p243_1: Mastering Linear Equations and Formulae | 掌握线性方程与公式

📚 p243_1: Mastering Linear Equations and Formulae | 掌握线性方程与公式

one of the most important algebraic skills in the Cambridge KS3 mathematics curriculum: solving linear equations and working with formulae. A linear equation is an equation in which the unknown appears only to the power of 1. Being able to solve equations confidently helps students to model real-life problems and prepares them for more advanced topics such as simultaneous equations and quadratics.

练习页 p243_1.pdf 聚焦于剑桥 KS3 数学课程中最重要的代数技能之一：解线性方程与运用公式。线性方程是指未知数只出现一次方（即指数为 1）的方程。能够熟练求解方程有助于学生建立现实问题的数学模型，并为后续学习联立方程和二次方程等更高级主题奠定基础。

1. Understanding Expressions, Equations and Formulae | 区分表达式、方程与公式

e relationship between two or more quantities, such as A = lw for the area of a rectangle.

表达式是由数字、字母和运算符号组成且不含等号的式子，例如 3x + 2。方程是说明两个表达式相等的陈述，例如 3x + 2 = 14。公式是一种特殊方程，用来表示两个或多个量之间的关系，例如矩形面积公式 A = lw。

In Cambridge KS3 examinations, students are often asked to identify whether a given mathematical statement is an expression, an equation or a formula. It is important not to call an expression 'an equation' just because it contains an x.

在剑桥 KS3 考试中，学生经常被要求判断给定的数学陈述是表达式、方程还是公式。不要因为含有 x 就把表达式误称为“方程”，这一点很重要。

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2. Balancing Method: The Golden Rule | 平衡法：黄金法则

The balancing method is based on the idea that an equation is like a balance scale. Whatever you do to one side of the equation, you must do to the other side to keep it balanced. This means you can add, subtract, multiply or divide by the same non-zero number on both sides.

平衡法基于方程就像天平的想法。对方程一边进行的任何操作，都必须对另一边进行同样的操作，以保持平衡。这意味着你可以在两边同时加上、减去、乘以或除以同一个非零数。

or example, to solve x + 5 = 12, subtract 5 from both sides: x + 5 − 5 = 12 − 5, so x = 7.

例如，解 x + 5 = 12。两边同时减去 5：x + 5 − 5 = 12 − 5，因此 x = 7。

x + 5 = 12 → x = 7

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3. Solving One-Step Equations | 解一步方程

to isolate the unknown. The four basic forms are x + a = b, x − a = b, ax = b and x/a = b.

一步方程只需一步运算就能分离未知数。四种基本形式是 x + a = b、x − a = b、ax = b 和 x/a = b。

For addition equations, subtract the constant from both sides. For multiplication equations, divide both sides by the coefficient of x.

对于加法方程，两边同时减去常数；对于乘法方程，两边同时除以 x 的系数。

x + 9 = 15 → x = 6

4x = 20 → x = 5

x/3 = 7 → x = 21

Always check your answer: substitute the value back into the original equation to make sure both sides are equal.

始终记得检验答案：把求得的值代回原方程，确认等号两边相等。

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4. Solving Two-Step Equations | 解两步方程

olve it, first remove the constant term by adding or subtracting, then divide by the coefficient of x.

两步方程包含两种运算。典型形式是 ax + b = c。解这类方程时，首先通过加减去掉常数项，然后除以 x 的系数。

olve 5x + 3 = 23. Subtract 3 from both sides: 5x = 20. Then divide both sides by 5: x = 4.

例如，解 5x + 3 = 23。两边减 3：5x = 20。再两边除以 5：x = 4。

5x + 3 = 23 → 5x = 20 → x = 4

racted. For 2x − 7 = 9, add 7 to both sides to get 2x = 16, then divide by 2 to get x = 8.

同样的方法也适用于常数项为减法的方程。例如 2x − 7 = 9，两边加 7，得到 2x = 16，再除以 2，得到 x = 8。

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5. Equations with Brackets | 带括号的方程

ve law: a(b + c) = ab + ac. Then solve the simplified equation using the balancing method.

当方程含有括号时，先用分配律展开：a(b + c) = ab + ac。然后使用平衡法求解简化后的方程。

27. Expand to get 3x + 12 = 27. Subtract 12 from both sides: 3x = 15. Divide by 3: x = 5.

例如，解 3(x + 4) = 27。展开得 3x + 12 = 27。两边减 12，得 3x = 15。再除以 3，得 x = 5。

3(x + 4) = 27 → 3x + 12 = 27 → x = 5

ful: −(x − 2) means −1 × (x − 2) = −x + 2. This is a common place where sign errors occur.

