Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Solving linear equations is a foundational skill in algebra. It involves finding the value of an unknown, usually represented by a letter like x, that makes the equation true. This skill unlocks problem-solving across mathematics and science, from balancing chemical equations to calculating distances. In this article, we will explore step-by-step methods to solve linear equations, starting from simple one-step equations and moving to more complex types involving brackets and variables on both sides.

解一元一次方程是代数的基础技能。它要求找出使等式成立的未知数（通常用字母如 x 表示）的值。这一技能能解锁数学与科学中的问题解决，从化学方程式的配平到距离的计算。本文我们将逐步探索解方程的方法，从简单的一步方程开始，逐步深入到含括号和两边都有未知数的复杂方程。

1. What is a Linear Equation? | 什么是线性方程？

A linear equation is an algebraic statement where the highest power of the variable is 1. It can often be written in the form ax + b = c, where a, b, and c are numbers and a ≠ 0. The graph of a linear equation in two variables is a straight line, which is why it is called “linear”. In Key Stage 3, we focus on equations with one variable.

线性方程是变量最高次幂为1的代数陈述。它通常可以写成 ax + b = c 的形式，其中 a、b、c 是数字且 a ≠ 0。含两个变量的线性方程的图像是一条直线，因此被称为“线性”。在Key Stage 3阶段，我们主要学习含一个变量的方程。

2. The Balance Method | 天平法

Think of an equation as a balance scale. Both sides must always remain equal. Whatever operation you do to one side, you must do exactly the same to the other side to keep it balanced. This principle is the heart of solving equations. For example, if we have x + 3 = 7, we can subtract 3 from both sides to isolate x, giving x = 4.

把方程想象成一个天平。两边必须始终保持相等。无论你对一边进行什么运算，必须对另一边做完全相同的运算以保持平衡。这一原则是解方程的核心。例如，如果给出 x + 3 = 7，我们可以从两边减去3，从而分离出 x，得到 x = 4。

3. One-Step Equations Using Inverse Operations | 使用逆运算的一步方程

To solve one-step equations, use the inverse (opposite) operation. If the equation has addition, subtract. If it has multiplication, divide. For instance, x − 5 = 2 becomes x = 7 after adding 5 to both sides. Similarly, 4x = 20 leads to x = 5 after dividing both sides by 4.

要解一步方程，使用逆运算。如果方程中有加法，就做减法。如果有乘法，就做除法。例如，x − 5 = 2 在两边加5后得 x = 7。类似地，4x = 20 在两边除以4后得 x = 5。

4. Two-Step Equations | 两步方程

Two-step equations involve two operations. The strategy is to undo addition or subtraction first, then undo multiplication or division. For example, 3x + 2 = 11. First subtract 2 from both sides: 3x = 9. Then divide by 3: x = 3. Always reverse the order of operations (BIDMAS backwards) – addition/subtraction before multiplication/division.

两步方程包含两种运算。策略是先消去加法或减法，再消去乘法或除法。例如，3x + 2 = 11。首先从两边减去2：3x = 9。然后除以3：x = 3。始终逆用运算顺序（BIDMAS的反向）—— 先加减后乘除。

5. Equations with Brackets | 带有括号的方程

When an equation contains brackets, expand them first using the distributive law, then solve as usual. For instance, 2(x + 3) = 12. Expand: 2x + 6 = 12. Subtract 6: 2x = 6. Divide by 2: x = 3. Alternatively, you could divide both sides by 2 first, but expanding is often simpler when the coefficient is an integer.

当方程含有括号时，先用分配律展开，然后照常求解。例如，2(x + 3) = 12。展开：2x + 6 = 12。减去6：2x = 6。除以2：x = 3。也可以先将两边除以2，但当系数为整数时展开通常更简单。

6. Unknown on Both Sides | 两边均含未知数

Eliminate the variable from one side by adding or subtracting the same term to both sides. For example, 5x − 3 = 2x + 6. Subtract 2x from both sides: 3x − 3 = 6. Add 3: 3x = 9. Divide by 3: x = 3. The goal is to collect all variable terms on one side and constants on the other.

通过向两边加或减去相同的项来消除一边的变量。例如，5x − 3 = 2x + 6。从两边减去 2x：3x − 3 = 6。加3：3x = 9。除以3：x = 3。目标是将所有含变量的项集中到一边，常数项集中到另一边。

7. Equations Involving Fractions | 涉及分数的方程

Multiply every term by the lowest common denominator (LCD) to clear fractions. For example, ½ x + ⅓ x = 5. The LCD of 2 and 3 is 6. Multiply through by 6: 6(½ x) + 6(⅓ x) = 6×5 → 3x + 2x = 30 → 5x = 30 → x = 6. Always check that the solution does not make any denominator zero.

将每一项乘以最小公分母以清除分数。例如，½ x + ⅓ x = 5。2和3的最小公分母是6。全式乘以6：6(½ x) + 6(⅓ x) = 6×5 → 3x + 2x = 30 → 5x = 30 → x = 6。始终检查解是否使任何分母为零。

8. Checking Your Solution | 检验你的解

Substitute your answer back into the original equation to verify it works. If both sides equal the same number, your solution is correct. For instance, for 2x + 1 = 9, if x = 4, then left side = 2(4) + 1 = 9, which equals the right side. This habit catches mistakes and builds confidence.

将你的答案代回原方程进行验证。如果两边相等，你的解就是正确的。例如，对于 2x + 1 = 9，若 x = 4，则左边 = 2(4) + 1 = 9，等于右边。这个习惯能找出错误并树立信心。

9. Forming Equations from Word Problems | 从文字题建立方程

Translate phrases into algebraic expressions. “Three more than a number” becomes x + 3; “twice a number decreased by 5” becomes 2x − 5. Set up an equation using the relationship given. Example: “I think of a number, double it, add 7, and the result is 19.” Equation: 2x + 7 = 19. Solve: 2x = 12, x = 6.

将短语翻译成代数表达式。“比一个数大3” 变成 x + 3；“一个数的两倍减去5” 变为 2x − 5。利用给定的关系建立方程。示例：“我想一个数，将它加倍，加7，结果是19。” 方程：2x + 7 = 19。求解：2x = 12，x = 6。

10. Common Mistakes and How to Avoid Them | 常见错误及其避免方法

Mistakes often happen when forgetting to do the same operation on both sides, expanding brackets incorrectly (e.g., 3(x+4) = 3x+4), or misapplying negative signs. Always write each step neatly, use the balance method, and check your answer. When a negative sign precedes a bracket, multiply all terms inside by −1, e.g., −(x−2) = −x + 2.

常见错误包括忘记对两边进行相同运算、括号展开错误（如 3(x+4) = 3x+4）、或负号使用不当。要工整地写出每一步，使用天平法，并检验答案。当括号前有负号时，将括号内所有项乘以 −1，如 −(x−2) = −x + 2。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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