Mastering Algebra: Expressions, Equations and Inequalities | 精通代数：表达式、方程与不等式

📚 Mastering Algebra: Expressions, Equations and Inequalities | 精通代数：表达式、方程与不等式

Algebra is the language of mathematics, and for the Edexcel IGCSE Mathematics examination, a secure command of algebraic manipulation is essential for success. This article breaks down the key skills you need, from simplifying expressions to solving linear and quadratic equations, with clear worked examples and common pitfalls to avoid.

代数是数学的语言，在 Edexcel IGCSE 数学考试中，扎实掌握代数运算技巧是取得好成绩的关键。本文将系统拆解核心技能，从化简表达式到求解线性方程与二次方程，提供清晰的例题和常见错误提醒。

1. Collecting Like Terms | 合并同类项

Like terms are terms whose variables (and their exponents) are identical. For example, 3x and 5x are like terms, while 3x² and 3x are not. To simplify an expression, add or subtract the coefficients of like terms only.

同类项是指变量（及其指数）完全相同的项。例如，3x 与 5x 是同类项，而 3x² 与 3x 不是。化简表达式时，只对同类项的系数进行加减。

4x + 3y − 2x + 5y = (4 − 2)x + (3 + 5)y = 2x + 8y

Be careful with negative signs: when subtracting a term, the sign belongs to the coefficient. Practice by grouping terms mentally before writing the simplified version.

注意负号：减去某一项时，符号属于该项的系数。建议先在心中分组，再写出化简结果。

2. Expanding Brackets | 展开括号

Expanding a bracket means multiplying every term inside the bracket by the term outside. The distributive law applies: a(b + c) = ab + ac.

展开括号是指将括号外的项乘以括号内的每一项。乘法分配律适用：a(b + c) = ab + ac。

3(2x − 4) = 3 × 2x + 3 × (−4) = 6x − 12

When a negative sign appears outside the bracket, remember to change the sign of every term inside. For example, −2(3x − 5) = −6x + 10.

当括号外是负号时，一定要改变括号内每一项的符号。例如，−2(3x − 5) = −6x + 10。

3. Expanding Two Brackets (FOIL) | 展开两个括号（FOIL 法）

To expand (a + b)(c + d), multiply each term in the first bracket by each term in the second bracket. The FOIL method helps: First, Outer, Inner, Last.

展开 (a + b)(c + d) 时，需将第一个括号中的每一项与第二个括号中的每一项相乘。FOIL 方法是一个口诀：First（首）、Outer（外）、Inner（内）、Last（末）。

(x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6

Notice that the middle terms, 2x and 3x, are collected together only after expansion. Do not combine them before you have multiplied everything.

请注意，中间项 2x 和 3x 只有在完整展开之后才能合并。在进行所有乘法之前，不要提前合并。

4. Factorising into Single Brackets | 提取公因式

Factorising is the reverse of expanding. Look for the highest common factor (HCF) of all terms, and write it outside the bracket, with the remaining factors inside.

因式分解是展开的逆过程。先找出所有项的最大公因式（HCF），将其写在括号外，剩余的部分写在括号内。

6x² + 9x = 3x(2x + 3)

Always check your answer by expanding it back. If you get the original expression, your factorisation is correct.

始终通过重新展开来验证答案。如果得到原表达式，说明因式分解正确。

5. Factorising Quadratics (ax² + bx + c) | 因式分解二次三项式

For a quadratic in the form x² + bx + c, find two numbers that multiply to give c and add to give b. If the coefficient of x² is not 1, use the “grouping” method or trial and error.

对于形如 x² + bx + c 的二次三项式，需要找到两个数，它们的乘积为 c，和为 b。如果 x² 的系数不是 1，则使用“分组”法或尝试法。

x² − 5x + 6 = (x − 2)(x − 3)

Here, (−2) + (−3) = −5 and (−2) × (−3) = 6. Always double-check the signs when reading the factors.

这里，(−2) + (−3) = −5，且 (−2) × (−3) = 6。读出因子时务必再次确认符号。

6. Solving Linear Equations | 解一元一次方程

A linear equation has the variable raised only to the power of 1. To solve it, perform inverse operations to isolate the variable. Remember: whatever you do to one side, you must do to the other.

一元一次方程中，变量的最高次数为 1。通过逆运算将变量孤立在等号的一侧。记住：对一边进行的操作，必须对另一边同样进行。

2x + 5 = 13 → 2x = 8 → x = 4

When the variable appears on both sides, first collect the variable terms on one side and constants on the other. Simplify fractions at the end if necessary.

