📚 Solving Linear Equations in One Variable | 一元一次方程求解
Linear equations are the foundation of algebra in KS3 Cambridge Mathematics. This revision guide covers solving one-step, two-step and multi-step linear equations, handling brackets, fractions and negative numbers, and forming equations from word problems. Each section pairs the key idea with a worked example so you can build confidence quickly.
一次方程是 KS3 剑桥数学代数部分的基础。本复习指南涵盖一步、两步和多步一元一次方程的解法，包括括号、分数和负数的处理，以及根据文字题列方程。每一节都配有核心概念与例题，帮助你快速建立解题信心。
1. What is a Linear Equation? | 什么是线性方程？
A linear equation in one variable can be written in the form ax + b = c, where a, b and c are numbers and a ≠ 0. The unknown, usually written as x, appears only to the power of 1. The solution is the value of x that makes the equation true. If the highest power of x were 2, the equation would be quadratic, not linear, and it would need a completely different method.
一元一次方程可以写成 ax + b = c 的形式，其中 a、b、c 是数，且 a ≠ 0。未知数通常写作 x，只以一次方出现。方程的解就是使等式成立的 x 值。如果 x 的最高次数是 2，那么方程就是二次方程，而不是一次方程，需要用完全不同的方法求解。
ared, cubed or put inside a square root. Expressions such as 4(x - 1) = 12 and x ÷ 5 + 2 = 9 are still linear, because the unknown is only multiplied or divided by a number.
有两个实用的判断标准。第一，未知数不能出现在分母中。第二，未知数不能被平方、立方或放在根号里。像 4(x - 1) = 12 和 x ÷ 5 + 2 = 9 这样的式子仍然是一次方程，因为未知数只是被一个数乘或除。
| Equation | 方程 | Linear? | 是否一次 | Reason | 原因 |
| 2x + 3 = 11 | Yes 是 | Highest power of x is 1 | x 的最高次数是 1 |
| 5 - 3x = 2 | Yes 是 | x appears once, power 1 | x 只出现一次，次数为 1 |
| x² + 1 = 10 | No 否 | Quadratic | 二次方程 |
| 1 ÷ x = 4 | No 否 | x is in the denominator | x 在分母中 |
Example: 2x + 3 = 11 is linear because the highest power of x is 1.
示例：2x + 3 = 11 是一次方程，因为 x 的最高次数是 1。
2. The Balance Method | 天平法
Think of an equation as a balanced pair of scales. Any operation you perform on one side must also be performed on the other side. Adding 3 on the left means adding 3 on the right. Dividing the left by 2 means dividing the right by 2. This keeps the equation balanced and does not change the solution.
把方程想象成保持平衡的天平。你对等号一边做任何运算，另一边也必须做同样的运算。左边加 3，右边也要加 3；左边除以 2，右边也要除以 2。这样方程始终保持平衡，解也不会改变。
ro number, dividing by a non-zero number, expanding brackets and collecting like terms. The goal is always to isolate x on one side so the equation reads x = something.
只有一种操作是禁止的：两边同时乘以或除以 0。其他操作都允许——加、减、乘以非零数、除以非零数、展开括号、合并同类项。目标始终是把 x 单独留在一边，使方程变成 x = 某个数 的形式。
Solve 2x + 3 = 11: subtract 3 → 2x = 8; divide by 2 → x = 4.
解 2x + 3 = 11：两边减 3 → 2x = 8；两边除以 2 → x = 4。
3. One-Step Equations | 一步方程
th division. Keep the equation balanced by doing the same thing on both sides, then write the answer clearly as x = ...
一步方程只需要一步运算就能解出。选择逆运算：加法用减法抵消，乘法用除法抵消。两边做同样的事情保持平衡，最后清楚地写出 x = …… 的答案。
- x + 5 = 12 → subtract 5 from both sides → x = 7
(x + 5 = 12 → 两边减 5 → x = 7) - x - 4 = 9 → add 4 to both sides → x = 13
(x - 4 = 9 → 两边加 4 → x = 13) - 4x = 20 → divide both sides by 4 → x = 5
(4x = 20 → 两边除以 4 → x = 5) - x ÷ 3 = 7 → multiply both sides by 3 → x = 21
(x ÷ 3 = 7 → 两边乘以 3 → x = 21) - -x = 6 → multiply both sides by -1 → x = -6
(-x = 6 → 两边乘以 -1 → x = -6)
racting 4 instead of dividing. Remember: the number written next to x is multiplying it, so the inverse operation is division, not subtraction.
