📚 Solving Linear Equations – Cambridge Checkpoint p46 Ex 1 | 解线性方程——剑桥Checkpoint第46页练习1
In this lesson, we focus on a typical linear equation problem from Cambridge Checkpoint Maths coursebook (page 46, exercise 1). The equation 3x + 5 = 2x + 11 serves as a starting point to build essential algebra skills for KS3 students. We will explore a step-by-step method, explain why each step works, and reinforce techniques that prevent common mistakes.
在本课中，我们将聚焦剑桥Checkpoint数学教材第46页练习1中的一道典型线性方程题：3x + 5 = 2x + 11。通过这道题，我们将建立KS3阶段必备的代数技能，逐步讲解每一步的原理，强化正确方法，从而避免常见错误。
1. The Problem Statement | 题目呈现
Consider the equation: 3x + 5 = 2x + 11. The goal is to find the value of x that makes the left-hand side equal to the right-hand side.
请看方程：3x + 5 = 2x + 11。目标是找到使得等号左右两边相等的未知数 x 的值。
3x + 5 = 2x + 11
2. What is a Linear Equation? | 什么是线性方程？
A linear equation is an algebraic statement in which the highest power of the variable is 1. It can have one or more terms on each side, and solving it means isolating the variable on one side of the equation.
线性方程是变量最高次幂为1的代数等式。它可以在等号两边各有一个或多个项，解方程意味着将变量单独移到方程的一边。
3. The Balancing Method | 平衡法原理
Think of an equation as a balanced scale. Whatever operation you do to one side, you must do exactly the same to the other side to maintain balance. This principle is central to solving any linear equation.
可以把方程想象成一座天平。你对一边进行的任何运算，必须对另一边完全同样地执行，才能保持平衡。这个原则是解所有线性方程的核心。
4. Step-by-Step Approach | 逐步解题步骤
For 3x + 5 = 2x + 11, we want all x terms on one side and numbers on the other. First, subtract 2x from both sides:
对于 3x + 5 = 2x + 11，我们希望将所有含 x 的项移到一边，常数项移到另一边。首先，两边同时减去 2x：
3x – 2x + 5 = 2x – 2x + 11
⇒ x + 5 = 11
Now subtract 5 from both sides to isolate x:
现在两边减去 5，分离出 x：
x + 5 – 5 = 11 – 5
⇒ x = 6
The solution is x = 6. We can verify by substituting back: 3(6) + 5 = 18 + 5 = 23, and 2(6) + 11 = 12 + 11 = 23. Both sides match.
解得 x = 6。我们可以代入验证：3(6) + 5 = 23，2(6) + 11 = 23，两边相等。
5. Handling Equations with Brackets | 处理带括号的方程
Sometimes equations include brackets, such as 2(x + 3) = 10. Use the distributive law first: multiply the term outside the bracket with each term inside.
有时方程会包含括号，例如 2(x + 3) = 10。应先使用分配律：将括号外的项乘以括号内的每一项。
2(x + 3) = 2x + 6 = 10
Then subtract 6 from both sides:
然后两边减6：
2x + 6 – 6 = 10 – 6 ⇒ 2x = 4
Finally, divide both sides by 2:
最后两边除以2：
x = 4 ÷ 2 = 2
Always expand brackets before collecting like terms.
务必在合并同类项前先展开括号。
6. Equations with Fractions | 含有分数的方程
When an equation includes a fraction like x/4 + 2 = 5, eliminate the fraction by multiplying every term by the denominator. Multiply all terms by 4:
当方程中出现分数，如 x/4 + 2 = 5，可以通过将每一项乘以分母来消去分数。将所有项乘以4：
(x/4)×4 + 2×4 = 5×4 ⇒ x + 8 = 20
Then subtract 8:
然后减8：
x = 20 – 8 = 12
This method works for any denominator. If there are multiple fractions, multiply by the least common multiple.
这种方法适用于任何分母。如果有多项分数，可乘以最小公倍数。
7. Why Checking Your Answer Matters | 验证解答的重要性
Substituting the solution back into the original equation confirms accuracy and helps catch arithmetic mistakes. Make it a habit to check, especially in exams.
把解代入原方程可以验证准确性，并有助于发现计算错误。养成检查的习惯，尤其在考试中。
For 3x + 5 = 2x + 11, plugging x = 6 gives 23 on both sides, so it is correct.
对于 3x + 5 = 2x + 11，代入 x = 6 得到两边均为23，证明正确。
8. Common Errors and How to Avoid Them | 常见错误与避免方法
Below are frequent pitfalls when solving linear equations, with tips to avoid them.
以下是解线性方程时的常见陷阱及避免方法。
| Common Mistake | How to Avoid |
|---|---|
| Forgetting to perform the same operation on both sides. | Always think ‘balance’ – write down each step clearly. |
| Adding instead of subtracting when moving terms. | Check the sign: if it is +2x, subtract 2x; if it is -3, add 3. |
| Misapplying the distributive law: e.g., 2(x+3) → 2x+3. | Multiply the outside term with every term inside: 2(x+3) = 2x+6. |
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