Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

This revision guide covers the key ideas in Year 8 Unit 4 on solving linear equations. You will learn how to use the balancing method, handle brackets and fractions, check your answers, and apply equations to word problems. Work through each section carefully and try the practice questions at the end.

本复习指南涵盖八年级第 4 单元关于解一元一次方程的核心内容。你将学习如何使用平衡法、处理括号和分数、检验答案，以及将方程应用于文字题。请仔细阅读每一节，并尝试最后的练习题。

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

A linear equation is an algebraic statement in which two expressions are equal, and the unknown variable is only raised to the power of 1. It can usually be written in the form ax + b = c, where x is the unknown and a, b, c are constants.

一元一次方程是一个代数等式，其中两个表达式相等，未知变量的次数仅为 1。它通常可以写成 ax + b = c 的形式，其中 x 是未知数，a、b、c 是常数。

For example, 2x + 5 = 13, 3(x – 4) = 9, and x/2 – 1 = 3 are all linear equations in one variable. If an equation contains x² or x³, it is not linear.

例如，2x + 5 = 13、3(x – 4) = 9 和 x/2 – 1 = 3 都是一元一次方程。如果方程含有 x² 或 x³，它就不是一次方程。

2. The Balancing Method | 平衡法

Think of an equation as a balanced set of scales. Whatever you do to one side, you must do the same to the other side to keep the equation balanced. This is the golden rule of solving equations.

把方程想象成一架平衡的天平。你对一边做什么操作，就必须对另一边做同样的操作，以保持方程平衡。这是解方程的黄金法则。

For example, to solve x + 7 = 12, subtract 7 from both sides:

例如，解方程 x + 7 = 12，两边同时减去 7：

x + 7 – 7 = 12 – 7 → x = 5

The same idea works for multiplication and division: to undo +7, use -7; to undo ×3, use ÷3.

同样的方法适用于乘法和除法：要消去 +7，就使用 -7；要消去 ×3，就使用 ÷3。

3. Solving One-Step Equations | 解一步方程

A one-step equation requires only one operation to isolate the variable. The operation may be addition, subtraction, multiplication, or division.

一步方程只需要一步运算就能求出未知数。运算可以是加法、减法、乘法或除法。

x + 9 = 20 → subtract 9 from both sides → x = 11
x – 5 = 8 → add 5 to both sides → x = 13
4x = 28 → divide both sides by 4 → x = 7
x/3 = 6 → multiply both sides by 3 → x = 18

中文配对：

x + 9 = 20 → 两边同时减 9 → x = 11
x – 5 = 8 → 两边同时加 5 → x = 13
4x = 28 → 两边同时除以 4 → x = 7
x/3 = 6 → 两边同时乘以 3 → x = 18

Always write the variable on the left side of the final answer, for example x = 5 rather than 5 = x.

最终答案总是把变量写在左边，例如写成 x = 5 而不是 5 = x。

4. Solving Two-Step Equations | 解两步方程

Two-step equations involve two operations, such as multiplication and addition. Use reverse order of operations: undo addition or subtraction first, then undo multiplication or division.

两步方程包含两种运算，例如乘法和加法。按照运算的逆序来解：先消去加法或减法，再消去乘法或除法。

Solve 3x + 5 = 20.

解方程 3x + 5 = 20。

3x + 5 – 5 = 20 – 5 → 3x = 15 → x = 15 ÷ 3 → x = 5

Another example: solve x/4 – 2 = 7. First add 2 to both sides, then multiply both sides by 4.

另一个例子：解方程 x/4 – 2 = 7。先两边同时加 2，再两边同时乘以 4。

x/4 = 9 → x = 9 × 4 → x = 36

5. Equations with Brackets | 含括号的方程

When an equation contains brackets, expand them first using the distributive law: a(b + c) = ab + ac. Then solve the resulting two-step or multi-step equation.

当方程含有括号时，先用分配律展开：a(b + c) = ab + ac。然后解得到的两步或多步方程。

Solve 3(x + 4) = 21.

解方程 3(x + 4) = 21。

3x + 12 = 21 → 3x = 9 → x = 3

If the bracket is divided by a number, such as (x + 5)/2 = 8, multiply both sides by 2 first to remove the fraction, then subtract 5.

如果括号除以一个数，例如 (x + 5)/2 = 8，可以两边先乘以 2 消去分数，再减去 5。

x + 5 = 16 → x = 11

6. Equations with Variables on Both Sides | 含两边变量的方程

When the unknown x appears on both sides of the equation, collect the variable terms on one side first. Move the smaller variable term to the other side by adding or subtracting.