常见错误是把 4x = 20 用减法去做。请记住：写在 x 旁边的数字与 x 是相乘关系，所以逆运算是除法，而不是减法。
4. Two-Step Equations | 两步方程
Two-step equations contain both a multiplication (or division) and an addition (or subtraction). Undo them in reverse order: deal with the addition or subtraction first, then the multiplication or division. This is the same order you would use to unpack a parcel - last thing packed, first thing removed.
两步方程同时含有乘法（或除法）与加法（或减法）。要按相反的顺序处理：先解决加法和减法，再解决乘法和除法。这就像拆包裹——最后放进去的东西最先拿出来。
Solve 3x - 4 = 17: add 4 → 3x = 21; divide by 3 → x = 7.
解 3x - 4 = 17：两边加 4 → 3x = 21；两边除以 3 → x = 7。
ich matches the right-hand side. Substitution takes ten seconds and catches almost every arithmetic error before it costs you marks.
把答案代回原方程检验。若 x = 7，则 3 × 7 - 4 = 21 - 4 = 17，与等号右边一致。代入检验只要十秒钟，却能在丢分之前发现几乎所有的计算错误。
Another useful layout is to write every step on its own line, with the equals signs lined up vertically. This makes long solutions readable and shows the examiner exactly what you did, which matters when marks are awarded for method.
另一个好习惯是每一步单独写一行，把等号上下对齐。这样长解答更易读，也让阅卷老师清楚看到你的思路——在按步骤给分时这非常重要。
5. Multi-Step Equations: Unknowns on Both Sides | 多步方程：两边都有未知数
When x appears on both sides, first collect all the x terms on one side and all the numbers on the other. Add or subtract the smaller x term from both sides; this usually keeps the coefficient positive and reduces sign errors.
当等号两边都出现 x 时，先把所有含 x 的项集中到一边，把所有数字集中到另一边。两边同时减去较小的 x 项，这样通常能让系数保持为正，减少符号错误。
Solve 5x - 3 = 2x + 9: subtract 2x → 3x - 3 = 9; add 3 → 3x = 12; divide by 3 → x = 4.
解 5x - 3 = 2x + 9：两边减 2x → 3x - 3 = 9；两边加 3 → 3x = 12；两边除以 3 → x = 4。
ution is worthwhile: 5 × 4 - 3 = 17 and 2 × 4 + 9 = 17, so the two sides really are equal.
把同类项整理清楚。像 5x 与 2x 是同类项，可以合并；5x 与 5 则不能。用代入法快速检验很有价值：5 × 4 - 3 = 17，2 × 4 + 9 = 17，两边确实相等。
6. Equations Containing Brackets | 含括号的方程
Two strategies work well with brackets. If the bracket is a simple factor, you may divide both sides by that factor first. Otherwise expand the brackets carefully, multiplying every term inside by the number outside, then collect like terms.
处理括号有两种好策略。如果括号前只是一个简单的因数，可以先两边同时除以这个因数。否则要小心展开括号，用括号外的数乘以括号内的每一项，然后合并同类项。
x + 4. Take an extra moment to distribute the factor to every term, including the sign in front of each term.
最常见的错误是只乘括号内的第一项。例如 3(x + 4) = 3x + 12，而不是 3x + 4。要花一点时间把因数分配给每一项，包括每一项前面的符号。
x - 5) = 39: expand → 8x + 4 - 3x + 15 = 39; collect → 5x + 19 = 39; so 5x = 20 and x = 4.
解 4(2x + 1) - 3(x - 5) = 39：展开 → 8x + 4 - 3x + 15 = 39；合并 → 5x + 19 = 39；所以 5x = 20，x = 4。
Notice how the minus sign before 3(x - 5) turned -15 into +15. Brackets preceded by a minus sign flip every sign inside, so write out the intermediate line rather than rushing straight to the answer.
注意 3(x - 5) 前面的负号使 -15 变成 +15。括号前是负号时，括号内每一项的符号都要改变，所以要老老实实写出中间那一步，不要急于跳到答案。
7. Equations Containing Fractions | 含分数的方程
Fractions look intimidating but disappear quickly. Multiply every term on both sides by the denominators, or by the lowest common multiple of the denominators. This clears all fractions in one step and leaves a familiar linear equation.
分数看起来吓人，其实很快就会消失。把两边每一项都乘以分母，或者乘以所有分母的最小公倍数。这一步可以一次性清掉所有分数，留下一个熟悉的一次方程。
2(x - 2) = 24; expand → 3x + 3 + 2x - 4 = 24; collect → 5x - 1 = 24; so 5x = 25 and x = 5.
解 (x + 1) ÷ 2 + (x - 2) ÷ 3 = 4：两边乘以 6 → 3(x + 1) + 2(x - 2) = 24；展开 → 3x + 3 + 2x - 4 = 24；合并 → 5x - 1 = 24；所以 5x = 25，x = 5。
Two details decide whether this method works. First, multiply every term, including the standalone number on the right-hand side. Second, when the numerator contains more than one term, keep the brackets until after you multiply - that is why 3(x + 1) appears in the line above.
