📚 Mastering Algebra Basics: Expressions, Equations and Formulae | 掌握代数基础：表达式、方程和公式
Algebra is a cornerstone of mathematics that unlocks the ability to solve a huge range of problems. In the Cambridge Lower Secondary and KS3 programmes, you build skills in using letters for unknowns, simplifying expressions, solving equations and applying formulae. This article covers the essential techniques with clear examples and step-by-step explanations, from the meaning of a variable to rearranging a formula and checking your final answer.
代数是数学的基石，它为你解决大量问题提供了可能。在剑桥初中与 KS3 课程中，你将逐步掌握用字母表示未知数、化简表达式、解方程和应用公式等技能。本文通过清晰的示例和分步解释，涵盖这些核心技巧，从变量的含义，到公式变形，再到检验最终答案，一应俱全。
1. What is Algebra? | 什么是代数？
Algebra is the part of mathematics where we use letters or symbols to stand for numbers that we do not know yet, or numbers that can vary. This allows us to write general rules and solve puzzles involving unknown values. Without algebra, we could only talk about one specific number at a time; with algebra, one short statement can describe millions of cases at once.
代数是数学中运用字母或符号代表未知或可变数字的部分。这使得我们可以书写通用规则，并解决涉及未知数值的谜题。没有代数，我们一次只能讨论一个具体的数字；有了代数，一句简短的表述就能同时描述千千万万种情况。
For instance, if you know that after buying a £3 pen you have £7 left, you can express the original amount you had as x and write the equation x - 3 = 7. Solving it tells you that you started with £10.
例如，如果你知道买了 3 英镑的钢笔后还剩 7 英镑，你就可以把你原来有的金额表示为 x，并写出方程 x - 3 = 7。解出它，你就知道一开始有 10 英镑。
square adds 3 sticks, the nth term is 3n + 1. Try n = 1: 3 × 1 + 1 = 4. Try n = 2: 3 × 2 + 1 = 7. The rule matches the pictures exactly.
代数还能帮助我们描述规律，例如一系列正方形中的火柴棍数量。如果第一个图形用 4 根，每多一个正方形就增加 3 根，那么第 n 项就是 3n + 1。试 n = 1：3 × 1 + 1 = 4；试 n = 2：3 × 2 + 1 = 7。这个规则与图形完全吻合。
2. Using Letters to Represent Numbers | 用字母表示数
Letters in algebra are called variables or unknowns. You often see statements like 'Let p be the price of a book' or 'The sum of two numbers is a + b'. A variable is simply a label for a number we have not fixed yet, or a number that is allowed to change.
代数中的字母被称为变量或未知数。你经常会看到这样的表述：“设 p 为一本书的价格”或“两数之和为 a + b”。变量不过是给一个尚未确定的数、或者一个允许变化的数贴上的标签。
Remember that 3a means 3 multiplied by a, not 3 plus a. If a standing causes more lost marks than almost any other error in early algebra, so say it out loud: 'three times a'.
请记住，3a 表示 3 乘以 a，而不是 3 加 a。如果 a = 5，那么 3a = 15。这个误解在代数入门阶段造成的失分几乎比任何其他错误都多，所以请大声读出来：“三乘 a”。
× a × b is written as 7ab. The order of letters does not matter: ab means the same as ba.
当数字和字母紧挨着书写时，乘号省略。因此 5 × y 就直接写成 5y，7 × a × b 写成 7ab。字母的顺序不影响结果：ab 与 ba 含义相同。
a fraction is usually the clearest choice, especially when the expression becomes longer.
代数中也会用到分数。记号 x/2 表示 x ÷ 2，等同于 ½ × x 或 ½x。把除法写成分数通常是最清晰的选择，尤其当表达式变得较长时。
3. Algebraic Terminology: Terms, Coefficients, Constants | 代数术语：项、系数、常数
An expression is made up of terms, which are separated by + and - signs. In 4x² + 3x - 7, there are three terms: 4x², 3x and -7. Notice that the minus sign travels with the 7; a term always keeps the sign in front of it.
