← 返回资讯屏轩教育 · 行业资讯

Cambridge KS3 Maths: Solving Linear Equations | 剑桥KS3数学：解一元一次方程

2026-08-21 11:17 · 本文部分内容由AI辅助创作

📚 Cambridge KS3 Maths: Solving Linear Equations | 剑桥KS3数学：解一元一次方程

h to Stage 9. Almost every later topic - substitution, rearranging formulae, simultaneous equations, straight-line graphs and sequences - depends on your ability to keep an equation balanced and isolate the unknown accurately. This revision guide accompanies the practice worksheet p42_2.pdf and works through the balance method, one-step and two-step equations, brackets, fractions, unknowns on both sides, word problems and checking your answers.

解一元一次方程是剑桥初中数学课程（第 7 至第 9 学段）中最重要的代数技能。后续几乎每一个主题——代入求值、公式变形、联立方程、直线图像与数列——都取决于你能否保持等式平衡并准确分离未知数。本复习指南配套练习 p42_2.pdf，系统梳理天平法、一步与两步方程、含括号与含分数的方程、未知数在两边的方程、应用题以及检验答案的方法。

1. What Is a Linear Equation? | 什么是一元一次方程？

A linear equation is an equation in which the unknown appears only to the first power. There are no squares, no cubes, no roots, and no unknown variables sitting in denominators. The unknown is usually written as x, but y, n, a or any other letter is equally acceptable.

线性方程是未知数只出现一次方的方程。它没有平方、没有立方、没有根号，也没有未知数出现在分母中。未知数通常写作 x，但写成 y、n、a 或任何其他字母都可以。

Most KS3 linear equations can be written in one of these standard forms, where a, b, c and d stand for numbers:

大多数 KS3 线性方程都可以写成下列标准形式之一，其中 a、b、c、d 代表数字：

ax + b = c   or   ax + b = cx + d

They are called 'linear' es a curve. That link between the algebra and the graph becomes very important in later stages.

它们被称为“线性”是因为它们的图像是直线。如果你画出 y = 2x + 1 的图像，会得到一条完全笔直的直线；而 y = x² 给出的是曲线。代数与图像之间的这种联系在后续学段会变得非常重要。

x + 5 = 12, 2x − 3 = 9 and 4(x − 1) = 12. Non-examples include x² + 1 = 10 and 6 ÷ x = 3.

线性方程的例子包括 x + 5 = 12、2x − 3 = 9 和 4(x − 1) = 12。反例包括 x² + 1 = 10 和 6 ÷ x = 3。

2. The Balance Method | 天平法

The balance method is the key idea behind solving any equation. Picture an equation as a pair of scales in perfect balance: whatever sits on the left must weigh exactly the same as whatever sits on the right.

天平法是解任何方程的关键思想。把方程想象成一架处于完美平衡的天平：左边的东西必须与右边的东西重量完全相同。

This gives you four legal moves, and you may use them as often as you like:

由此得到四种“合法操作”，你可以随意反复使用：

As long as you perform exactly the same operation on the left-hand side (LHS) and the right-hand side (RHS), the equation stays balanced and the solution never changes. The aim is to isolate the unknown, leaving x alone on one side and a single number on the other.

只要你对左边（LHS）和右边（RHS）执行完全相同的运算，方程就会保持平衡，解也不会改变。目标是把未知数孤立出来，让 x 单独留在一边，另一边是一个数。

One useful mental check: if your scales tip, you have done something to one side that you did not do to the other. Rewind and repeat the step.

一个有用的心理检查：如果天平倾斜了，说明你对一边做了某件事却没有对另一边做同样的事。退回这一步重新来过。

3. Vocabulary and Notation | 关键术语与符号

Exams reward precise language, so learn these terms and use them in your written answers. Confusing them is one of the most common ways KS3 students lose easy marks.

