📚 Cambridge KS3 p270.1 Solving Linear Equations | 剑桥KS3 p270.1 解一元一次方程
This worksheet covers the essential KS3 Cambridge algebra skill of solving linear equations. The questions on p270.1 require you to use inverse operations, balance both sides of an equation, and check solutions carefully.
本练习页涵盖 KS3 剑桥代数中解一元一次方程的核心技能。p270.1 中的题目要求你运用逆运算、保持等式两边平衡并仔细检验结果。
1. What Is a Linear Equation? | 什么是一元一次方程？
A linear equation is an algebraic statement in which the highest power of the variable is 1. It can usually be written in the form ax + b = c, where a, b and c are constants and x is the unknown.
一元一次方程是变量的最高次数为 1 的代数等式。它通常可以写成 ax + b = c 的形式，其中 a、b、c 是常数，x 是未知数。
For example, 3x + 4 = 16 is a linear equation because x is only raised to the power 1. The goal is to find the value of x that makes the equation true.
例如，3x + 4 = 16 是一个一元一次方程，因为 x 只是一次方。我们的目标是找到使等式成立的 x 的值。
3x + 4 = 16
In a linear equation, you will never see terms such as x², x³, or 1/x. If you do, it is not a linear equation at this level.
在一元一次方程中，你不会看到 x²、x³ 或 1/x 这样的项。如果看到了，它就不是本阶段的一元一次方程。
2. Key Vocabulary | 核心词汇
You should be confident with the words equation, expression, variable, coefficient, constant, and solution. An equation contains an equals sign; an expression does not.
你应该熟悉以下词汇：方程、表达式、变量、系数、常数和解。方程包含等号，而表达式不包含。
- Variable 变量: a letter such as x or y that stands for an unknown number
- Coefficient 系数: the number multiplying a variable, such as 4 in 4x
- Constant 常数: a number on its own, such as 3 in 4x + 3
- Solution 解: the value that makes the equation true
For example, in the equation 4x + 3 = 15, the coefficient is 4, the constant is 3, and the solution is x = 3.
例如，在方程 4x + 3 = 15 中，系数是 4，常数是 3，解是 x = 3。
3. The Balance Method | 天平法
Think of an equation as a balanced scale. Whatever you do to one side, you must do to the other side. This rule keeps the equation true while you isolate the variable.
把方程想象成一个平衡的天平。无论你对一边做什么，必须对另一边做同样的操作。这条规则能保证在你分离未知数时方程仍然成立。
For example, to solve x + 5 = 12, subtract 5 from both sides so that x is left by itself.
例如，要解 x + 5 = 12，两边同时减去 5，使 x 单独留在一边。
x + 5 = 12
x + 5 − 5 = 12 − 5
x = 7
Always write the same operation on both sides before simplifying. This shows your method clearly and reduces mistakes.
在化简之前，一定要先把同样的运算写在两边。这样可以清晰展示你的解题步骤，减少错误。
4. Inverse Operations | 逆运算
To undo an operation, use its inverse. Addition and subtraction are inverses. Multiplication and division are inverses. This idea is central to solving equations.
要撤销一种运算，就使用它的逆运算。加法和减法互为逆运算，乘法和除法互为逆运算。这个思想是解方程的核心。
| Operation 运算 | Inverse 逆运算 |
|---|---|
| + | − |
| − | + |
| × | ÷ |
| ÷ | × |
When you solve an equation, look at the order of operations applied to the variable, then undo them in reverse order. For 2x + 3, x is first multiplied by 2, then 3 is added.
解方程时，先看变量经历了哪些运算，再按相反顺序逐步撤销。例如 2x + 3 中，x 先乘以 2，再加 3。
To solve, undo the addition first, then undo the multiplication.
求解时，先撤销加法，再撤销乘法。
5. Solving Two-Step Equations | 解两步方程
A two-step equation involves two operations. A common example is 2x + 3 = 11. The variable x is multiplied by 2 and then 3 is added.
两步方程包含两种运算。常见例子是 2x + 3 = 11。变量 x 先乘以 2，然后加上 3。
2x + 3 = 11
Step 1: subtract 3 from both sides.
第一步：两边同时减去 3。
2x = 8
Step 2: divide both sides by 2.
第二步：两边同时除以 2。
x = 4
Write each step on a new line so that an examiner can follow your reasoning. Keep the equals signs aligned vertically.
每个步骤另起一行书写，这样阅卷人能够清楚地跟随你的推理过程。并保持等号上下对齐。
6. Equations with Brackets | 带括号的方程
If an equation contains brackets, expand them first. Then solve using the balance method and inverse operations.
如果方程中含有括号，先展开括号。然后使用天平法和逆运算来求解。
For example, consider 3(x + 2) = 18.
例如，看方程 3(x + 2) = 18。
3x + 6 = 18
Subtract 6 from both sides, then divide by 3.
