Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations are a core part of the Cambridge Key Stage 3 Mathematics syllabus. They involve finding an unknown value, usually written as x or n, that makes an equation true. This topic builds on arithmetic and introduces algebraic thinking, so it is essential for higher-level problem solving.

一元一次方程是剑桥 KS3 数学课程的核心内容。它们涉及找出使等式成立的未知数，通常用 x 或 n 表示。这一主题以算术为基础，同时引入代数思维，因此对更高层次的解题能力至关重要。

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

A linear equation is an algebraic statement in which the unknown has a power of 1. The general form of a linear equation in one variable is ax + b = c, where a, b and c are known numbers and x is the unknown. For example, 2x + 3 = 11 is a linear equation because the highest power of x is 1.

一元一次方程是一种代数等式，其中未知数的最高次数为 1。一元一次方程的一般形式为 ax + b = c，其中 a、b 和 c 是已知数，x 是未知数。例如，2x + 3 = 11 就是一元一次方程，因为 x 的最高次数是 1。

2x + 3 = 11

2. The Balance Method | 天平法

Think of an equation as a balanced set of scales. The left-hand side and the right-hand side must always be equal. Whatever operation you perform on one side must also be performed on the other side. This is called the balance method, and it is the foundation of solving equations correctly.

可以把等式想象成一台平衡的天平。等式的左边和右边必须始终保持相等。你对等式的一边进行任何运算，也必须对另一边进行同样的运算。这种方法称为天平法，它是正确解方程的基础。

If you add, subtract, multiply or divide one side by a number, you must do exactly the same to the other side. This rule keeps the equation balanced and allows you to isolate the unknown.

如果你对等式的一边加、减、乘或除以一个数，你必须对另一边进行完全相同的操作。这一规则保持等式平衡，并帮助你分离未知数。

3. Solving One-Step Equations | 解一步方程

A one-step equation only needs one inverse operation to find the solution. For example, if x + 5 = 12, subtract 5 from both sides to get x = 7. If 3x = 18, divide both sides by 3 to get x = 6. If x ÷ 4 = 5, multiply both sides by 4 to get x = 20.

一步方程只需要一次逆运算即可求出答案。例如，如果 x + 5 = 12，两边同时减去 5，得到 x = 7。如果 3x = 18，两边同时除以 3，得到 x = 6。如果 x ÷ 4 = 5，两边同时乘以 4，得到 x = 20。

x + 5 = 12 → x = 7

3x = 18 → x = 6

Always use the inverse operation: addition is undone by subtraction, multiplication is undone by division, and subtraction is undone by addition.

一定要使用逆运算：加法用减法抵消，乘法用除法抵消，减法用加法抵消。

4. Solving Two-Step Equations | 解两步方程

A two-step equation requires two inverse operations in the correct order. For 2x + 3 = 11, first subtract 3 from both sides to get 2x = 8, then divide both sides by 2 to get x = 4. For 5x − 7 = 18, first add 7 to both sides to get 5x = 25, then divide by 5 to get x = 5.

两步方程需要按照正确顺序进行两次逆运算。对于 2x + 3 = 11，先两边同时减去 3，得到 2x = 8，然后两边同时除以 2，得到 x = 4。对于 5x − 7 = 18，先两边同时加 7，得到 5x = 25，再除以 5，得到 x = 5。

2x + 3 = 11

2x = 8

x = 4

The order matters: undo addition or subtraction first, then undo multiplication or division. This follows the reverse order of operations.

顺序非常重要：先处理加减法，再处理乘除法。这正好是运算顺序的逆过程。

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

Some equations have the unknown on both sides, such as 3x + 2 = x + 10. To solve it, collect the x terms on one side and the numbers on the other. Subtract x from both sides: 2x + 2 = 10. Then subtract 2 from both sides: 2x = 8. Finally divide by 2: x = 4.

有些方程的未知数出现在等号两边，例如 3x + 2 = x + 10。解这类方程时，要先把含 x 的项移到一边，把数字移到另一边。两边同时减去 x，得到 2x + 2 = 10。再两边同时减去 2，得到 2x = 8。最后除以 2，得到 x = 4。

3x + 2 = x + 10

2x + 2 = 10

x = 4

A common mistake is to move a term without changing its sign. Remember that subtracting x from both sides removes x from the right-hand side and reduces 3x to 2x: sign changes happen because you are performing an inverse operation, not just moving a term.

一个常见错误是移项时没有改变符号。记住，两边同时减去 x，右边就消去了 x，左边从 3x 变成 2x。符号的变化是因为你做了逆运算，而不是简单地移动项。

6. Expanding Brackets First | 先去括号

When an equation contains brackets, expand them before applying the balance method. For 2(x + 3) = 14, expand the left side to get 2x + 6 = 14. Then subtract 6 from both sides: 2x = 8. Divide by 2 to get x = 4.

当方程中含有括号时，要先展开括号，再使用天平法。对于 2(x + 3) = 14，展开左边得到 2x + 6 = 14。然后两边同时减 6，得到 2x = 8。再除以 2，得到 x = 4。

2(x + 3) = 14

2x + 6 = 14

x = 4

For a more complex example, solve 3(2x − 1) = 2(x + 4). Expand both sides: 6x − 3 = 2x + 8. Subtract 2x: 4x − 3 = 8. Add 3: 4x = 11. Divide by 4: x = 11 ÷ 4 = 2.75.

