Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations form the backbone of algebra at Key Stage 3. Understanding how to find the unknown value in an equation like 2x + 5 = 13 is an essential skill that will carry you through all future mathematics topics. This guide will break down the process step by step, from simple one-step equations to more complex problems with brackets and variables on both sides. By the end, you will be confident in solving any linear equation you encounter in your Cambridge KS3 syllabus.

线性方程是KS3阶段代数的基石。理解如何求解像 2x + 5 = 13 这样的方程中的未知数，是一项至关重要的技能，它将贯穿你未来所有的数学学习。本指南将逐步分解求解过程，从简单的一步方程到含有括号和变量在等式两边的更复杂问题。最终，你将能够自信地解决剑桥KS3课程中遇到的任何线性方程。

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

A linear equation is a mathematical statement that shows two expressions are equal, and the highest power of the variable is 1. For example, x + 3 = 7 and 4y – 2 = 10 are both linear equations. The word ‘linear’ comes from the fact that if you graph these equations, they form a straight line. In KS3, you will typically work with one variable, often denoted by a letter like x, y, or a. The goal is to find the value of that variable that makes the equation true.

线性方程是一个数学语句，表明两个表达式相等，且变量的最高次幂为1。例如 x + 3 = 7 和 4y – 2 = 10 都是线性方程。“线性”一词源于将这些方程绘制成图时会形成一条直线。在KS3阶段，你通常处理的是含有一个变量的方程，该变量常用 x、y 或 a 等字母表示。目标是找出使方程成立的该变量的值。

2. The Balance Method | 天平法

Think of an equation as a perfectly balanced scale. Whatever you do to one side, you must do exactly the same to the other side to keep it balanced. This is the golden rule of solving equations. If you add 3 to the left-hand side, you must add 3 to the right-hand side. If you divide the left side by 2, you must divide the right side by 2. Following this principle ensures that the equality remains true while you isolate the variable.

把方程想象成一个完全平衡的天平。无论你对一边做什么操作，都必须对另一边做完全相同的操作，以保持平衡。这是解方程的黄金法则。如果你在左边加3，就必须在右边也加3。如果你把左边除以2，就必须把右边也除以2。遵循这一原则，可以确保在分离变量的过程中等式始终成立。

3. Solving One-Step Equations | 解一步方程

A one-step equation requires only one operation to isolate the variable. For example, x + 5 = 12. To solve it, subtract 5 from both sides: x + 5 – 5 = 12 – 5, so x = 7. Another example: 3y = 15. Divide both sides by 3: 3y ÷ 3 = 15 ÷ 3, giving y = 5. The key is to perform the inverse operation: addition undoes subtraction, multiplication undoes division, and vice versa.

一步方程只需一次运算即可分离出变量。例如 x + 5 = 12。求解时，两边同时减去5：x + 5 – 5 = 12 – 5，得到 x = 7。另一个例子：3y = 15。两边同时除以3：3y ÷ 3 = 15 ÷ 3，得到 y = 5。关键是进行逆运算：加法抵消减法，乘法抵消除法，反之亦然。

4. Solving Two-Step Equations | 解两步方程

Two-step equations involve two operations. Take 2x + 3 = 11. First, undo the addition by subtracting 3 from both sides: 2x + 3 – 3 = 11 – 3 → 2x = 8. Second, undo the multiplication by dividing both sides by 2: 2x ÷ 2 = 8 ÷ 2 → x = 4. Always reverse the order of operations: deal with addition/subtraction before multiplication/division.

两步方程涉及两种运算。以 2x + 3 = 11 为例。首先，通过两边同时减3来抵消加法：2x + 3 – 3 = 11 – 3 → 2x = 8。然后，通过两边同时除以2来抵消乘法：2x ÷ 2 = 8 ÷ 2 → x = 4。始终要逆用运算顺序：先处理加减，再处理乘除。

5. Dealing with Negative Variables | 处理负变量

Sometimes you might end up with a negative variable, like -x = 4. To find x, multiply or divide both sides by -1: (-x) × (-1) = 4 × (-1) → x = -4. Alternatively, you can avoid negative variables by adding x to both sides early on. For instance, if you have 5 – x = 3, you can add x to both sides: 5 – x + x = 3 + x → 5 = 3 + x, then subtract 3: x = 2.

有时你可能会得到像 -x = 4 这样的负变量。要求出 x，可以将两边同时乘以或除以 -1：(-x) × (-1) = 4 × (-1) → x = -4。或者，你可以通过在早期两边同时加上 x 来避免负变量的出现。例如，如果遇到 5 – x = 3，可以在两边同时加上 x：5 – x + x = 3 + x → 5 = 3 + x，然后再减去3：x = 2。

6. Equations with Brackets | 含括号的方程

When an equation contains brackets, such as 3(x + 2) = 15, your first step is to expand the brackets. Multiply each term inside the bracket by the number outside: 3 × x + 3 × 2 = 3x + 6. So the equation becomes 3x + 6 = 15. Then solve as a two-step equation: subtract 6 → 3x = 9, divide by 3 → x = 3. If a negative number is outside the bracket, remember to multiply carefully with signs, e.g., -2(x – 4) = -2x + 8.