如果括号前有负号，要特别小心：−(x − 2) 表示 −1 × (x − 2) = −x + 2。这是常见的符号错误点。

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6. Equations with Unknowns on Both Sides | 未知数在等号两边的方程

If the unknown appears on both sides of the equation, collect all the unknown terms on one side and the number terms on the other side. You can do this by adding or subtracting the same unknown term from both sides.

如果未知数出现在等号两边，应把所有含未知数的项集中到一边，把数字项集中到另一边。可以通过在两边同时加减相同的未知数项来实现。

x + 18. Subtract 3x from both sides: 4x + 2 = 18. Subtract 2: 4x = 16. Divide by 4: x = 4.

例如，解 7x + 2 = 3x + 18。两边减 3x，得 4x + 2 = 18。两边减 2，得 4x = 16。两边除以 4，得 x = 4。

7x + 2 = 3x + 18 → 4x + 2 = 18 → x = 4

= 4 into the original equation: 7(4) + 2 = 30 and 3(4) + 18 = 30, so both sides are equal.

检验时，将 x = 4 代入原方程：7(4) + 2 = 30，3(4) + 18 = 30，两边相等。

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7. Equations Involving Fractions | 含分数的方程

Equations with fractions can be solved by multiplying every term by the lowest common denominator to clear the fractions. This makes the equation easier to work with.

含分数的方程可以通过将每一项都乘以最小公分母来消去分数。这会让方程更容易处理。

Solve x/4 + 1 = 3. Multiply all terms by 4: x + 4 = 12. Subtract 4: x = 8.

例如，解 x/4 + 1 = 3。所有项乘以 4，得 x + 4 = 12。再减 4，得 x = 8。

x/4 + 1 = 3 → x + 4 = 12 → x = 8

2x + 1)/5 = 3, multiply both sides by 5 to get 2x + 1 = 15, then solve: 2x = 14, so x = 7.

当分母更复杂时，例如 (2x + 1)/5 = 3，两边乘以 5，得 2x + 1 = 15，然后求解：2x = 14，所以 x = 7。

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8. Constructing Equations from Word Problems | 从文字题构建方程

Many Cambridge KS3 questions ask students to form an equation from a written problem. Look for key words such as 'total', 'difference', 'product' and 'equals'. Let the unknown quantity be x, then translate the sentence into algebra.

许多剑桥 KS3 题目要求学生根据文字题写出方程。注意“总共”“差”“乘积”“等于”等关键词。设未知量为 x，然后把句子翻译成代数式。

For example, the sum of a number and 7 is 15. Let the number be x: x + 7 = 15, so x = 8.

例如，一个数与 7 的和是 15。设这个数为 x：x + 7 = 15，所以 x = 8。

h + width) = 2(2x + 1 + x) = 26. Simplify: 2(3x + 1) = 26 → 6x + 2 = 26 → 6x = 24 → x = 4.

另一个例子：一个矩形的长为 2x + 1 cm，宽为 x cm。如果周长为 26 cm，建立并求解方程。周长 = 2(长 + 宽) = 2(2x + 1 + x) = 26。化简：2(3x + 1) = 26，进而 6x + 2 = 26，6x = 24，所以 x = 4。

2(2x + 1 + x) = 26 → 6x + 2 = 26 → x = 4

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9. Substitution into Formulae | 代入公式

A formula shows how variables are related. Substitution means replacing a letter with a given number and calculating the value using the correct order of operations.

公式表示变量之间的关系。代入是指用给定的数字替换字母，并按照正确的运算顺序计算值。

when u = 2, t = 3 and a = 4. Substitute: s = 2 × 3 + ½ × 4 × 3² = 6 + 2 × 9 = 6 + 18 = 24.

例如，已知 s = ut + ½at²，当 u = 2、t = 3、a = 4 时，求 s 的值。代入计算：s = 2 × 3 + ½ × 4 × 3² = 6 + 2 × 9 = 6 + 18 = 24。

s = 2 × 3 + ½ × 4 × 3² = 24

Remember the order of operations: brackets, indices, division, multiplication, addition, then subtraction. For substitution, calculate powers before multiplying.

记住运算顺序：括号、指数、除法、乘法、加法、减法。代入时，先计算幂再相乘。

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10. Changing the Subject of a Formula | 公式变形（改换主项）

Changing the subject means rearranging a formula so that a different variable is isolated on one side. Use inverse operations in reverse order, just as when solving linear equations.

公式变形（改换主项）是指重新排列公式，使另一个变量单独位于等号一侧。要像解线性方程一样，按照相反的顺序使用逆运算。

y = 3x + 5. Subtract 5 from both sides: y − 5 = 3x. Divide both sides by 3: x = (y − 5)/3.

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