对于更复杂的例子，解 3(2x − 1) = 2(x + 4)。先展开两边：6x − 3 = 2x + 8。两边减 2x：4x − 3 = 8。两边加 3：4x = 11。再除以 4：x = 11 ÷ 4 = 2.75。

7. Equations with Fractions | 含分数的一元一次方程

Equations can also include fractions. For x ÷ 3 + 2 = 5, subtract 2 from both sides first: x ÷ 3 = 3. Then multiply both sides by 3: x = 9. If the equation is (2x) ÷ 3 = 4, multiply both sides by 3 first to get 2x = 12, then divide by 2: x = 6.

方程中也可能含有分数。对于 x ÷ 3 + 2 = 5，先两边同时减去 2，得到 x ÷ 3 = 3。然后两边同时乘以 3，得到 x = 9。如果方程是 (2x) ÷ 3 = 4，先两边同时乘以 3，得到 2x = 12，再除以 2，得到 x = 6。

x ÷ 3 + 2 = 5 → x = 9

(2x) ÷ 3 = 4 → x = 6

Multiplying both sides by the denominator clears the fraction and makes the equation easier to solve. Always apply multiplication to every term on both sides of the equation.

两边同时乘以分母可以消去分数，使方程更容易求解。一定要对等式两边的每一项都进行乘法运算。

8. Checking Your Solution | 检验答案

After finding a value for the unknown, substitute it back into the original equation. For example, if you solved 2x + 3 = 11 and got x = 4, check: 2 × 4 + 3 = 8 + 3 = 11, which matches the right-hand side. This confirms that x = 4 is correct.

求出未知数的值后，要把它代回原方程进行检验。例如，如果你解 2x + 3 = 11 得到 x = 4，检验：2 × 4 + 3 = 8 + 3 = 11，与右边相等。这就能确认 x = 4 是正确的。

Checking is especially useful in exams because it helps catch sign errors, arithmetic mistakes or incorrectly expanded brackets. If the left-hand side does not equal the right-hand side, review each step until you find the error.

在考试中，检验特别有用，因为它能帮助发现符号错误、算术错误或括号展开错误。如果左边不等于右边，就一步一步回顾，直到找出错误。

9. Word Problems into Equations | 文字题转化为方程

Word problems ask you to form and solve an equation from a real-life situation. For example: ‘I think of a number, double it and add 5. The answer is 17. What is the number?’ Let n be the number, then 2n + 5 = 17. Subtract 5: 2n = 12. Divide by 2: n = 6.

文字题要求你根据实际情境列出方程并求解。例如：“我想一个数，把它乘以 2 再加 5，结果是 17。这个数是多少？”设 n 是这个数，那么 2n + 5 = 17。两边减 5：2n = 12。再除以 2：n = 6。

Another example: the perimeter of a rectangle is 26 cm. Its length is 3 cm more than its width. Let the width be w, so length is w + 3. The perimeter equation is 2(w + w + 3) = 26, or 2(2w + 3) = 26. Expand: 4w + 6 = 26. Solve: 4w = 20, so w = 5 cm and length = 8 cm.

另一个例子：一个长方形的周长是 26 厘米，长比宽多 3 厘米。设宽为 w，那么长为 w + 3。周长方程是 2(w + w + 3) = 26，即 2(2w + 3) = 26。展开：4w + 6 = 26。求解：4w = 20，所以 w = 5 厘米，长为 8 厘米。

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

A frequent mistake is performing an operation on only one side of the equation. Always use the balance method and write each new line clearly. Another common error is losing a negative sign when subtracting a term from both sides, for example turning x − 3 = 5 into x = 2 instead of x = 8.

一个常见错误是只对等式的一边进行运算。一定要使用天平法，并把每一步清晰地写下来。另一个常见错误是两边同时减去一个带负号的项时丢掉了负号，例如把 x − 3 = 5 算成 x = 2，而正确答案是 x = 8。

When brackets are involved, expand them fully before collecting like terms. When fractions are involved, multiply every term by the denominator. In exams, show your working step by step because even if your final answer is wrong, method marks are often available.

遇到括号时，要先完全展开再合并同类项。遇到分数时，要把每一项都乘以分母。在考试中，一步一步展示解题过程，因为即使最终答案错了，也常常能获得方法分。

11. Practice Questions | 练习

Try these questions to test your understanding. Solve each equation and check your answer by substitution. (a) 4x + 7 = 31; (b) 5x − 9 = 26; (c) 2(x + 6) = 22; (d) 3x + 4 = x + 18; (e) (3x) ÷ 5 = 6.

试试以下题目来检验你的理解。解出每个方程，并通过代入检验答案。(a) 4x + 7 = 31；(b) 5x − 9 = 26；(c) 2(x + 6) = 22；(d) 3x + 4 = x + 18；(e) (3x) ÷ 5 = 6。

Solutions: (a) subtract 7: 4x = 24, so x = 6. (b) add 9: 5x = 35, so x = 7. (c) expand: 2x + 12 = 22, subtract 12: 2x = 10, so x = 5. (d) subtract x: 2x + 4 = 18, subtract 4: 2x = 14, so x = 7. (e) multiply by 5: 3x = 30, so x = 10.

答案：(a) 两边减 7：4x = 24，所以 x = 6。(b) 两边加 9：5x = 35，所以 x = 7。(c) 展开：2x + 12 = 22，两边减 12：2x = 10，所以 x = 5。(d) 两边减 x：2x + 4 = 18，两边减 4：2x = 14，所以 x = 7。(e) 两边乘以 5：3x = 30，所以 x = 10。

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Solving Linear Equations | 解一元一次方程