📚 KS3 Maths: Algebra and Functions – Key Revision Points | KS3 数学：代数和函数考点精讲
Algebra and functions form the foundation of KS3 mathematics. This guide walks you through the essential topics: simplifying expressions, expanding brackets, solving equations, using function machines, and plotting straight-line graphs. Each section is designed to help you understand the core concepts and apply them confidently in class assessments and exams.
代数与函数是 KS3 数学的基石。本指南将带你梳理关键知识点：化简表达式、展开括号、解方程、使用函数机器以及绘制直线图像。每个部分都旨在帮助你理解核心概念，并在课堂评估和考试中自信地运用。
1. Algebraic Expressions and Terms | 代数表达式与项
An algebraic expression is a combination of numbers, letters (variables) and operation signs. For example, 3x + 2y − 7 is an expression. The parts separated by + or − are called ‘terms’. In 4a²b, the number 4 is the coefficient, ‘a’ and ‘b’ are variables, and the small 2 is the exponent.
代数表达式是由数字、字母（变量）和运算符号组成的式子。例如，3x + 2y − 7 就是一个表达式。由加号或减号分隔的部分称为“项”。在 4a²b 中，数字 4 是系数，’a’ 和 ‘b’ 是变量，小号 2 是指数。
‘Like terms’ have exactly the same variable parts and exponents. For instance, 5x and −3x are like terms, while 2x² and 3x are not, because the exponents differ. Constants like 7 and −9 are also like terms.
“同类项”具有完全相同的变量部分和指数。例如 5x 和 −3x 是同类项，而 2x² 和 3x 不是，因为指数不同。常数项如 7 和 −9 也属于同类项。
2. Simplifying Expressions | 化简表达式
To simplify an expression, collect all like terms. Add or subtract their coefficients while keeping the variable part unchanged. Always pay attention to the signs in front of each term.
化简表达式时，先合并所有同类项。系数相加减，变量部分保持不变。要特别留意每一项前面的正负号。
Simplify 4a + 3b − 2a + 5b. First, group the a terms: 4a − 2a = 2a. Then group the b terms: 3b + 5b = 8b. The simplified expression is 2a + 8b. The table below shows the steps clearly.
化简 4a + 3b − 2a + 5b。首先合并 a 项：4a − 2a = 2a；然后合并 b 项：3b + 5b = 8b。化简结果为 2a + 8b。下表清晰地展示了步骤。
| Original expression | 4a + 3b − 2a + 5b |
| Group like terms | (4a − 2a) + (3b + 5b) |
| Simplify | 2a + 8b |
When negative signs appear, be careful. For example, 5x − 3y − 2x + y becomes (5x − 2x) + (−3y + y) = 3x − 2y.
当出现负号时要仔细。例如 5x − 3y − 2x + y 变为 (5x − 2x) + (−3y + y) = 3x − 2y。
3. Expanding Brackets | 展开括号
Expanding brackets means multiplying each term inside the bracket by the term outside. This uses the distributive law: a(b + c) = ab + ac. The same rule applies when a minus sign or a negative factor sits in front of the bracket.
展开括号就是将括号外的项与括号内的每一项相乘。这使用了分配律：a(b + c) = ab + ac。当括号前是减号或负因数时，同样遵循这个法则。
Expand 3(x + 4). Multiply 3 by x and 3 by 4: 3 × x + 3 × 4 = 3x + 12. For −2(3y − 5), multiply −2 by 3y and −2 by −5, giving −6y + 10.
展开 3(x + 4)：3 乘 x 和 3 乘 4，得 3x + 12。对于 −2(3y − 5)，用 −2 乘以 3y 和 −2 乘以 −5，得到 −6y + 10。
For two brackets, such as (x + 2)(x + 3), multiply each term in the first bracket by each term in the second: x·x + x·3 + 2·x + 2·3 = x² + 3x + 2x + 6. Then simplify to x² + 5x + 6.
对于两个括号，如 (x + 2)(x + 3)，将第一个括号的每一项分别乘以第二个括号的每一项：x·x + x·3 + 2·x + 2·3 = x² + 3x + 2x + 6。然后化简得 x² + 5x + 6。
4. Factorising | 因式分解
Factorising is the reverse of expanding: you take out the highest common factor (HCF) of all terms and rewrite the expression as a product. Always check your answer by expanding again.
因式分解是展开的反向操作：将各项的最大公因数 (HCF) 提取出来，把表达式改写为乘积形式。完成后一定要再展开验证。
Factorise 6x + 9. Both terms share a factor of 3, so 6x + 9 = 3(2x + 3). A more complex example: 4x² − 8x. The HCF is 4x, so the factorised form is 4x(x − 2).
因式分解 6x + 9。两项的公因数是 3，所以 6x + 9 = 3(2x + 3)。更复杂的例子：4x² − 8x，HCF 是 4x，因此因式分解后为 4x(x − 2)。
For expressions like x² + 5x + 6, you look for two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3, so the factorised form is (x + 2)(x + 3).
对于 x² + 5x + 6 这类表达式，需要找到两个相乘得 6、相加得 5 的数。这两个数是 2 和 3，因此因式分解为 (x + 2)(x + 3)。
5. Substitution and Formulae | 代入与公式
Substitution means replacing letters with given numbers and working out the value. Always follow the order of operations – brackets, indices, division/multiplication, addition/subtraction (BIDMAS).