当变量出现在等号两侧时，首先将含变量的项移到一边，常数项移到另一边。最后如有必要，化简分数。

7. Solving Simultaneous Equations | 解二元一次方程组

Simultaneous equations are two equations sharing two unknowns. The elimination method involves adding or subtracting equations to eliminate one variable. The substitution method involves rearranging one equation and substituting it into the other.

方程组是两个方程共享两个未知数。消元法通过加减两个方程来消去一个变量；代入法先将一个方程变形，再代入另一个方程。

x + y = 7
x − y = 3

Adding the two equations gives 2x = 10, so x = 5. Substituting back gives y = 2. Always check both equations with your solution.

两式相加得 2x = 10，因此 x = 5。代回原式得 y = 2。务必用两个原方程检验答案。

8. Solving Quadratic Equations by Factorising | 用因式分解解二次方程

A quadratic equation is in the form ax² + bx + c = 0. If it can be factorised, set each factor equal to zero and solve the resulting linear equations.

二次方程的一般形式为 ax² + bx + c = 0。如果能够因式分解，则将每个因式分别设为零，并解出相应的一元一次方程。

x² + 3x − 10 = 0 → (x + 5)(x − 2) = 0 → x = −5 or x = 2

Do not divide both sides by x unless you know x ≠ 0; you may lose a solution. Always rearrange the equation to one side first before factorising.

除非已知 x ≠ 0，否则不要两边同时除以 x，否则可能丢失一个解。务必先将方程整理到一边，再因式分解。

9. Solving Inequalities | 解不等式

Inequalities are solved like equations, with one critical difference: multiplying or dividing by a negative number reverses the inequality sign.

不等式与方程的解法类似，但有一个关键区别：乘以或除以负数时，不等号方向必须反转。

−2x < 6 → x > −3

Graphing solutions on a number line is common in IGCSE. An open circle (○) means “not included”, and a closed circle (●) means “included”.

在 IGCSE 中，常用数轴表示解集。空心圆（○）表示“不包含该点”，实心圆（●）表示“包含该点”。

10. Algebraic Fractions | 代数分式

Algebraic fractions follow the same rules as numerical fractions. To add or subtract, first find a common denominator. To simplify, cancel common factors from the numerator and denominator.

代数分式的运算规则与数值分数相同。加减时需要先找公分母；化简时需要约去分子分母中的公因式。

2/x + 3/(x+1) = [2(x+1) + 3x] / [x(x+1)] = (5x + 2) / [x(x+1)]

Never cancel individual terms that are separated by addition or subtraction signs. Only common factors of the entire numerator and the entire denominator can be cancelled.

切勿约掉被加减号分隔的单独项。只有整个分子和整个分母的公因式才能被约去。

11. Changing the Subject of a Formula | 变换公式的主项

To make a different variable the subject of a formula, treat it as a linear equation and perform inverse operations. Use factorising if the variable appears in more than one term.

要把另一个变量变为公式的主项，可以将其视为一元一次方程，进行逆运算。如果变量出现在多个项中，则需要提取公因式。

Given A = P(1 + rt), make t the subject: t = (A/P − 1) / r

Keep the order of operations in mind: clearing fractions first, then expanding brackets, then collecting the desired variable terms.

牢记运算顺序：先去分母，再展开括号，最后合并所需变量的各项。

12. Exam Tips and Common Traps | 考试技巧与常见陷阱

Always show your working; method marks can save you even if the final answer is wrong. Check your answers by substituting them back into the original equation or inequality.

务必写出解题步骤；即使最终答案错误，方法分也可能帮助你得分。将答案代回原方程或不等式进行检验。

Beware of sign errors when expanding brackets. | 展开括号时注意符号错误。
Do not forget the negative sign when factorising. | 因式分解时不要漏掉负号。
When solving inequalities, reverse the sign if multiplying by a negative. | 解不等式时，乘以负数需变号。
Read questions carefully: solve, simplify, factorise, or expand? | 仔细审题：是求解、化简、因式分解还是展开？

Consistent practice with past papers will help you recognise common question styles. Focus on accuracy first, then speed.

定期练习历年真题有助于你熟悉常见题型。先保证准确率，再提高速度。

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Mastering Algebra: Expressions, Equations and Inequalities | 精通代数：表达式、方程与不等式