Solving Linear Equations and Formulae | 解一元一次方程与运用公式

📚 Solving Linear Equations and Formulae | 解一元一次方程与运用公式

Linear equations are one of the most important building blocks in KS3 mathematics. They appear in almost every Cambridge checkpoint test, often as straightforward ‘solve for x’ questions or hidden inside word problems. This article explains the key methods step by step, from one-step equations to rearranging formulae, and gives you practice questions with full working.

一元一次方程是 KS3 数学中最重要的基础模块之一。它们几乎出现在每一次剑桥 checkpoint 测试中，有时是直接的 ‘求 x’ 题型，有时隐藏在应用题里。本文将逐步讲解从一步方程到公式变形的关键方法，并提供带完整过程的练习题。

---

1. Understanding Linear Equations | 理解一元一次方程

A linear equation is an equation in which the unknown, usually written as x, appears only to the power of 1. For example, 2x + 3 = 11 is linear because x is not squared, cubed, or placed in a denominator. The word ‘linear’ tells us that if we drew the expression as a graph, it would form a straight line.

一元一次方程是指未知数（通常写作 x）只以一次幂出现的方程。例如 2x + 3 = 11 就是线性方程，因为 x 没有被平方、立方，也没有出现在分母中。’线性’ 这个词告诉我们，如果把表达式画成图像，它会形成一条直线。

In KS3 Cambridge mathematics, you need to be able to solve equations such as 3x − 7 = 14, 5(x + 2) = 35, and 4x + 1 = 2x + 13. These all describe a single value of x that makes the equation true.

在 KS3 剑桥数学中，你需要能够求解像 3x − 7 = 14、5(x + 2) = 35 和 4x + 1 = 2x + 13 这样的方程。这些方程都描述了一个使等式成立的唯一 x 值。

---

2. Balancing Both Sides | 等式两边平衡

The most powerful idea in solving linear equations is the balance method. Think of an equation as a set of scales. Whatever you do to one side, you must also do to the other side to keep the scales balanced. If you add 5 to the left, add 5 to the right. If you divide the left by 3, divide the right by 3.

解一元一次方程最有力的思想是天平法。把方程想象成一架天平。你对一边做的任何操作，必须对另一边也做同样的操作，才能保持天平平衡。如果左边加 5，右边也要加 5；如果左边除以 3，右边也要除以 3。

For example, to solve x + 9 = 15, subtract 9 from both sides:

例如，解 x + 9 = 15，需要从两边减去 9：

x + 9 − 9 = 15 − 9 → x = 6

This does not ‘move’ the 9 to the other side by magic. It is simply applying the inverse operation to both sides.

这并不是通过 ‘魔法’ 把 9 移到另一边，而是对两边同时进行了逆运算。

---

3. Solving One-Step Equations | 解一步方程

A one-step equation requires only one inverse operation to isolate x. There are four basic types: addition, subtraction, multiplication, and division.

一步方程只需要一次逆运算就可以把 x 单独留在一边。基本类型有四种：加法、减法、乘法和除法。

For x + 4 = 10, subtract 4 from both sides to get x = 6. For x − 4 = 10, add 4 to both sides to get x = 14. For 4x = 20, divide both sides by 4 to get x = 5. For x ÷ 4 = 3, multiply both sides by 4 to get x = 12.

对于 x + 4 = 10，两边同时减去 4，得到 x = 6。对于 x − 4 = 10，两边同时加上 4，得到 x = 14。对于 4x = 20，两边同时除以 4，得到 x = 5。对于 x ÷ 4 = 3，两边同时乘以 4，得到 x = 12。

Always check your answer by substituting it back into the original equation. If the left-hand side equals the right-hand side, your solution is correct.

一定要把答案代回原方程检验。如果左边等于右边，说明你的解是正确的。

---

4. Solving Two-Step Equations | 解两步方程

Two-step equations involve two operations attached to x. A common example is 2x + 3 = 15. The order of operations when evaluating this expression would be: multiply x by 2 first, then add 3. To solve, we reverse this order: subtract 3 first, then divide by 2.

两步方程涉及与 x 相关的两种运算。常见例子是 2x + 3 = 15。计算这个表达式时的运算顺序是：先把 x 乘以 2，再加 3。解方程时则要颠倒顺序：先减 3，再除以 2。

2x + 3 − 3 = 15 − 3 → 2x = 12 → x = 12 ÷ 2 → x = 6

Another example is x/5 − 2 = 4. Add 2 to both sides first, then multiply both sides by 5:

另一个例子是 x/5 − 2 = 4。先两边加 2，再两边乘以 5：

x/5 − 2 + 2 = 4 + 2 → x/5 = 6 → x = 6 × 5 → x = 30

Remember: undo addition or subtraction first, then undo multiplication or division. This is because addition and subtraction are on the outside of the expression.

记住：先逆运算加法或减法，再逆运算乘法或除法。因为加法和减法位于表达式的外层。

---

5. Removing Brackets First | 先去括号

When an equation contains brackets, such as 3(x + 4) = 27, the first step is to expand the brackets. Multiply each term inside the bracket by the number outside.

当方程中含有括号时，例如 3(x + 4) = 27，第一步是去括号。用括号外的数乘以括号内的每一项。

3(x + 4) = 27 → 3x + 12 = 27 → 3x = 15 → x = 5

You can also divide both sides by the outside number first if the number divides evenly into the right-hand side. In this case, divide by 3: x + 4 = 9, then x = 5. Both methods give the same answer.

如果括号外的数能整除右边，你也可以先两边同时除以这个数。比如这里先除以 3：x + 4 = 9，然后 x = 5。两种方法得到相同答案。

For equations like 2(x − 3) + 5 = 17, expand first: 2x − 6 + 5 = 17, then simplify to 2x − 1 = 17, then solve.

对于像 2(x − 3) + 5 = 17 这样的方程，先展开：2x − 6 + 5 = 17，再合并为 2x − 1 = 17，然后求解。

---

6. Unknowns on Both Sides | 未知数在两边

Sometimes x appears on both sides of the equation, for example 5x + 2 = 2x + 14. Start by collecting all x terms on one side and all constant terms on the other side. Subtract 2x from both sides:

有时 x 同时出现在方程两边，例如 5x + 2 = 2x + 14。先收集所有含 x 的项到一边，所有常数项到另一边。两边同时减去 2x：

5x − 2x + 2 = 2x − 2x + 14 → 3x + 2 = 14 → 3x = 12 → x = 4

It is often easier to subtract the smaller x term from both sides so that you keep a positive coefficient of x. If you end up with a negative coefficient, such as −3x = 12, divide both sides by −3 to get x = −4.

通常从两边减去较小的 x 项，以保持 x 的系数为正。如果你最终得到负系数，例如 −3x = 12，两边同时除以 −3，得到 x = −4。

Always substitute your solution back into both sides of the original equation to check that they produce the same value.

一定要把解代回原方程的两边，检查两边得到相同的值。

---

7. Substitution into Formulae | 代入公式

A formula shows how one quantity relates to others, such as A = lw for the area of a rectangle or v = u + at for speed. Substitution means replacing letters with given numbers, then evaluating the result using the correct order of operations.

公式表示一个量与另一些量之间的关系，例如矩形面积 A = l

Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）