KS3 Cambridge Maths: Page 74 Q1 – Solving Equations with Variables on Both Sides | KS3 剑桥数学：第74页第1题 – 解含双边变量的方程

📚 KS3 Cambridge Maths: Page 74 Q1 – Solving Equations with Variables on Both Sides | KS3 剑桥数学：第74页第1题 – 解含双边变量的方程

In the Cambridge KS3 curriculum, pupils encounter algebraic equations where the unknown appears on both sides of the equals sign. A typical example is Question 1 on page 74 of the practice book: Solve 5x – 2 = 3x + 6. This article walks through the method in detail and builds up to more challenging variations, helping students develop confidence in solving linear equations.

在剑桥 KS3 课程中，学生会遇到等号两边都含有未知数的代数方程。练习册第 74 页第 1 题就是一个典型例子：解方程 5x – 2 = 3x + 6。本文将详细讲解求解方法，并逐步深入到更具挑战性的变体，帮助学生建立解方程的信心。

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1. Understanding the Problem | 理解问题

Before we start, observe that the left side contains the term 5x and a constant –2, while the right side contains 3x and +6. The goal is to find the single value of x that makes both sides equal. We need to move the x-terms together, keeping the equation balanced at every step.

在开始之前，先观察左侧有 5x 和常数项 –2，右侧有 3x 和 +6。目标是找出使两边相等的唯一 x 值。我们需要把含 x 的项移到一起，同时每一步都要保持等式平衡。

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2. Step 1: Eliminate the Variable from One Side | 第1步：将变量从一边消去

Subtract 3x from both sides to remove the x-term from the right. This uses the balance method:
5x – 2 – 3x = 3x + 6 – 3x
This simplifies to 2x – 2 = 6. Now the variable appears only on the left.

两边同时减去 3x，消去右边的含 x 项。这就运用了天平法：
5x – 2 – 3x = 3x + 6 – 3x
化简得 2x – 2 = 6。此时变量只出现在左边。

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3. Step 2: Remove the Constant Term | 第2步：移走常数项

Add 2 to both sides to isolate the term containing x. The equation becomes:
2x – 2 + 2 = 6 + 2
which simplifies to 2x = 8.

两边同时加 2，将含 x 的项单独留在左边。方程变为：
2x – 2 + 2 = 6 + 2
化简得 2x = 8。

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4. Step 3: Solve for x | 第3步：解出 x

Divide both sides by the coefficient of x. Here the coefficient is 2, so we divide by 2:
2x ÷ 2 = 8 ÷ 2
Therefore, x = 4.

两边同时除以 x 的系数。这里系数是 2，所以两边除以 2：
2x ÷ 2 = 8 ÷ 2
得到 x = 4。

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5. Checking Your Answer | 检验答案

Always substitute your solution back into the original equation. Left side: 5(4) – 2 = 20 – 2 = 18. Right side: 3(4) + 6 = 12 + 6 = 18. Both sides equal 18, so x = 4 is correct.

一定要把解代回原方程检验。左边：5(4) – 2 = 20 – 2 = 18。右边：3(4) + 6 = 12 + 6 = 18。两边都等于 18，所以 x = 4 是正确解。

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6. Equations with Negative Coefficients | 系数为负的方程

Consider –2x + 5 = x + 11. To avoid errors, add 2x to both sides to keep the coefficient of x positive: 5 = 3x + 11. Then subtract 11 from both sides: –6 = 3x, so x = –2.

考虑 –2x + 5 = x + 11 这样的方程。为了避免出错，可以同时在两边加 2x，使 x 的系数为正：5 = 3x + 11。然后两边减 11：–6 = 3x，得 x = –2。

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7. Equations Involving Brackets | 含括号的方程

If the equation includes brackets, expand them first. For example, 3(2x – 1) = 4x + 5 becomes 6x – 3 = 4x + 5. Then subtract 4x from both sides: 2x – 3 = 5. Add 3: 2x = 8, so x = 4.