当未知数 x 出现在方程两边时，先把含变量的项移到同一边。通过加减将较小的变量项移到另一边。

Solve 5x + 2 = 2x + 14.

解方程 5x + 2 = 2x + 14。

5x – 2x = 14 – 2 → 3x = 12 → x = 4

Always check your answer by substituting x = 4 back into the original equation: left side = 22, right side = 22.

一定要把 x = 4 代回原方程检验：左边 = 22，右边 = 22。

7. Equations with Fractions | 含分数的方程

To solve equations with fractions, find the least common denominator (LCD) of all fractions and multiply every term by it. This clears the fractions and gives a simpler equation.

解含分数的方程时，先找出所有分母的最小公分母，并将每一项都乘以它。这样可以消去分数，得到更简单的方程。

Solve x/3 + x/4 = 7.

解方程 x/3 + x/4 = 7。

LCD = 12 → 12(x/3) + 12(x/4) = 12 × 7 → 4x + 3x = 84 → 7x = 84 → x = 12

If the variable appears in the numerator only, this method is fast and reliable. If the variable is in the denominator, cross-multiplication may be needed.

如果变量只出现在分子中，这个方法又快又可靠。如果变量在分母中，可能需要用到交叉相乘。

8. Checking Solutions | 检验解

After solving an equation, always substitute your answer back into the original equation to confirm that both sides are equal. This habit catches sign errors and arithmetic mistakes.

解完方程后，一定要把答案代回原方程，确认两边相等。这个习惯能帮助发现符号错误和计算错误。

For example, if you solve 2x – 3 = 9 and get x = 6, check: 2(6) – 3 = 12 – 3 = 9. Since the left side equals the right side, the solution is correct.

例如，解方程 2x – 3 = 9 得到 x = 6，检验：2(6) – 3 = 12 – 3 = 9。左边等于右边，所以解正确。

If the two sides are not equal, go back and review your working step by step.

如果两边不相等，请回到解题过程逐步复查。

9. Word Problems Leading to Linear Equations | 列方程解文字题

Many real-life problems can be solved by forming a linear equation. Read the problem carefully, define the unknown with a letter, translate the words into mathematical symbols, solve the equation, and write a sentence answer.

许多实际问题可以通过列一元一次方程来解决。仔细读题，用字母设未知数，把文字转化为数学符号，解方程，最后写出完整的答案句。

Example: Three times a number plus 8 is 29. Find the number.

例题：一个数的 3 倍加 8 等于 29。求这个数。

Let the number be x → 3x + 8 = 29 → 3x = 21 → x = 7

Always answer in context: “The number is 7”, not just “x = 7”.

一定要结合题意作答：”这个数是 7″，而不是只写 “x = 7″。

10. Common Mistakes to Avoid | 常见错误及避免方法

One common error is forgetting to apply an operation to both sides of the equation. Another is making a sign error when moving terms across the equals sign.

一个常见错误是忘记对方程两边同时进行运算。另一个错误是在移项时出现符号错误。

Mistake: 2x + 3 = 9 → 2x = 9 + 3 → wrong
Correct: 2x + 3 = 9 → 2x = 9 – 3 → 2x = 6 → x = 3

中文配对：

错误：2x + 3 = 9 → 2x = 9 + 3 → 错误
正确：2x + 3 = 9 → 2x = 9 – 3 → 2x = 6 → x = 3

Also avoid dividing by the wrong coefficient. In 3x = 15, divide by 3, not by 15.

还要避免除以错误的系数。在 3x = 15 中，应该除以 3，而不是除以 15。

11. Practice Questions | 练习题

Try solving these equations. Check each answer by substitution.

尝试解下列方程。每个答案都用代入法检验。

4x + 7 = 31
5(x – 2) = 35
2x + 9 = 5x – 6
x/2 + 3 = 8
3x/4 – 1 = 5

Answers: x = 6, x = 9, x = 5, x = 10, x = 8.

答案：x = 6，x = 9，x = 5，x = 10，x = 8。

12. Summary | 单元总结

To solve a linear equation, keep the equation balanced by doing the same operation to both sides. Expand brackets first, clear fractions using the LCD, collect variable terms on one side, and isolate the variable using inverse operations. Always check your solution in the original equation.

解一元一次方程时，要通过对两边进行相同运算来保持方程平衡。先展开括号，用最小公分母消去分数，把变量项移到一边，再利用逆运算求出未知数。一定要把解代回原方程进行检验。

Mastering these techniques in Unit 4 will prepare you for more advanced algebra topics, including simultaneous equations and quadratic equations.

掌握第 4 单元的这些方法，将为你学习更高级的代数内容（如联立方程和二次方程）打下基础。

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Solving Linear Equations | 解一元一次方程