考试会给精确的表述加分，所以要掌握这些术语并在书面答案中使用它们。混淆这些概念是 KS3 学生白白丢分的最常见原因之一。

Operation in the equation Inverse operation to undo it
+ 7 − 7
− 5 + 5
× 4 ÷ 4
÷ 3 × 3

n invisible coefficient of 1. Writing 3x rather than 3 × x is standard algebraic notation.

还要注意，3x 表示 3 × x，而单独的 x 其实有一个看不见的系数 1。写作 3x 而不是 3 × x 是标准的代数写法。

4. Solving One-Step Equations | 解一步方程

A one-step equation needs only one inverse operation to solve it. Undo addition with subtraction, subtraction with addition, multiplication with division, and division with multiplication. Always write one step per line so the marker can follow your reasoning.

一步方程只需一次逆运算即可求解。用减法抵消加法，用加法抵消减法，用除法抵消乘法，用乘法抵消除法。每一步单独写一行，阅卷老师才能跟上你的思路。

Example 1: Solve x + 7 = 15.

例 1：解方程 x + 7 = 15。

x + 7 − 7 = 15 − 7 → x = 8

Example 2: Solve x − 9 = 4.

例 2：解方程 x − 9 = 4。

x − 9 + 9 = 4 + 9 → x = 13

Example 3: Solve 5x = 35.

例 3：解方程 5x = 35。

5x ÷ 5 = 35 ÷ 5 → x = 7

Example 4: Solve x ÷ 6 = 4.

例 4：解方程 x ÷ 6 = 4。

x ÷ 6 × 6 = 4 × 6 → x = 24

Example 5: Solve −3x = 12. Divide both sides by −3, and be careful with the sign:

例 5：解方程 −3x = 12。两边同时除以 −3，注意符号：

−3x ÷ (−3) = 12 ÷ (−3) → x = −4

In every case, read the equation as a sentence: 'something plus seven equals fifteen', so the something must be eight. That verbal reading catches careless errors before they reach the page.

在每一种情况下，都把方程当作一句话来读：“某数加七等于十五”，那么这个数必定是八。这种口头阅读能在错误落到纸上之前就抓住它。

5. Solving Two-Step Equations | 解两步方程

Most KS3 equations need two inverse operations. The golden rule is to undo operations in the reverse order of BIDMAS: deal with addition or subtraction first, then multiplication or division. Think of putting on socks and shoes - you take off the shoes first.

大多数 KS3 方程需要两次逆运算。黄金法则是按 BIDMAS 的相反顺序来抵消运算：先处理加减，再处理乘除。想一想穿袜子和穿鞋——脱的时候要先脱鞋。

Example 1: Solve 3x + 4 = 19.

例 1：解方程 3x + 4 = 19。

3x + 4 − 4 = 19 − 4 → 3x = 15 → x = 5

Example 2: Solve 2x − 7 = 11.

例 2：解方程 2x − 7 = 11。

2x − 7 + 7 = 11 + 7 → 2x = 18 → x = 9

Example 3: Solve x ÷ 5 + 3 = 10.

例 3：解方程 x ÷ 5 + 3 = 10。

x ÷ 5 + 3 − 3 = 10 − 3 → x ÷ 5 = 7 → x = 35

Here the unknown term is subtracted, so it is often safest to add 2x to both sides first:

例 4：解方程 8 − 2x = 2。这里含未知数的项是被减去的，所以最稳妥的做法是先在两边加上 2x：

8 − 2x + 2x = 2 + 2x → 8 = 2 + 2x → 6 = 2x → x = 3

line always reads x = ... rather than 5 = x, even though both are mathematically correct.

一个可靠的习惯是最后把含未知数的项写在左边，让最后一行总是写成 x = …… 而不是 5 = x，虽然两者在数学上都正确。

6. Equations with Brackets | 含括号的方程

When a bracket is multiplied by a number, you have two valid routes: expand the bracket first, or divide both sides by the multiplier first. Choose whichever produces smaller numbers.