一个表达式由若干项组成，项与项之间用 + 和 - 号分隔。在 4x² + 3x - 7 中，有三项：4x²、3x 和 -7。注意负号是和 7 一起移动的；一项总是带着它前面的符号。
The number multiplying a letter is called the coefficient. In 4x², the coefficient is 4; in 3x, it is 3. A term with no letter, such as -7, is called a constant because its value never changes.
乘以字母的那个数字叫作系数。在 4x² 中系数是 4，在 3x 中系数是 3。没有字母的项，例如 -7，叫作常数项，因为它的值从不改变。
| Term 项 | Coefficient 系数 | Type 类型 |
| 4x² | 4 | x² term x² 项 |
| 3x | 3 | x term x 项 |
| -7 | -7 | constant 常数项 |
Terms with exactly the same letter part are called like terms. For example, 5a and -2a are like terms, and 3x² and 8x² are like terms. But 3x² and 3x are not like terms, because the powers are different. Only like terms can be added or subtracted together.
字母部分完全相同的项叫作同类项。例如 5a 与 -2a 是同类项，3x² 与 8x² 也是同类项。但 3x² 与 3x 不是同类项，因为幂次不同。只有同类项才能相加或相减。
4. Simplifying Expressions by Collecting Like Terms | 通过合并同类项化简表达式
Simplifying an expression means writing it in the shortest equivalent form, without changing its value. The main tool is collecting like terms: add the coefficients of the terms that share the same letter part, and keep the letter part unchanged.
化简表达式，就是在不改变其值的前提下，把它写成最短的等价形式。主要工具就是合并同类项：把字母部分相同的项的系数相加，字母部分保持不变。
3a + 5a = 8a 7b - 2b = 5b 4x² + 2x² = 6x²
the x terms give 3x - 5x = -2x, and the constant stays as -7. The answer is 6x² - 2x - 7.
举一个更长的例子：化简 4x² + 3x - 7 + 2x² - 5x。先分组同类项：x² 项相加得 4x² + 2x² = 6x²，x 项相加得 3x - 5x = -2x，常数项保持 -7。答案是 6x² - 2x - 7。
ul with negatives: subtracting 5x means adding -5x, so 3x - 5x really is 3x + (-5x) = -2x.
把各项按幂次从高到低排列，会让答案更易读，也更便于批改。注意负号：减去 5x 等于加上 -5x，所以 3x - 5x 实际上就是 3x + (-5x) = -2x。
5. Expanding and Factorising Brackets | 展开与因式分解括号
Expanding brackets means multiplying each term inside the bracket by the term outside. This is the distributive law:
展开括号，就是用括号外的项去乘括号内的每一项。这就是分配律：
a(b + c) = ab + ac
0. Notice what happens with the negative sign: multiplying -5 by -2 gives +10, not -10. A useful check is to count the terms: two terms inside the bracket should give two terms before you simplify.
因此 3(x + 4) = 3x + 12，-2(x - 5) = -2x + 10。注意负号的处理：-5 乘以 -2 得到 +10，而不是 -10。一个实用的检查方法是数项数：括号内有两项，展开后（化简前）就应该有两项。
use x is common to both terms. Similarly 6a + 9 = 3(2a + 3), since 3 divides both 6 and 9.
因式分解是相反的过程：提取最大公因式，放到括号外面。例如 x² + 5x = x(x + 5)，因为 x 是两项的公因式。同样地，6a + 9 = 3(2a + 3)，因为 3 能同时整除 6 和 9。
Always expand your factorised answer back out to check it. If x(x + 5) expands to x² + 5x, the factorisation is correct. This takes a few seconds and catches most mistakes.
一定要把因式分解的结果再展开回去检验。如果 x(x + 5) 展开得 x² + 5x，说明分解正确。这样做只需几秒钟，却能发现绝大多数错误。
6. Substitution: Finding the Value of an Expression | 代入求值：求表达式的值
Substitution means replacing each letter with a given number and working out the result. Write the expression again with brackets around every substituted value, then evaluate. The brackets are essential when the number is negative.
