Solving Linear Equations for KS3 | KS3 解一元一次方程

📚 Solving Linear Equations for KS3 | KS3 解一元一次方程

Linear equations are one of the most important foundations in KS3 mathematics. They appear in almost every topic, from geometry to word problems, and mastering them now will make GCSE algebra much easier. In this article, you will learn how to solve equations step by step using the balance method, including equations with brackets, fractions, and unknowns on both sides.

一元一次方程是 KS3 数学最重要的基础之一。它们几乎出现在每个主题中，从几何到应用题，现在掌握它们会让未来的 GCSE 代数轻松很多。本文将教你如何使用平衡法一步一步解方程，包括含括号、分数和两边都有未知数的方程。

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

A linear equation is an equation where the unknown variable, usually written as x, is only raised to the power of 1. For example, x + 3 = 7 and 2x – 5 = 9 are linear equations. There are no terms like x², x³, or 1/x in a linear equation.

线性方程是指未知数（通常写作 x）的指数只为 1 的方程。例如 x + 3 = 7 和 2x – 5 = 9 都是线性方程。线性方程中不会出现 x²、x³ 或 1/x 这样的项。

The word “linear” comes from the fact that the graph of these equations is a straight line. Solving a linear equation means finding the value of x that makes the equation true. This value is called the solution or the root of the equation.

“线性” 这个词来源于这些方程的图像是一条直线。解线性方程意味着找到使方程成立的 x 的值。这个值称为方程的解或根。

2. The Balance Method | 天平法（平衡法）

Think of an equation as a set of balance scales. The left side and the right side must always have the same value. Whatever you do to one side, you must also do to the other side to keep the scales balanced. This is the golden rule for solving all linear equations.

把方程想象成一台天平。左边和右边必须始终具有相同的值。无论你对一边做什么，也必须对另一边做同样的操作，以保持天平平衡。这是解所有线性方程的黄金法则。

If you add, subtract, multiply, or divide one side by a number, you must do exactly the same to the other side. This idea allows you to change the equation into a simpler form without changing its solution.

如果你对一边加上、减去、乘以或除以一个数，你必须对另一边做完全相同的操作。这个思想可以让你把方程变成更简单的形式，而不会改变它的解。

Equation: 3x + 2 = 11 → Subtract 2 from both sides → 3x = 9 → Divide both sides by 3 → x = 3

3. Solving One-Step Equations | 解一步方程

A one-step equation only needs one operation to find the solution. If the equation is x + 5 = 12, you subtract 5 from both sides to get x = 7. If the equation is x – 4 = 9, you add 4 to both sides to get x = 13.

一步方程只需要一次运算就能求出解。如果方程是 x + 5 = 12，你从两边减去 5，得到 x = 7。如果方程是 x – 4 = 9，你给两边加上 4，得到 x = 13。

For multiplication and division, you use the inverse operation. If 4x = 20, divide both sides by 4 to get x = 5. If x / 6 = 3, multiply both sides by 6 to get x = 18. Always use the opposite operation to undo the one attached to x.

对于乘法和除法，你需要使用逆运算。如果 4x = 20，两边都除以 4，得到 x = 5。如果 x / 6 = 3，两边都乘以 6，得到 x = 18。总是使用相反的运算来撤销与 x 相连的运算。

x + 8 = 15 → x = 15 – 8 → x = 7
x – 3 = 10 → x = 10 + 3 → x = 13
7x = 42 → x = 42 ÷ 7 → x = 6
x ÷ 5 = 9 → x = 9 × 5 → x = 45

4. Solving Two-Step Equations | 解两步方程

A two-step equation requires two operations to solve, usually one addition or subtraction and then one multiplication or division. For example, in 2x + 3 = 15, you first subtract 3 from both sides to get 2x = 12, then divide both sides by 2 to get x = 6.

两步方程需要两次运算才能求解，通常是一次加法或减法，然后是一次乘法或除法。例如，在 2x + 3 = 15 中，你首先从两边减去 3，得到 2x = 12，然后两边都除以 2，得到 x = 6。

Always undo the addition or subtraction first, and then undo the multiplication or division. This order follows the reverse of the order of operations. Do not divide first unless the whole side is already a single term.

总是先撤销加法或减法，然后再撤销乘法或除法。这个顺序遵循运算顺序的逆序。除非整边已经是一个单项，否则不要先做除法。

5x – 7 = 18 → 5x = 18 + 7 → 5x = 25 → x = 25 ÷ 5 → x = 5

5. Equations with Brackets | 含括号的方程

When an equation has brackets, you must expand them first using the distributive law. For example, 3(x + 4) = 21 becomes 3x + 12 = 21. Then solve as a normal two-step equation: subtract 12 from both sides to get 3x = 9, and divide by 3 to get x = 3.