有两个细节决定这个方法是否奏效。第一，要乘每一项，包括右边单独的数字。第二，当分子不止一项时，乘完之前要保留括号——这正是上面出现 3(x + 1) 的原因。
y. There is no need to look for a common denominator when only one denominator is present.
如果只有一个分数，例如 x ÷ 4 = 3，两边直接乘以 4 就得到 x = 12。当只有一个分母时，不必去找公分母。
8. Negative Numbers and Signs | 负数与符号
Negative numbers cause more lost marks than any other topic in this chapter. Treat a minus sign as "multiply by -1" and apply it to every term it touches. Writing the working line by line is the simplest defence against sign errors.
在本章中，负数造成的丢分比其他任何知识点都多。把负号看作"乘以 -1"，并把它作用到所涉及的每一项。把过程一行行写清楚，是防止符号错误最简单的方法。
- -3x = 12 → divide both sides by -3 → x = -4
(-3x = 12 → 两边除以 -3 → x = -4) - 5 - x = 9 → subtract 5 → -x = 4 → multiply by -1 → x = -4
(5 - x = 9 → 两边减 5 → -x = 4 → 乘以 -1 → x = -4) - x ÷ (-2) = 6 → multiply by -2 → x = -12
(x ÷ (-2) = 6 → 两边乘以 -2 → x = -12) - 3(x - 2) = -15 → divide by 3 → x - 2 = -5 → x = -3
(3(x - 2) = -15 → 除以 3 → x - 2 = -5 → x = -3)
en 10 - x = 4, giving x = 6. Substituting back: 7 - (6 - 3) = 7 - 3 = 4, which is correct.
要小心像 7 - (x - 3) 这样的式子。这里括号被减去，所以 7 - (x - 3) = 7 - x + 3 = 10 - x。如果方程是 7 - (x - 3) = 4，则 10 - x = 4，得 x = 6。代回检验：7 - (6 - 3) = 7 - 3 = 4，正确。
9. Forming Equations from Word Problems | 根据文字题列方程
Word problems reward a calm, step-by-step approach. Read the question twice, choose a letter to stand for the unknown, then translate each sentence into symbols. The equation usually appears once you have written down what each quantity is.
文字题需要冷静、分步处理。先把题目读两遍，用一个字母表示未知量，再把每句话翻译成符号。当你把每个量表示清楚之后，方程通常就会自然出现。
| Words | 文字 | Symbols | 符号 |
| a number | 一个数 | x |
| 5 more than a number | 比一个数多 5 | x + 5 |
| 3 less than a number | 比一个数少 3 | x - 3 |
| twice a number | 一个数的两倍 | 2x |
| a number divided by 4 | 一个数除以 4 | x ÷ 4 |
| 3 times a number, then 4 subtracted | 一个数乘 3 再减 4 | 3x - 4 |
Example: "A number is multiplied by 3 and then 4 is subtracted; the result is 17." riting x = 7 - especially if the question asks for a length, an age or an amount of money.
例题："一个数乘以 3 再减去 4，结果是 17。"设这个数为 x，则 3x - 4 = 17，所以 3x = 21，x = 7。最后一定要用文字回答问题——不要只写 x = 7——尤其是题目问的是长度、年龄或金额时。
the perimeter is 2(w + w + 3) = 34. This gives 4w + 6 = 34, so 4w = 28 and w = 7. The width is 7 cm and the length is 10 cm.
几何题也很常见。一个长方形的长比宽多 3 cm，周长是 34 cm。设宽为 w，则长为 w + 3，周长为 2(w + w + 3) = 34，即 4w + 6 = 34，所以 4w = 28，w = 7。宽为 7 cm，长为 10 cm。
10. Special Cases: No Solution or Infinitely Many Solutions | 特殊情况：无解或无穷多解
statement is false, such as 0 = 4, the equation has no solution. If the remaining statement is always true, such as 0 = 0, every value of x works.
并非每个一次方程都有唯一解。有时 x 项会完全消掉，剩下的式子会告诉你答案。如果剩下的是假命题，比如 0 = 4，则方程无解；如果剩下的是恒真命题，比如 0 = 0，则任何 x 值都成立。
- 2x + 3 = 2x + 7 → subtract 2x → 3 = 7, which is false → no solution
(2x + 3 = 2x + 7 → 两边减 2x → 3 = 7，不成立 → 无解) - + 1) = 2x + 2 → expand → 2x + 2 = 2x + 2 → 2 = 2, always true → infinitely many solutions
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