代入意味着用给定的数字替换字母并计算出数值。始终遵循运算顺序——括号、指数、乘除、加减 (BIDMAS)。
If a = 2 and b = −1, evaluate 3a + 4b − 5. Replace a with 2 and b with −1: 3 × 2 + 4 × (−1) − 5 = 6 − 4 − 5 = −3.
若 a = 2，b = −1，计算 3a + 4b − 5。将 a 换成 2，b 换成 −1：3 × 2 + 4 × (−1) − 5 = 6 − 4 − 5 = −3。
A formula links two or more variables. For instance, the perimeter of a rectangle is P = 2l + 2w. If l = 5 cm and w = 3 cm, then P = 2×5 + 2×3 = 10 + 6 = 16 cm. Substitution is used in science and everyday situations too.
公式将两个或多个变量联系起来。例如，矩形周长公式为 P = 2l + 2w。若 l = 5 厘米、w = 3 厘米，则 P = 2×5 + 2×3 = 10 + 6 = 16 厘米。代入也常用于科学和日常生活中。
6. Solving Linear Equations | 解线性方程
To solve an equation, find the value of the unknown that makes the equation true. Keep the equation balanced by performing the same operation on both sides. Always aim to get the variable on its own.
解方程就是求出使等式成立的未知数的值。在方程两边同时进行相同的运算，保持平衡。目标总是将变量单独留在一边。
One-step example: x + 7 = 12. Subtract 7 from both sides: x = 5. Two-step example: 2x − 3 = 9. Add 3 to both sides: 2x = 12. Then divide by 2: x = 6.
一步方程示例：x + 7 = 12。两边减 7 得 x = 5。两步方程：2x − 3 = 9。两边加 3 得 2x = 12，再除以 2 得 x = 6。
When the variable appears on both sides, like 5x + 2 = 3x + 10, collect the x terms on one side: 5x − 3x = 10 − 2, giving 2x = 8, so x = 4.
当变量出现在等式两边时，如 5x + 2 = 3x + 10，将含 x 的项移到一边：5x − 3x = 10 − 2，得 2x = 8，所以 x = 4。
7. Inequalities | 不等式
An inequality compares two expressions using < (less than), > (greater than), ≤ (less than or equal to) and ≥ (greater than or equal to). Solving inequalities works like solving equations, but you must reverse the inequality sign when multiplying or dividing by a negative number.
不等式用 <（小于）、>（大于）、≤（小于等于）、≥（大于等于）比较两个表达式。解不等式与解方程类似，但当两边同时乘以或除以一个负数时，必须反转不等号方向。
Solve 2x + 3 ≤ 9. Subtract 3: 2x ≤ 6. Divide by 2: x ≤ 3. The solution can be shown on a number line with a solid dot at 3 and shading to the left.
解 2x + 3 ≤ 9：两边减 3 得 2x ≤ 6，除以 2 得 x ≤ 3。解集可在数轴上用实心点标在 3 处并向左画线表示。
If you have −2x > 6, dividing by −2 reverses the sign: x < −3. Always remember to flip the direction when the coefficient of x is negative.
若为 −2x > 6，两边除以 −2 后不等号反转：x < −3。当 x 的系数为负时，务必记得改变方向。
8. Function Machines | 函数机器
A function machine takes an input, applies a rule and produces an output. You can represent it with a diagram or an algebraic expression, such as y = 2x + 1. The input is often called x, and the output is often called y or f(x).
函数机器接收一个输入、应用某个规则后输出一个结果。可以用图表或像 y = 2x + 1 这样的代数式来表示。输入通常称为 x，输出常称为 y 或 f(x)。
If the rule is ‘multiply by 3, then add 4’, the output for an input x is 3x + 4. For input 5, the output is 3 × 5 + 4 = 19. To find the input from an output, work backwards using inverse operations.
若规则为“乘 3 再加 4”，输入 x 对应的输出为 3x + 4。输入 5 则输出 3 × 5 + 4 = 19。若要从输出求输入，则逆向使用逆运算。
Given an output 22 with the same machine, the input satisfies 3x + 4 = 22. Solve: 3x = 18, x = 6. Reversing the machine gives the same result: subtract 4, then divide by 3.
若同一机器的输出为 22，则输入满足 3x + 4 = 22。解得 3x = 18，x = 6。反向操作：先减 4，再除以 3，结果相同。
9. Sequences | 序列
A sequence is an ordered list of numbers following a rule. In a linear (arithmetic) sequence, the difference between consecutive terms is constant – this is the common difference. You can describe the sequence using the term-to-term rule or the nth term expression.
序列是按一定规则排列的有序数字列表。在线性（等差）序列中，相邻项的差恒定，称为公差。你可以用邻项规则或第 n 项表达式来描述序列。
Example: 5, 8, 11, 14, 17, … The first term is 5 and the common difference is +3. The nth term expression is 3n + 2, because when n = 1, 3(1) + 2 = 5; n = 2 gives 8, and so on. To find the 10th term, substitute n = 10: 3×10 + 2 = 32.
示例：5, 8, 11, 14, 17, … 首项为 5，公差为 +3。第 n 项表达式为 3n + 2，因为 n = 1 时，3×1 + 2 = 5；n =
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