当括号被一个数相乘时，你有两条有效路径：先展开括号，或先在两边除以这个乘数。选择能让数字更简单的那一条。

Example 1: Solve 4(x − 1) = 12. Dividing first is quicker here because 12 ÷ 4 is exact.

例 1：解方程 4(x − 1) = 12。这里先除更快，因为 12 ÷ 4 是整数。

4(x − 1) ÷ 4 = 12 ÷ 4 → x − 1 = 3 → x = 4

广告 / Sponsored

海外直订Higher Mathematics for AQA GCSE 高等数学AQA GCSE
海外直订Higher Mathematics for AQA GCSE 高等数学AQA GCSE
¥645.0
📱 去京东购买
京东精选

Example 2: Solve 3(2x + 5) = 27.

例 2：解方程 3(2x + 5) = 27。

3(2x + 5) ÷ 3 = 27 ÷ 3 → 2x + 5 = 9 → 2x = 4 → x = 2

= 3(x + 4). Here the brackets must be expanded, and every term inside must be multiplied:

例 3：解方程 5(x − 2) = 3(x + 4)。这里必须展开括号，且括号内每一项都要乘到：

5x − 10 = 3x + 12 → 2x − 10 = 12 → 2x = 22 → x = 11

rgetting to multiply the second term. Write the expansion on its own line before you start solving, and check it term by term.

经典错误是写成 5(x − 2) = 5x − 2，忘记乘第二项。在开始求解之前，把展开式单独写一行，逐项检查。

7. Unknowns on Both Sides | 未知数在两边的方程

When the unknown appears on both sides, your first job is to collect all the x terms on one side and all the numbers on the other. It does not matter which side you choose, but moving the smaller x term usually keeps the coefficient positive.

当未知数出现在两边时，你的首要任务是把所有含 x 的项集中到一边，把所有数字集中到另一边。选择哪一边都可以，但通常移动较小的 x 项能让系数保持为正。

Example 1: Solve 7x − 3 = 4x + 9.

例 1：解方程 7x − 3 = 4x + 9。

7x − 3 − 4x = 4x + 9 − 4x → 3x − 3 = 9 → 3x = 12 → x = 4

Example 2: Solve 5x + 2 = 2x − 10.

例 2：解方程 5x + 2 = 2x − 10。

3x + 2 = −10 → 3x = −12 → x = −4

9 − 2x = 5x − 5. Adding 2x to both sides avoids a negative coefficient on the answer side:

例 3：解方程 9 − 2x = 5x − 5。在两边加上 2x 可以避免答案一边出现负系数：

9 = 7x − 5 → 14 = 7x → x = 2

4 is not a mistake; it simply means the line crosses the x-axis to the left of the origin.

在 KS3 中，负解是完全正常的。答案是 x = −4 并不代表出错，它只说明这条直线在原点左侧穿过 x 轴。

8. Equations with Fractions | 含分数的方程

Fractional equations look harder than they are. The trick is to clear the fractions immediately by multiplying every term by the lowest common multiple of the denominators. Once the fractions are gone, you are back to a familiar two-step problem.

含分数的方程看起来比实际难。诀窍是立刻消去分数：把每一项都乘以各分母的最小公倍数。分数消失之后，你就回到了熟悉的两步方程。

Example 1: Solve (x + 3) ÷ 4 = 5.

例 1：解方程 (x + 3) ÷ 4 = 5。

4 × (x + 3) ÷ 4 = 4 × 5 → x + 3 = 20 → x = 17

olve x ÷ 3 + x ÷ 4 = 7. The LCM of 3 and 4 is 12, and every term must be multiplied by it.

例 2：解方程 x ÷ 3 + x ÷ 4 = 7。3 和 4 的最小公倍数是 12，每一项都必须乘 12。

4x + 3x = 84 → 7x = 84 → x = 12

+ 1) ÷ 2 = (x − 2) ÷ 3. This is a proportion, so cross-multiplying is the fastest method.