代入求值，是把每个字母替换成给定的数字并算出结果。先带着括号把每个数字写进表达式，然后再计算。当数字为负数时，括号是必不可少的。
a = 4 and b = -2, find the value of 3a + 2b. Substitute to get 3(4) + 2(-2) = 12 - 4 = 8.
示例：若 a = 4，b = -2，求 3a + 2b 的值。代入得 3(4) + 2(-2) = 12 - 4 = 8。
ld be read as -(3²) = -9, which is a completely different answer. Never skip the brackets.
再举一例：若 x = -3，求 x²。写成 (-3)² 得 9，因为负负得正。如果不加括号，-3² 会被读作 -(3²) = -9，这是完全不同的答案。千万不要省略括号。
ide 5(4) - 3 = 17, right side 2(4) + 9 = 17. Both sides match, so the answer is confirmed.
代入也是检验答案是否正确的方法。如果你认为 x = 4 是 5x - 3 = 2x + 9 的解，就把它代进去：左边 5(4) - 3 = 17，右边 2(4) + 9 = 17。两边相等，答案得到确认。
7. What is an Equation? Solving One-Step Equations | 什么是方程？解一步方程
fference from an expression: an expression such as 3x + 5 has no equals sign and cannot be 'solved'; an equation such as 3x + 5 = 20 can be solved to find the value of x.
方程是说明两个表达式相等的陈述，用等号连接。这正是它与表达式的关键区别：像 3x + 5 这样的表达式没有等号，无法“求解”；而像 3x + 5 = 20 这样的方程可以解出 x 的值。
To solve an equation, keep it balanced and undo the operations one at a time, using inverse operations. Addition is undone by subtraction, multiplication by division.
解方程时，要保持等式平衡，每次用一个逆运算抵消原有的运算。加法用减法抵消，乘法用除法抵消。
- x - 3 = 7 → add 3 to both sides → x = 10
- x + 5 = 12 → subtract 5 from both sides → x = 7
- 3x = 18 → divide both sides by 3 → x = 6
- x/4 = 5 → multiply both sides by 4 → x = 20
Whatever you do to one side, you must do to the other. That single rule guarantees the two sides remain equal, and it is the reason we can 'move' a term across by doing the opposite operation.
你对等式一边做什么，就必须对另一边做同样的事。这条唯一的规则保证了等式两边始终相等，也正是我们可以通过逆运算把一项“移”到另一边的原因。
8. Solving Two-Step and Multi-Step Equations | 解两步与多步方程
Most exam questions need more than one step. Work in reverse order of operations: deal with addition and subtraction first, then multiplication and division.
大多数考题都不止一步。请按运算的相反顺序来做：先处理加减，再处理乘除。
1: solve 2x + 5 = 17. Subtract 5 from both sides: 2x = 12. Divide both sides by 2: x = 6.
例 1：解 2x + 5 = 17。两边同时减 5：2x = 12。两边同时除以 2：x = 6。
- 6 = 12, giving 3x = 18 and x = 6. Both routes are valid; choose the one you find safer.
例 2：解 3(x - 2) = 12。可以先两边同除以 3，得 x - 2 = 4，于是 x = 6；也可以先展开为 3x - 6 = 12，得 3x = 18，x = 6。两条路径都正确，选你更有把握的那条。
- 3 = 9. Add 3: 3x = 12. Divide by 3: x = 4. Always finish by substituting back to check.
例 3：解 5x - 3 = 2x + 9。两边同时减 2x，把 x 项集中到一边：3x - 3 = 9。再加 3：3x = 12。除以 3：x = 4。最后一定要代回原方程检验。
9. Formulae and Changing the Subject | 公式与改变主项
ngle is given by P = 2(l + w), where P is the subject because it stands alone on one side.
公式是一种特殊的方程，它把两个或多个量联系起来，并且在所有情况下都成立。长方形的周长由 P = 2(l + w) 给出，其中 P 是主项，因为它单独位于等号一边。
ubject of P = 2(l + w), divide both sides by 2: P/2 = l + w. Then subtract l: w = P/2 - l.