当方程含有括号时，你必须先用分配律展开它们。例如，3(x + 4) = 21 变成 3x + 12 = 21。然后按普通的两步方程求解：从两边减去 12，得到 3x = 9，再除以 3，得到 x = 3。

Be careful with negative signs outside brackets. For example, -2(x – 5) = 8 expands to -2x + 10 = 8 because -2 × -5 = +10. A very common mistake is to write -2x – 10, so always double-check the signs when you expand.

括号外有负号时要小心。例如，-2(x – 5) = 8 展开为 -2x + 10 = 8，因为 -2 × -5 = +10。一个非常常见的错误是写成 -2x – 10，所以展开时一定要再次检查符号。

4(2x – 3) = 20 → 8x – 12 = 20 → 8x = 32 → x = 4

6. Equations with Unknowns on Both Sides | 两边都含未知数的方程

If a variable appears on both sides of the equation, your first step is to collect all the x terms on one side and all the number terms on the other side. For example, in 5x + 2 = 3x + 10, subtract 3x from both sides to get 2x + 2 = 10. Then subtract 2 from both sides to get 2x = 8, so x = 4.

如果变量出现在方程的两边，你的第一步是把所有的 x 项移到一边，把所有的数字项移到另一边。例如，在 5x + 2 = 3x + 10 中，从两边减去 3x，得到 2x + 2 = 10。然后从两边减去 2，得到 2x = 8，所以 x = 4。

You can choose to move the x terms to either side, but it is usually easier to keep the coefficient of x positive. If the equation is 2x + 7 = 5x – 2, subtract 2x from both sides to get 7 = 3x – 2, then add 2 to both sides to get 9 = 3x, so x = 3.

你可以选择把 x 项移到任意一边，但通常保持 x 的系数为正更容易。如果方程是 2x + 7 = 5x – 2，从两边减去 2x，得到 7 = 3x – 2，然后两边加 2，得到 9 = 3x，所以 x = 3。

4x – 5 = 2x + 9 → 4x – 2x = 9 + 5 → 2x = 14 → x = 7

7. Equations with Fractions | 含分数的方程

Equations with fractions can look tricky, but you can remove the fractions by multiplying every term on both sides by the lowest common denominator. For example, in x/4 + 1 = 3, multiply every term by 4 to get x + 4 = 12, then subtract 4 to get x = 8.

含分数的方程可能看起来很棘手，但你可以通过将两边每一项都乘以最小公分母来去掉分数。例如，在 x/4 + 1 = 3 中，将每一项都乘以 4，得到 x + 4 = 12，然后减去 4，得到 x = 8。

If the equation has more than one fraction, such as (x + 1)/3 = (x – 2)/2, cross-multiplication is a fast method. Multiply the left numerator by the right denominator and the right numerator by the left denominator: 2(x + 1) = 3(x – 2). Then expand and solve: 2x + 2 = 3x – 6, so x = 8.

如果方程有不止一个分数，例如 (x + 1)/3 = (x – 2)/2，交叉相乘是一种快速方法。将左边分子乘以右边分母，将右边分子乘以左边分母：2(x + 1) = 3(x – 2)。然后展开并求解：2x + 2 = 3x – 6，所以 x = 8。

(2x + 3)/5 = 7 → 2x + 3 = 7 × 5 → 2x + 3 = 35 → 2x = 32 → x = 16

8. Word Problems Leading to Equations | 从应用题到方程

Many real-life problems can be solved by setting up a linear equation. The first step is to let the unknown quantity be x. Then translate the words into mathematical symbols. For example, “three times a number plus five equals twenty” becomes 3x + 5 = 20.

许多现实生活中的问题都可以通过建立线性方程来解决。第一步是设未知量为 x。然后把文字翻译成数学符号。例如，”一个数的三倍加五等于二十” 变成 3x + 5 = 20。

Let us look at a complete example: The perimeter of a rectangle is 36 cm. Its length is 4 cm more than its width. Find the width. Let the width be x, so the length is x + 4. The perimeter equation is 2(x + x + 4) = 36, which simplifies to 2(2x + 4) = 36. Divide by 2 to get 2x + 4 = 18, subtract 4 to get 2x = 14, so x = 7. The width is 7 cm.

我们来看一个完整的例子：一个矩形的周长是 36 厘米。它的长比宽多 4 厘米。求宽。设宽为 x，那么长为 x + 4。周长方程是 2(x + x + 4) = 36，化简为 2(2x + 4) = 36。除以 2，得到 2x + 4 = 18，减去 4，得到 2x = 14，所以 x = 7。宽为 7 厘米。

Always check that your answer makes sense in the original problem. If the width is 7 cm, the length is 11 cm, and the perimeter is 2(7 + 11) = 36 cm, which is correct.