如果方程含有括号，先展开。例如，3(2x – 1) = 4x + 5 变成 6x – 3 = 4x + 5。然后两边减 4x：2x – 3 = 5。加 3：2x = 8，得 x = 4。

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8. Equations with Fractional Coefficients | 含分数系数的方程

When fractions appear, multiply every term by the lowest common denominator. For (x/2) + 3 = (x/4) + 5, multiply through by 4: 2x + 12 = x + 20. Then solve: 2x – x = 20 – 12, giving x = 8.

当出现分数时，可以用最小公分母乘以每一项。对于 (x/2) + 3 = (x/4) + 5，两边乘 4：2x + 12 = x + 20。再解：2x – x = 20 – 12，得 x = 8。

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

Watch out for these frequent pitfalls when solving equations with variables on both sides.

解答变量在两侧的方程时，要注意以下常见错误。

Common Mistake 常见错误	Why It Happens 原因	How to Avoid 如何避免
Forgetting to change the sign when moving a term across the equals sign. 移项时忘记变号	Inverse operations are not applied consistently. 逆运算运用不一致	Always perform the same operation (add, subtract, multiply, divide) on both sides. 始终对等式两边执行相同的运算。
Losing a negative sign when collecting like terms. 合并同类项时丢掉负号	Subtracting a negative term is confused. 减去一个负数项容易混淆	Re-write the step clearly: e.g. –2x + 7 = 3x becomes 7 = 5x. 清晰地写出步骤：如 –2x + 7 = 3x 变为 7 = 5x。
Only moving the variable, but not the constant. 只移变量项，不动常数	Learners focus only on the x-terms and forget to balance constants. 学生只关注 x 项而忘记平衡常数	Treat the equation like a balanced scale; every term must be moved correctly. 把方程当作天平，每一项都要正确移动。

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10. Practice Questions with Worked Solutions | 练习题及解答步骤

Try these equations to strengthen your skills.

尝试解下列方程，强化你的技巧。

• Question 1: 7x + 4 = 3x + 20
  Subtract 3x: 4x + 4 = 20. Subtract 4: 4x = 16. x = 4.

  第1题：7x + 4 = 3x + 20
  减 3x：4x + 4 = 20。减 4：4x = 16，x = 4。

• Question 2: 2(x – 5) = x + 3
  Expand: 2x – 10 = x + 3. Subtract x: x – 10 = 3. Add 10: x = 13.

  第2题：2(x – 5) = x + 3
  展开：2x – 10 = x + 3。减 x：x – 10 = 3。加 10：x = 13。

• Question 3: 5x – 9 = –2x + 12
  Add 2x: 7x – 9 = 12. Add 9: 7x = 21. x = 3.

  第3题：5x – 9 = –2x + 12
  加 2x：7x – 9 = 12。加 9：7x = 21，x = 3。

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11. Why Mastering These Equations Matters | 掌握这些方程的重要性

Solving equations with variables on both sides is a cornerstone of algebra. It trains logical thinking and prepares you for more advanced topics such as simultaneous equations, inequalities, and functions. In KS3 assessments, you will often see these questions in both non-calculator and calculator papers.

解两边都有变量的方程是代数的基石。它训练逻辑思维，并为后续更高级的课题——如联立方程、不等式和函数——打下基础。在 KS3 考核中，无论是使用计算器还是不允许使用计算器的试卷里，这类题目都经常出现。

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12. Tips for Exam Success | 考试通关建议

• Always write each step clearly. Clear working helps you spot mistakes and earns method marks.

  每一步都要写清楚。清晰的演算有助于发现错误，也能拿到步骤分。

• Check your answer by substitution, especially if time allows. It is the only way to be certain.

  时间允许的话，一定要用代入法检验答案。这是唯一能确定的办法。

• If you get stuck, try adding or subtracting x-terms so that the variable ends up on the side where the coefficient is positive. This often reduces sign errors.

  如果卡住了，可以尝试把 x 项移到系数为正的一边。这常常能减少符号错误。

• Practice with negatives and fractions regularly—they appear frequently in exam papers.

  经常练习带负数和分数的方程——它们在试卷中出现频率很高。

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