两边同时减去 6，再同时除以 3。
3x = 12
x = 4
A common error is to forget that the number outside the bracket multiplies every term inside. Make sure to multiply both terms by the coefficient.
一个常见错误是忘记括号外的数要乘以括号内的每一项。一定要把括号中的每一项都乘以该系数。
7. Equations with Unknowns on Both Sides | 未知数在等式两边
Sometimes the variable appears on both sides, such as in 5x − 2 = 2x + 7. Your first job is to collect all x terms on one side.
有时候变量出现在等式两边，例如 5x − 2 = 2x + 7。你的第一步是把所有含 x 的项移到同一边。
5x − 2 = 2x + 7
Subtract 2x from both sides.
两边同时减去 2x。
3x − 2 = 7
Then add 2 to both sides.
然后两边同时加上 2。
3x = 9
Divide both sides by 3.
两边同时除以 3。
x = 3
You can move either the smaller or the larger x term first. Choose the step that keeps coefficients positive if possible, as this reduces sign errors.
你可以先移较小或较大的 x 项。尽量选择能让系数保持为正的步骤，这样可以减少符号错误。
8. Equations with Fractions | 含分数的方程
If an equation has a fraction such as x/3 + 1 = 5, the quickest first step is to eliminate the denominator by multiplying every term by it.
如果方程中含有分数，例如 x/3 + 1 = 5，最快的做法是先乘以分母，消去分数。
x/3 + 1 = 5
Multiply every term by 3.
每一项都乘以 3。
x + 3 = 15
Subtract 3 from both sides.
两边同时减去 3。
x = 12
Always multiply every term, not just the fraction. This is another common error that can change the whole equation.
一定要乘以每一项，而不只是分数项。这是另一个容易改变整个方程的常见错误。
9. Checking Your Solution | 检验方程的解
Checking is not optional: it is a key step in algebraic accuracy. Substitute your answer back into the original equation to see if both sides are equal.
检验不是可有可无的步骤，而是保证代数计算准确的关键。将你的答案代入原方程，看看两边是否相等。
For example, if you solve 2x + 3 = 11 and get x = 4, substitute 4 into the left-hand side.
例如，你解 2x + 3 = 11 得到 x = 4，把 4 代入等号左边。
2 × 4 + 3 = 8 + 3 = 11
Since the left-hand side equals the right-hand side, the solution is correct. If they are not equal, go back and find the error.
因为左边等于右边，所以这个解是正确的。如果两边不相等，就返回去检查错误。
10. Common Mistakes | 常见错误
Knowing what can go wrong helps you avoid it. Watch out for the following mistakes when solving linear equations.
了解容易出错的地方有助于避免犯错。解一元一次方程时，请注意以下常见问题。
- Forgetting to do the same thing to both sides 忘记两边同时进行相同的操作
- Using the wrong inverse operation 使用错误的逆运算
- Expanding brackets incorrectly 展开括号时计算错误
- Losing a negative sign 漏掉负号
- Multiplying one term by the denominator instead of all terms 只把一项乘以分母，而不是所有项
If your final answer does not check, do not simply change the answer. Trace back through each step to find where the balance was broken.
如果最终答案检验不成立，不要只修改答案。倒推每一步，找出天平是在哪一步被破坏的。
11. Word Problems Leading to Equations | 从文字题到方程
Many KS3 exam questions describe a situation in words. You must translate the words into an equation. Look for phrases such as ‘more than’, ‘less than’, ‘times’, and ‘altogether’.
很多 KS3 考试题目会用文字描述情景。你必须把文字转换成方程。注意像“比……多”、“比……少”、“乘以”、“总共”这样的词语。
For example: ‘Three more than twice a number is 15’. Let the unknown number be n. Then twice the number is 2n, and three more is 2n + 3.
例如：“一个数的两倍加 3 等于 15”。设未知数为 n。这个数的两倍是 2n，再加 3 就是 2n + 3。
2n + 3 = 15
Solve to find n = 6. Always define your variable clearly before writing the equation.
解方程得到 n = 6。在写方程之前，一定要清楚地定义变量。
12. Exam-Style Practice | 考试风格练习
Try these questions, then check your answers with the method shown. This mirrors the style of the p270.1 worksheet.
尝试下面的题目，然后用所示方法核对答案。这些题目模拟 p270.1 练习页的风格。
- Solve 4x − 7 = 9 解方程 4x − 7 = 9
- Solve 2(x + 3) = 14 解方程 2(x + 3) = 14
- Solve 6x + 4 = 2x + 20 解方程 6x + 4 = 2x + 20
For the first equation, add 7 to both sides to get 4x = 16, then divide by 4 to get x = 4.
对于第一个方程，两边同时加 7 得到 4x = 16，然后除以 4 得到 x = 4。
For the second
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