Solving Linear Equations | 解一次方程

📚 Solving Linear Equations | 解一次方程

Linear equations are the building blocks of algebra. They allow us to find unknown values by performing the same operation on both sides of an equals sign. Mastering this skill will help you solve problems in geometry, number patterns, and real-life situations.

一次方程是代数的基石。通过在等号两边执行相同的运算，我们可以求出未知数的值。掌握这个技能将帮助你解决几何、数字规律和实际生活中的各种问题。

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

A linear equation is an equation where the variable (often x) is raised to the power of 1. It contains an equals sign and the highest exponent is 1. Examples include 3x + 2 = 11 and 5y − 7 = 3y + 9. The graph of a linear equation is a straight line.

一次方程是指未知数（通常是 x）的指数为 1 的方程。它包含等号，最高指数为 1。例如 3x + 2 = 11 和 5y − 7 = 3y + 9。一次方程的图像是一条直线。

2. The Balancing Method | 平衡法原理

Think of an equation as a balanced scale. Whatever you do to one side, you must do to the other to keep the equation true. This means adding, subtracting, multiplying, or dividing both sides by the same non-zero number. The goal is to isolate the variable on one side.

把方程想象成一座平衡的天平。你对一边做什么，就必须对另一边做同样的操作，才能保持等式成立。这意味着在同一侧可以进行加减乘除（除以非零数），目标是把未知数单独放在一侧。

3. Solving One-Step Equations | 解一步方程

One-step equations require only one operation to solve. For example, x + 5 = 12. Subtract 5 from both sides: x = 12 − 5, so x = 7. For 4y = 20, divide both sides by 4: y = 5. Always perform the inverse operation.

一步方程只需进行一步运算即可求解。例如 x + 5 = 12，两边同时减去 5：x = 12 − 5，得到 x = 7。对于 4y = 20，两边同时除以 4：y = 5。始终使用逆运算。

4. Solving Two-Step Equations | 解两步方程

Two-step equations involve two operations. Take 2x + 3 = 11. First, subtract 3 from both sides to get 2x = 8. Then divide both sides by 2 to find x = 4. The order of undoing operations is the reverse of BODMAS: handle addition or subtraction first, then multiplication or division.

两步方程涉及两种运算。以 2x + 3 = 11 为例：首先两边同时减去 3，得到 2x = 8；然后两边除以 2，解得 x = 4。解除运算的顺序与 BODMAS 相反：先处理加减，再处理乘除。

5. Equations with Brackets | 含括号的方程

When an equation contains brackets, the first step is to expand them. Use the distributive law: 3(x + 4) = 21 becomes 3x + 12 = 21. Then solve the two-step equation: subtract 12 to get 3x = 9, and divide by 3 to obtain x = 3. Always multiply every term inside the brackets by the term outside.

当方程含有括号时，第一步是去括号。使用分配律：3(x + 4) = 21 展开为 3x + 12 = 21。然后解两步方程：减去 12 得 3x = 9，除以 3 得 x = 3。务必用括号外的项乘以括号内的每一项。

6. Equations with Variables on Both Sides | 含两边未知数的方程

For equations like 5x + 2 = 3x + 10, collect the variable terms on one side and constants on the other. Subtract 3x from both sides: 2x + 2 = 10. Then subtract 2: 2x = 8. Finally divide by 2: x = 4. You can also move terms from right to left if you prefer.

对于 5x + 2 = 3x + 10 这类方程，把含未知数的项移到一边，常数项移到另一边。两边同时减去 3x：2x + 2 = 10。再减去 2：2x = 8。最后除以 2 得 x = 4。你也可以根据需要从右往左移项。

7. Equations Involving Fractions | 含分数的方程

When fractions appear, the easiest method is to multiply every term by the lowest common denominator (LCD) to clear the fractions. For x/3 + 1 = 5, multiply through by 3: x + 3 = 15, so x = 12. For more complex fractions, always find the LCD first, then simplify.

出现分数时，最简单的方法是用所有分母的最小公倍数 (LCD) 乘以每一项，消去分母。以 x/3 + 1 = 5 为例，两边乘以 3：x + 3 = 15，因此 x = 12。遇到更复杂的分数，先找出 LCD，再化简。

8. Checking Your Solution | 检验你的解

After finding a solution, always substitute it back into the original equation to verify your answer. If 2x + 7 = 15 and you found x = 4, check: 2(4) + 7 = 8 + 7 = 15, which matches. This habit catches mistakes and builds confidence.

求出解后，务必将它代回原方程进行检验。如果 2x + 7 = 15 且你得到 x = 4，检验：2(4) + 7 = 8 + 7 = 15，等式成立。养成这个习惯可以避免错误，增强信心。

9. Word Problems Leading to Linear Equations | 列一次方程解应用题

Many real-life situations can be modelled with linear equations. Read the problem carefully, define the unknown (let x be …), turn the words into an equation, then solve. For example: “Three more than twice a number is 11.” Let the number be n: 2n + 3 = 11, n = 4.

许多实际问题都可以用一次方程来建模。仔细阅读题目，设出未知数（设 x 为…），把文字转化为方程，然后求解。例如：“一个数的两倍再多 3 等于 11。”设该数为 n：2n + 3 = 11，n = 4。

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

Students often forget to perform the same operation on both sides, or they mishandle negative signs. Others expand brackets incorrectly, e.g. −2(x − 3) becomes −2x − 6 instead of −2x + 6. Take your time and double-check each step, especially when subtracting a negative.

同学们常常忘记对两边执行同样的运算，或者处理负号时出错。还有人在去括号时出错，例如 −2(x − 3) 写成 −2x − 6，但正确是 −2x + 6。要耐心检查每一步，尤其是减去负数时。

11. Summary of Key Steps | 关键步骤总结

Here is a quick guide for solving any linear equation:

以下是解任何一次方程的快速指南：

• Expand brackets if present. / 如果有括号，先去括号。
• Clear fractions by multiplying by the LCD. / 通过乘以 LCD 消去分母。
• Collect variable terms on one side, constants on the other. / 将未知数项移到一边，常数项移到另一边。
• Combine like terms. / 合并同类项。
• Use inverse operations to isolate the variable. / 使用逆运算将未知数隔离。
• Check your solution by substitution. / 用代入法检验解。

12. Practice Makes Perfect | 熟能生巧

Solving linear equations becomes automatic with practice. Start with simple one-step and two-step problems, then gradually move to brackets, fractions, and word problems. Use online quizzes and past paper questions to build speed and accuracy.

解一次方程会随着练习变得越来越熟练。从简单的一步、两步问题入手，逐步过渡到含括号、分数和应用题。通过在线测验和往年真题来提升速度和准确率。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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