当方程中含有括号时，例如 3(x + 2) = 15，第一步是展开括号。用括号外的数乘以括号内的每一项：3 × x + 3 × 2 = 3x + 6。于是方程变为 3x + 6 = 15。然后将其作为两步方程来解：减6 → 3x = 9，除以3 → x = 3。如果括号外是负数，记得要小心处理符号相乘，例如 -2(x – 4) = -2x + 8。

7. Variables on Both Sides | 变量在等式两边

For equations like 5x + 2 = 3x + 10, you need to collect the variable terms on one side and the constant terms on the other. Subtract the smaller variable term from both sides: 5x – 3x + 2 = 3x – 3x + 10 → 2x + 2 = 10. Then subtract 2 from both sides: 2x = 8, and divide by 2: x = 4. The aim is to get all the x’s on one side and all the numbers on the other.

对于像 5x + 2 = 3x + 10 这样的方程，需要将变量项移到一边，常数项移到另一边。在两边同时减去较小的变量项：5x – 3x + 2 = 3x – 3x + 10 → 2x + 2 = 10。然后两边同时减去2：2x = 8，再除以2：x = 4。目标是将所有含 x 的项放在一边，所有数字放在另一边。

8. Checking Your Solution | 检查结果

Always substitute your answer back into the original equation to verify it. If you found x = 4 for 2x + 3 = 11, check: 2(4) + 3 = 8 + 3 = 11. Since 11 = 11, the solution is correct. This habit will save you from careless mistakes in exams and ensures you haven’t missed any sign errors.

始终将求得的答案代回原方程进行验证。如果你求出 2x + 3 = 11 的解为 x = 4，检验一下：2(4) + 3 = 8 + 3 = 11。因为 11 = 11，所以解是正确的。这个习惯可以避免你在考试中因粗心犯错，并确保没有遗漏任何符号错误。

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

One frequent error is forgetting to apply an operation to both sides. For instance, in x/2 = 5, some might incorrectly write x = 5/2 instead of multiplying both sides by 2 to get x = 10. Another mistake is mishandling negative signs, especially when expanding brackets like -3(x – 2) to get -3x – 6 instead of -3x + 6. Also, avoid combining like terms incorrectly, such as adding 2x and 3 to get 5x—these are not like terms.

一个常见错误是忘记对等式两边同时进行运算。例如，在 x/2 = 5 中，有些人可能会错误地写成 x = 5/2，而不是两边同乘以2得到 x = 10。另一个错误是处理负号不当，尤其是在展开括号时，比如 -3(x – 2) 错误地展开为 -3x – 6，而正确应为 -3x + 6。此外，要避免错误地合并同类项，比如将 2x 和 3 相加得到 5x——它们不是同类项。

10. Practice Problems | 练习题

Try solving these equations step by step:

4x – 7 = 13
2(x + 5) = 16
6x + 3 = 2x + 19
5 – 2x = 9
3(x – 1) + 2x = 12

After attempting them, check your answers: x = 5, x = 3, x = 4, x = -2, x = 3. Regular practice builds fluency and confidence.

尝试分步求解下列方程：

4x – 7 = 13
2(x + 5) = 16
6x + 3 = 2x + 19
5 – 2x = 9
3(x – 1) + 2x = 12

解答后核对答案：x = 5，x = 3，x = 4，x = -2，x = 3。经常练习可以提升熟练度和信心。

11. Real-Life Applications | 实际应用

Linear equations are not just abstract schoolwork—they model real situations. For example, if a taxi charges a £3 base fare plus £2 per mile, the total cost for m miles is C = 2m + 3. To find how many miles you traveled if the fare was £17, solve 2m + 3 = 17. Subtract 3: 2m = 14, divide by 2: m = 7 miles. Equations help you answer every day ‘how many’ or ‘how much’ questions.

线性方程不仅仅是抽象的学校作业——它们模拟了真实情境。例如，如果出租车起步价为3英镑，之后每英里收费2英镑，行驶 m 英里的总费用为 C = 2m + 3。如果车费是17英镑，想知道行驶了多少英里，就解方程 2m + 3 = 17。减去3：2m = 14，除以2：m = 7 英里。方程帮助你解答日常生活中“多少”之类的问题。

12. Summary | 总结

Solving linear equations is a skill built on the balance method and inverse operations. Always follow these steps: simplify each side if needed (expand brackets, combine like terms), collect variable terms on one side and constants on the other, perform inverse operations to isolate the variable, and finally check your answer. With consistent practice, you will master this fundamental topic and be well-prepared for more advanced algebra.

解线性方程是建立在天平法和逆运算基础上的一项技能。始终遵循以下步骤：必要时化简两边（展开括号、合并同类项），将变量项移到一边、常数项移到另一边，执行逆运算分离变量，最后检验答案。通过持续练习，你将掌握这个基础课题，并为学习更高级的代数做好充分准备。

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Solving Linear Equations | 解一元一次方程