📚 Solving Linear Equations and Using Formulae | 解一元一次方程与公式运用
Linear equations and formulae are core algebra skills in the Cambridge KS3 mathematics programme. This revision guide explains how to solve one-step, two-step, and multi-step linear equations, how to substitute into formulae, and how to rearrange formulae to change the subject. Work through the examples and checkpoint-style questions to build confidence for class tests and Checkpoint assessments.
一元一次方程和公式是剑桥 KS3 数学课程中的核心代数技能。本复习指南讲解如何解一步、两步和多步一元一次方程，如何代入公式求值，以及如何变形公式以转换主项。通过例题和 Checkpoint 风格练习题，你可以为课堂测验和 Checkpoint 评估建立信心。
1. What is a Linear Equation? | 什么是一元一次方程？
ally be written in the form ax + b = c, where a, b and c are numbers and x is the unknown.
一元一次方程是指未知数的最高次数为 1 的方程。它通常可以写成 ax + b = c 的形式，其中 a、b 和 c 是数，x 是未知数。
- Examples of linear equations: x + 3 = 7, 2x - 5 = 13, 4x = 20, x/3 + 2 = 6.
- 一元一次方程的例子：x + 3 = 7，2x - 5 = 13，4x = 20，x/3 + 2 = 6。
- The word 'linear' tells us that the graph of the equation is a straight line.
- “线性”一词告诉我们该方程的图像是一条直线。
2. Keeping the Equation Balanced | 保持方程平衡
An equation is like a balance scale. Whatever you do to one side, you must do to the other side to keep it balanced. This is the golden rule of solving equations.
方程就像一座天平。你对一边所做的任何操作，都必须对另一边做相同的操作，以保持平衡。这是解方程的金科玉律。
If x + 5 = 9, subtract 5 from both sides: x + 5 - 5 = 9 - 5, so x = 4.
如果 x + 5 = 9，两边同时减去 5：x + 5 - 5 = 9 - 5，因此 x = 4。
- You may add, subtract, multiply or divide both sides by the same non-zero number.
- 你可以对两边同时加、减、乘或除以同一个非零的数。
- Never perform an operation on only one side, or the equation becomes false.
- 切勿只对一边进行运算，否则方程就不再成立。
3. Solving by Inverse Operations | 用逆运算解方程
To solve a linear equation, undo the operations that have been applied to the variable. Work in reverse order, using inverse operations: addition and subtraction are inverses, and multiplication and division are inverses.
解一元一次方程时，要逆着运算顺序撤销对变量所做的运算。加法和减法互为逆运算，乘法和除法互为逆运算。
2x + 3 = 11. First subtract 3 from both sides: 2x = 8. Then divide both sides by 2: x = 4.
解方程 2x + 3 = 11。首先两边同时减去 3：2x = 8。然后两边同时除以 2：x = 4。
| Equation 方程 | Step 步骤 | Solution 解 |
| x - 6 = 9 | Add 6 to both sides 两边加 6 | x = 15 |
| 5x = 35 | Divide both sides by 5 两边除以 5 | x = 7 |
| x/4 = 3 | Multiply both sides by 4 两边乘以 4 | x = 12 |
4. Solving Equations with Brackets | 解含括号的方程
When an equation contains brackets, expand them first using the distributive law, then solve as usual. Alternatively, if the whole bracket is multiplied by a number, you can divide both sides by that number first.
当方程中含有括号时，先运用分配律展开括号，然后按常规方法求解。或者，如果整个括号都乘以一个数，也可以先两边同时除以这个数。
Solve 3(x + 2) = 21. Expand: 3x + 6 = 21. Subtract 6: 3x = 15. Divide by 3: x = 5.
解方程 3(x + 2) = 21。展开：3x + 6 = 21。减去 6：3x = 15。除以 3：x = 5。
- Alternative method: 3(x + 2) = 21, divide both sides by 3: x + 2 = 7, so x = 5.
- 另一种方法：3(x + 2) = 21，两边除以 3：x + 2 = 7，所以 x = 5。
- Be careful with negative signs, such as -2(x - 3) = -2x + 6.
- 注意负号，例如 -2(x - 3) = -2x + 6。
5. Equations with Variables on Both Sides | 变量在方程两边的情况
If the same unknown appears on both sides of the equation, collect the variable terms on one side and the constant terms on the other side. Choose the side that gives a positive coefficient of x where possible.
如果同一个未知数出现在方程两边，就把含变量的项移到一边，常数项移到另一边。尽量选择能让 x 的系数为正的一边。
x + 14. Subtract 2x from both sides: 3x + 2 = 14. Subtract 2: 3x = 12. Divide by 3: x = 4.
解方程 5x + 2 = 2x + 14。两边同时减去 2x：3x + 2 = 14。再减去 2：3x = 12。除以 3：x = 4。
- Always check by substituting the value back into the original equation.
