: subtract 6 from both sides, then divide by 3.

展开后，像平常一样解得到的两步方程：两边减去 6，然后除以 3。

You can also divide both sides by the outside number first if it divides evenly. For 4(x – 3) = 32, dividing by 4 gives x – 3 = 8, then x = 11. Both methods give the same answer.

如果括号外的数字能整除右边，也可以先除以那个数字。对于 4(x – 3) = 32，先除以 4 得到 x – 3 = 8，然后 x = 11。两种方法的结果相同。

7. Equations with the Unknown on Both Sides | 未知数在等号两边的方程

If an equation has x on both sides, such as 5x + 2 = 3x + 10, you need to collect the variable terms on one side and the constant terms on the other side.

如果方程两边都有 x，例如 5x + 2 = 3x + 10，你需要把所有含变量的项移到一边，把常数项移到另一边。

5x + 2 = 3x + 10

5x – 3x = 10 – 2

2x = 8

x = 4

Subtract 3x from both sides to move the variable terms to the left, and subtract 2 from both sides to move the constant terms to the right. Always apply the same operation to both sides.

两边减去 3x 把变量项移到左边，两边减去 2 把常数项移到右边。一定要对两边同时进行相同的运算。

Remember that when you subtract a term, its sign changes on the other side. If the equation had 7x – 5 = 2x + 15, subtracting 2x and adding 5 gives 5x = 20, so x = 4.

记住，当你把一项移到另一边时，它的符号会改变。如果方程是 7x – 5 = 2x + 15，两边减去 2x 并加上 5 得到 5x = 20，所以 x = 4。

8. Equations Containing Fractions | 含分数的方程

To solve an equation with a fraction, multiply both sides by the denominator to remove the fraction. For example, if x/4 + 3 = 7, first subtract 3 from both sides to get x/4 = 4, then multiply both sides by 4 to get x = 16.

解含分数的方程时，两边乘以分母以消去分数。例如，如果 x/4 + 3 = 7，首先两边减去 3 得到 x/4 = 4，然后两边乘以 4 得到 x = 16。

x/4 + 3 = 7

x/4 = 4

x = 16

If the fraction is on one side and has a numerator other than 1, such as 2x/3 = 8, multiply both sides by 3 to get 2x = 24, then divide by 2 to get x = 12.

如果分数一边的分子不是 1，例如 2x/3 = 8，两边乘以 3 得到 2x = 24，再除以 2 得到 x = 12。

When there are two fractions, you can multiply every term by the lowest common denominator. This removes all fractions in one step and makes the equation easier to solve.

当方程中有两个分数时，可以把每一项都乘以最小公分母。这样能一步消去所有分数，使方程更容易求解。

9. Checking Your Solution | 检验你的解

After you find a value for x, substitute it back into the original equation to make sure the left-hand side equals the right-hand side. This is called checking your solution.

求出 x 的值后，把它代回原方程，确认左边等于右边。这叫做检验你的解。

For example, if you solved 5x + 3 = 28 and got x = 5, substitute 5 into the equation: 5 × 5 + 3 = 25 + 3 = 28. Since 28 = 28, the solution is correct.

例如，如果你解 5x + 3 = 28 得到 x = 5，把 5 代入方程：5 × 5 + 3 = 25 + 3 = 28。因为 28 = 28，所以解是正确的。

If your value does not make both sides equal, you have made an arithmetic error or an incorrect inverse operation. Go back and check each step carefully.

如果你的值不能使两边相等，说明你出现了算术错误或使用了错误的逆运算。回去仔细检查每一步。

10. Common Mistakes to Avoid | 常见错误与避免方法

Many students lose marks because of small sign errors or forgetting to apply an operation to both sides. The most common mistakes are listed below:

许多学生因为小的符号错误或忘记对两边同时进行运算而失分。下面列出最常见的错误：

Mistake 1: adding or subtracting from only one side. Every operation must be done to both sides at the same time.

错误 1：只对一边进行加减。每一歩运算都必须同时作用于两边。

Mistake 2: error when expanding brackets with negative numbers. For example, -2(x + 3) = -2x – 6, not -2x + 6.

错误 2：展开带负号的括号时出错。例如，-2(x + 3) = -2x – 6，而不是 -2x + 6。

Mistake 3: reversing the order of inverse operations. With 3x – 7 = 14, add 7 first, then divide by 3. Do not divide first.

错误 3：逆运算的顺序颠倒。对于 3x – 7 = 14，先加 7，再除以 3。不要先除。

Mistake 4: forgetting to check the answer. Checking takes little time and can catch many errors before you finish.

错误 4：忘记检验答案。检验花不了多少时间，却能在你完成前发现许多错误。

11. Practice Questions with Worked Solutions | 练习题与详细解答

Try these questions before looking at the solutions. Write each line clearly and keep the equals signs aligned.

在看答案之前先试做以下题目。每一行都要写清楚，并保持等号对齐。

Solve: 6x + 4 = 46

Solution: 6x = 42, so x = 7

Solve: 4(x – 5) = 32

Solution: 4x – 20 = 32, so 4x = 52, x = 13

Solve: 3x + 7 = 2x + 18

Solution: x + 7 = 18, so x = 11

Solve: x/6 – 2 = 1

Solution: x/6 = 3, so x = 18

Work through these examples step by step. If you can solve them confidently, you are ready for a wide range of KS3 equation problems.

一步一步完成这些例子。如果你能自信地解出这些题，就能应对 KS3 中各种方程问题。

12. Key Summary | 重点总结

A linear equation has an unknown to the power of 1. Solve it by applying inverse operations to both sides in reverse order. Expand brackets first, remove fractions by multiplying, and always check your final answer by substitution.

一元一次方程中未知数的次数为 1。解方程时，按逆序对两边同时使用逆运算。先展开括号，通过乘以分母消去分数，并始终用代入法检验最终答案。

With regular practice, the steps become automatic. Focus on neat working, correct signs and careful checking, and you will build strong algebra skills for later topics.

通过经常练习，这些步骤会变得熟练。注意书写整齐、符号正确和仔细检验，你将为后续主题打下扎实的代数基础。

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Solving Linear Equations: Balancing and Inverse Operations | 解一元一次方程：等式平衡与逆运算