改变主项就是重新整理公式，让另一个字母单独位于一边。若要把 w 变为 P = 2(l + w) 的主项，先两边同除以 2：P/2 = l + w；再减 l，得 w = P/2 - l。
a: t = (v - u)/a. Notice that we treat every letter except t as if it were a known number.
另一个常见例子：把 v = u + at 变形成以 t 为主项。两边同时减 u：v - u = at。两边同时除以 a：t = (v - u)/a。注意，我们把除 t 以外的每个字母都当作已知数来处理。
Students often ask why we can divide by a letter. The answer is that in a formula the letters stand for numbers, and as long as that number is not zero, the division is valid. This is exactly the same operation as dividing both sides of an equation by 3.
学生常问，为什么可以除以一个字母？答案是：在公式中字母代表数字，只要这个数字不为零，除法就是合法的。这与方程两边同时除以 3 是完全相同的运算。
10. Algebra in Context: Patterns and Word Problems | 情境中的代数：规律与实际应用题
Exam questions rarely say 'simplify' or 'solve' in isolation; they wrap the algebra inside a story. The skill is to translate the words into symbols before doing any arithmetic.
考题很少孤立地写“化简”或“求解”，它们通常把代数藏在一个情境里。关键在于先翻译成符号，再做任何计算。
each time, so the nth term is 3n + 1. To test it, put n = 10 and get 31 sticks, which you could verify by drawing.
再看火柴棒规律：一连串共用边的正方形。第 1 个图形用 4 根，第 2 个用 7 根，第 3 个用 10 根。每次增加 3 根，所以第 n 项是 3n + 1。检验一下：令 n = 10，得 31 根，你可以通过画图验证。
Word problems usually follow a reliable routine. Read the question twice, define the unknown with a letter and a sentence such as 'Let x be the number of small boxes', write an equation, solve it, and answer the actual question asked. Many marks are lost simply because the student finds x but never states the final quantity in words.
应用题通常有一套可靠的流程：把题目读两遍；用一个字母加一句话来定义未知数，例如“设 x 为小盒子的数量”；列出方程；解方程；最后回答题目真正问的问题。很多失分只是因为学生求出 x 后，没有用文字写出最终的答案。
= 4x + 6. Expanding and collecting like terms are the two skills doing all the work here.
几何题也使用同样的思路。如果一个长方形的长是 x + 3、宽是 x，那么面积是 x(x + 3) = x² + 3x，周长是 2(x + 3) + 2x = 4x + 6。在这里真正起作用的，就是展开括号和合并同类项这两项技能。
11. Common Mistakes to Avoid | 需要避免的常见错误
Most algebra errors are not caused by difficult mathematics; they come from a handful of predictable slips. Knowing them in advance is the fastest way to raise your mark.
大多数代数错误并不是因为数学难，而是来自几类可预见的疏忽。提前了解它们，是提分最快的途径。
- Reading 3a as 3 + a. It means 3 × a. 把 3a 读成 3 + a。它表示 3 × a。
- Combining unlike terms, such as writing 3x + 4 as 7x. 合并不同类项，例如把 3x + 4 写成 7x。
- Multiplying only the first term in a bracket: 3(x + 4) = 3x + 4 is wrong. 只乘括号内第一项：3(x + 4) = 3x + 4 是错的。
- Losing a minus sign when moving terms across the equals sign. 移项时丢掉负号。
- Substituting a negative number without brackets, so (-3)² becomes -9. 代入负数时忘加括号，把 (-3)² 算成 -9。
- Solving for x but forgetting to answer the question actually asked. 求出 x 却忘记回答题目真正问的内容。
Three habits prevent nearly all of these. First, write one step per line so the equals signs line up. Second, keep negative signs attached to the number that follows them. Third, always substitute your final answer back into the original equation or formula.
三个习惯几乎可以避免上述全部错误。第一，每行只写一步，让等号上下对齐。第二，把负号与它后面的数字绑在一起书写。第三，永远把最终答案代回原方程或原公式检验。
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