- 始终通过将值代回原方程进行检验。
- If the coefficient of x becomes negative, multiply both sides by -1.
- 如果 x 的系数变为负数，两边同时乘以 -1。
6. Working with Formulae | 使用公式
A formula is a rule that shows the relationship between two or more variables. In KS3 mathematics, common formulae include those for area, perimeter, speed, and temperature conversion.
公式是表示两个或多个变量之间关系的规则。在 KS3 数学中，常见公式包括面积、周长、速度和温度转换公式。
| Formula 公式 | Meaning 含义 |
| A = lw | Area of rectangle 矩形面积 |
| C = 2πr | Circumference of circle 圆的周长 |
| v = u + at | Final velocity 最终速度 |
| F = (9/5)C + 32 | Celsius to Fahrenheit 摄氏转华氏 |
7. Substitution into Formulae | 代入公式求值
Substitution means replacing each variable in a formula with a given numerical value, then carrying out the operations in the correct order using BIDMAS or BODMAS.
代入是指用给定的数值替换公式中的每个变量，然后按照 BIDMAS 或 BODMAS 规则按正确顺序进行运算。
Given A = lw, find A when l = 6 cm and w = 4 cm. A = 6 x 4 = 24 cm².
已知 A = lw，当 l = 6 厘米、w = 4 厘米时求 A。A = 6 x 4 = 24 平方厘米。
- Include units in the final answer, such as cm, m², km/h or °C.
- 最终答案要写明单位，如 cm、m²、km/h 或 °C。
- Be careful with negative numbers: if x = -3, then x² = 9, but -x² = -9.
- 注意负数：如果 x = -3，那么 x² = 9，但 -x² = -9。
8. Changing the Subject of a Formula | 公式变形
Changing the subject of a formula means rearranging the formula so that a different variable is on its own. Use inverse operations just as you would when solving an equation.
公式变形是指重新整理公式，使另一个变量单独位于等号一侧。就像解方程一样使用逆运算。
y = 2x + 3. Subtract 3 from both sides: y - 3 = 2x. Divide both sides by 2: x = (y - 3)/2.
将 y = 2x + 3 变形为以 x 为主项。两边减去 3：y - 3 = 2x。两边除以 2：x = (y - 3)/2。
Make r the subject of C = 2πr. Divide both sides by 2π: r = C / 2π.
将 C = 2πr 变形为以 r 为主项。两边同时除以 2π：r = C / 2π。
- The subject must be on its own, with a coefficient of 1.
- 主项必须单独在一边，且系数为 1。
- When the subject appears in a denominator, multiply both sides by the denominator first.
- 当主项出现在分母中时，先两边乘以分母。
9. Common Misconceptions and How to Avoid Them | 常见误区与规避方法
Many errors in linear equations come from forgetting the balance principle, misusing negative signs, or applying operations in the wrong order. Recognising these mistakes will improve accuracy.
解一元一次方程时的许多错误源于忘记平衡原则、误用负号或运算顺序错误。识别这些错误可以提高准确性。
- 4 = 10 by subtracting 3 from both sides. Correct: subtract 4 first, so 3x = 6, then x = 2.
- 误区：解方程 3x + 4 = 10 时先用两边减 3。正确做法：先减 4，得到 3x = 6，然后 x = 2。
- Misconception: writing -2(x - 3) as -2x - 6. Correct: -2(x - 3) = -2x + 6.
- 误区：把 -2(x - 3) 写成 -2x - 6。正确：-2(x - 3) = -2x + 6。
- t wrongly, such as y = x/3 + 2 becoming x = 3y - 2. Correct: y - 2 = x/3, so x = 3(y - 2).
- 误区：主项变形错误，例如 y = x/3 + 2 变成 x = 3y - 2。正确：y - 2 = x/3，所以 x = 3(y - 2)。
10. Checkpoint-style Practice Questions | Checkpoint 风格练习题
Try these questions without looking at the solutions first. Then check your answers step by step to identify any gaps in understanding.
先尝试完成这些题目，不要看答案。然后逐步核对答案，找出理解上的漏洞。
- Solve 4x - 7 = 25. Answer: x = 8.
- 解方程 4x - 7 = 25。答案：x = 8。
- Solve 2(x + 3) = 16. Answer: x = 5.
- 解方程 2(x + 3) = 16。答案：x = 5。
- Solve 6x + 1 = 4x + 13. Answer: x = 6.
- 解方程 6x + 1 = 4x + 13。答案：x = 6。
- Given A = ½bh, find A when b = 8 cm and h = 5 cm. Answer: A = 20 cm².
- 已知 A = ½bh，当 b = 8 厘米、h = 5 厘米时求 A。答案：A = 20 平方厘米。
- Make x the subject of y = 4x - 1. Answer: x = (y + 1)/4.
- 将 y = 4x - 1 变形为以 x 为主项。答案：x = (y + 1)/4。
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