始终检查你的答案在原始问题中是否合理。如果宽为 7 厘米，长为 11 厘米，周长是 2(7 + 11) = 36 厘米，这是正确的。

9. Common Mistakes and How to Avoid Them | 常见错误及避免方法

One of the most common mistakes is forgetting to do the same operation on both sides. For example, going from x + 3 = 7 to x = 7 + 3 gives x = 10, which is wrong. The correct step is x = 7 – 3, so x = 4. Always keep the balance method in mind.

最常见的错误之一是忘记在两边做相同的运算。例如，从 x + 3 = 7 得到 x = 7 + 3，得出 x = 10，这是错误的。正确的步骤是 x = 7 – 3，所以 x = 4。始终记住天平法。

Another common mistake is mishandling negative signs. When you divide both sides by a negative number, the sign of the answer changes. For example, -3x = 12 gives x = 12 ÷ (-3), which is x = -4, not x = 4.

另一个常见错误是处理负号不当。当你两边都除以一个负数时，答案的符号会改变。例如，-3x = 12 得到 x = 12 ÷ (-3)，即 x = -4，而不是 x = 4。

The table below summarises typical errors and corrections.

下表总结了典型错误及纠正方法。

Common mistake	Correct method
2x + 3 = 11 → 2x = 8 → x = 8	2x + 3 = 11 → 2x = 8 → x = 4
-2(x – 3) = 8 → -2x – 6 = 8 → -2x = 14 → x = -7	-2(x – 3) = 8 → -2x + 6 = 8 → -2x = 2 → x = -1
x/3 = 6 → x = 2	x/3 = 6 → x = 18

10. Practice Questions with Worked Solutions | 练习题与详细解答

Try these questions before looking at the solutions. Write down each step using the balance method, and always check your answer by substituting it back into the original equation.

在看答案之前先尝试这些问题。使用天平法写下每一步，并始终将答案代回原方程进行检查。

Question 1: Solve 4x + 7 = 31.

问题 1： 解方程 4x + 7 = 31。

Solution: Subtract 7 from both sides: 4x = 24. Divide both sides by 4: x = 6. Check: 4(6) + 7 = 24 + 7 = 31.

解答： 从两边减去 7：4x = 24。两边都除以 4：x = 6。检验：4(6) + 7 = 24 + 7 = 31。

Question 2: Solve 2(x – 3) = 5x + 6.

问题 2： 解方程 2(x – 3) = 5x + 6。

Solution: Expand the left side: 2x – 6 = 5x + 6. Subtract 2x from both sides: -6 = 3x + 6. Subtract 6 from both sides: -12 = 3x. Divide by 3: x = -4. Check: 2(-4 – 3) = 2(-7) = -14 and 5(-4) + 6 = -20 + 6 = -14.

解答： 展开左边：2x – 6 = 5x + 6。从两边减去 2x：-6 = 3x + 6。从两边减去 6：-12 = 3x。除以 3：x = -4。检验：2(-4 – 3) = 2(-7) = -14，且 5(-4) + 6 = -20 + 6 = -14。

Question 3: Solve (3x – 1)/4 = 5.

问题 3： 解方程 (3x – 1)/4 = 5。

Solution: Multiply both sides by 4: 3x – 1 = 20. Add 1 to both sides: 3x = 21. Divide by 3: x = 7. Check: (3(7) – 1)/4 = (21 – 1)/4 = 20/4 = 5.

解答： 两边都乘以 4：3x – 1 = 20。两边加 1：3x = 21。除以 3：x = 7。检验：(3(7) – 1)/4 = (21 – 1)/4 = 20/4 = 5。

Question 4: Three times a number decreased by 8 is equal to 2 times the number plus 5. Find the number.

问题 4： 一个数的三倍减去 8 等于这个数的两倍加上 5。求这个数。

Solution: Let the number be x. The equation is 3x – 8 = 2x + 5. Subtract 2x from both sides: x – 8 = 5. Add 8 to both sides: x = 13. Check: 3(13) – 8 = 39 – 8 = 31 and 2(13) + 5 = 26 + 5 = 31.

解答： 设这个数为 x。方程是 3x – 8 = 2x + 5。从两边减去 2x：x – 8 = 5。两边加 8：x = 13。检验：3(13) – 8 = 39 – 8 = 31，且 2(13) + 5 = 26 + 5 = 31。

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Solving Linear Equations for KS3 | KS3 解一元一次方程