Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations appear throughout the Cambridge KS3 mathematics curriculum. They describe a balance where one unknown number, usually written as x, is hidden inside a simple expression. This article explains how to solve one-step, two-step, and bracket equations with confidence.

一元一次方程贯穿剑桥 KS3 数学课程。它们描述一种平衡关系：一个未知数（通常写成 x）隐藏在简单代数式中。本文将讲解如何自信地求解一步、两步以及含括号的方程。

---

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

A linear equation is an algebraic statement that uses an equals sign and contains the unknown variable raised only to the power of 1. For example, 2x + 3 = 11 is linear because x is not squared or cubed.

一元一次方程是含有等号、并且未知数的次数仅为 1 的代数等式。例如 2x + 3 = 11 是一元一次方程，因为 x 没有被平方或立方。

In KS3, linear equations usually have one solution. Your job is to find the value of the variable that makes the left-hand side exactly equal to the right-hand side.

在 KS3 阶段，一元一次方程通常只有一个解。你的任务就是找到使左边恰好等于右边的未知数的值。

---

2. The Balancing Method | 天平法

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

把方程想象成一架平衡的天平。无论你对一边做什么，都必须对另一边做同样的操作，才能保持平衡。这是解方程的金科玉律。

If you add, subtract, multiply, or divide, apply the operation to both sides. For example, to undo +3 in x + 3 = 10, subtract 3 from both sides: x = 7.

如果你加、减、乘或除，都要对等式两边同时进行。例如，要消去 x + 3 = 10 中的 +3，就在两边同时减去 3，得到 x = 7。

---

3. Inverse Operations | 逆运算

Solving an equation means reversing the operations that have been applied to the unknown. Addition is undone by subtraction, and multiplication is undone by division.

解方程就是逆转施加在未知数上的运算。加法用减法来抵消，乘法用除法来抵消。

The reverse order is important: when the unknown has been multiplied and then increased by a number, you undo addition first, then multiplication.

运算顺序的逆转很重要：当未知数先被乘、再加一个数时，你应该先抵消加法，再抵消乘法。

---

4. Solving Two-Step Equations | 解两步方程

Consider the equation below. The unknown x has been multiplied by 2 and then increased by 3. To solve it, reverse these two steps.

考虑下面的方程。未知数 x 先被乘以 2，再加 3。要解这个方程，就需要逆转这两步。

2x + 3 = 11

First subtract 3 from both sides to get 2x = 8. Then divide both sides by 2 to get x = 4.

首先两边同时减去 3，得到 2x = 8。然后两边同时除以 2，得到 x = 4。

2x = 8 → x = 4

---

5. Equations with Fractions | 含分数的方程

When an equation contains a fraction, multiply both sides by the denominator to clear it. This makes the equation much easier to handle.

当方程含有分数时，两边同时乘以分母来去掉分母。这会使方程更容易处理。

For example, in x ÷ 5 = 3, multiply both sides by 5 to obtain x = 15. If the equation is (x + 2) ÷ 4 = 3, multiply by 4 first, then subtract 2.

例如，在 x ÷ 5 = 3 中，两边同时乘以 5，得到 x = 15。如果方程是 (x + 2) ÷ 4 = 3，先乘以 4，再减去 2。

Always write the fraction line clearly as a division. Keeping steps tidy helps you avoid arithmetic mistakes in Cambridge tests.

始终把分数线清楚地写成除号。保持步骤整洁有助于你在剑桥考试中避免计算错误。

---

6. Expanding Brackets First | 先去括号

If an equation has brackets, you usually expand them first. Use the distributive property to multiply the term outside by every term inside.

如果方程含有括号，通常要先展开括号。使用分配律，把括号外的项乘以括号内的每一项。

For 3(x + 2) = 21, expand to 3x + 6 = 21. Then subtract 6 and divide by 3 to find x = 5.

对于 3(x + 2) = 21，展开得到 3x + 6 = 21。然后减去 6，再除以 3，得到 x = 5。

3x + 6 = 21 → 3x = 15 → x = 5

Sometimes you may also need to expand two sets of brackets, then collect like terms before using inverse operations.

有时你可能还需要展开两组括号，然后先合并同类项，再使用逆运算。

---

7. Unknowns on Both Sides | 未知数在等式两边

When the variable appears on both sides, collect all variable terms on one side and all number terms on the other side.

当未知数出现在等式两边时，把含未知数的项移到一边，把数字项移到另一边。

Example: 5x + 2 = 2x + 14. Subtract 2x from both sides to get 3x + 2 = 14, then subtract 2 to get 3x = 12, so x = 4.

例如：5x + 2 = 2x + 14。两边同时减去 2x，得到 3x + 2 = 14，再减去 2，得到 3x = 12，所以 x = 4。

更多咨询请联系16621398022（同微信）