例 3：解方程 (x + 1) ÷ 2 = (x − 2) ÷ 3。这是一个比例式，交叉相乘是最快的方法。

3(x + 1) = 2(x − 2) → 3x + 3 = 2x − 4 → x = −7

e 4x + x ÷ 4 = 84, which is wrong. Draw a small tick above every term you have multiplied.

要警惕只把第一项乘以最小公倍数的诱惑。在例 2 中，若只把 x ÷ 3 乘以 12，就会得到 4x + x ÷ 4 = 84，这是错误的。在你乘过的每一项上方画一个小勾。

9. Forming Equations from Word Problems | 从应用题列方程

Cambridge KS3 papers frequently ask you to build the equation yourself. Follow a fixed four-step routine and the marks become straightforward: define the unknown, translate the words into algebra, solve, then answer the question that was actually asked.

剑桥 KS3 试题经常要求你自己列出方程。遵循固定的四步流程，得分就变得简单：设未知数、把文字翻译成代数、求解，最后回答题目真正问的问题。

Worked example: A rectangle is 3 cm longer than it is wide. Its perimeter is 26 cm. Find the length and the width.

例题：一个长方形的长比宽多 3 cm，周长是 26 cm。求它的长和宽。

Step 1 - Let the width be w cm, so the length is (w + 3) cm.

第一步——设宽为 w cm，则长为 (w + 3) cm。

Step 2 - Perimeter: 2w + 2(w + 3) = 26.

第二步——周长：2w + 2(w + 3) = 26。

Step 3 - Solve: 2w + 2w + 6 = 26 → 4w = 20 → w = 5.

第三步——求解：2w + 2w + 6 = 26 → 4w = 20 → w = 5。

Step 4 - Interpret: the width is 5 cm and the length is 8 cm. Always write the units and always return to the original question.

第四步——作答：宽是 5 cm，长是 8 cm。一定要写上单位，并且一定要回到原题的问题上。

Other phrases you should be able to translate instantly: 'sum of' means +, 'difference between' means −, 'product of' means ×, 'twice a number' means 2x, and 'I think of a number, multiply it by 4 and subtract 5' means 4x − 5.

其他应当能立刻翻译的表述：“……的和”表示 +，“……与……之差”表示 −，“……的积”表示 ×，“某数的两倍”表示 2x，“我想一个数，把它乘以 4 再减去 5”表示 4x − 5。

10. Checking Your Solution | 检验你的解

Every equation you solve can be checked by substitution, and this takes about ten seconds. Substitute your value back into each side separately, work out both numbers, and confirm they match.

你解出的每一个方程都可以用代入法检验，这大约只需要十秒钟。把你的答案分别代入左右两边，算出两个数值，确认它们相等。

Check for 7x − 3 = 4x + 9 with x = 4:

对 7x − 3 = 4x + 9，用 x = 4 检验：

LHS = 7(4) − 3 = 28 − 3 = 25    RHS = 4(4) + 9 = 16 + 9 = 25 ✓

Check for x ÷ 3 + x ÷ 4 = 7 with x = 12:

对 x ÷ 3 + x ÷ 4 = 7，用 x = 12 检验：

12 ÷ 3 + 12 ÷ 4 = 4 + 3 = 7 ✓

4 = 19, something has gone badly wrong, so estimate the answer mentally before you begin.

在不可使用计算器的试卷中，快速的情理判断也很有价值。如果像 3x + 4 = 19 这样的方程解出 x = 400，那一定出了大问题，所以在动笔前先在心里估一估答案。

Marking schemes often award a method mark for a correct checking statement even when the original answer was wrong, so never leave this step out.

评分标准常常会给正确的检验步骤一个方法分，即使原答案错了也一样，所以千万不要省略这一步。

11. Common Mistakes and Exam Tips | 常见错误与考试技巧

The same handful of errors appears again and again in KS3 scripts. Learn to recognise them and you can protect several marks per paper.

KS3 试卷中反复出现的就是那么几个错误。学会识别它们，你就能每份试卷保住好几分。