📚 Forming and Solving Linear Equations from Word Problems | 图文示意与列方程解应用题
Many IGCSE and A-Level preparatory word problems ask you to find an unknown number from a real-life situation: ages, money, lengths, shares or consecutive numbers. Drawing a simple picture or bar model and then translating that situation into a linear equation is one of the most reliable problem-solving strategies in school mathematics, because it turns vague sentences into a clear structure you can actually work with.
许多 IGCSE 及 A-Level 预备阶段的应用题要求你从实际情境中求未知数，例如年龄、钱款、长度、分配比例或连续整数。先画出示意图或条形模型，再把情境翻译成一元一次方程，是中学数学中最可靠的解题策略之一，因为它把含糊的文字变成清晰、可操作的结构。
1. Why Draw a Picture First? | 为什么先画图
A picture turns words into a visual relationship. If you can see the parts and the total at a glance, you can write an equation far more easily than by staring at the text again and again.
图形把文字变成直观关系。若你能一眼看出部分与总量，列方程就比反复死盯文字容易得多。
Drawing also makes hidden information visible. It shows which quantities are equal, which are parts of a whole, and which are being compared with each other, so you can choose the correct operation with confidence instead of guessing.
画图还能让隐藏信息显现出来。它能显示哪些量相等、哪些是整体的组成部分、哪些量在互相比较，从而让你有把握地选择正确的运算，而不是靠猜。
Finally, a sketch reduces careless reading errors. When the diagram sits on the page in front of you, it is much harder to forget a condition such as "in five years' time", "twice as many" or "5 less than the other number".
最后，示意图能减少粗心读题造成的错误。当图形就摆在眼前时，你很难忘记"五年后""是……的两倍""比另一个数少 5"这类条件。
2. From Picture to Equation | 从示意图到方程
A bar model uses rectangles to represent known and unknown quantities. The length of each bar stands for the size of that quantity, so equal lengths mean equal amounts and a longer bar means a larger amount.
条形模型用矩形表示已知量与未知量。每条矩形的长度代表该数量的大小，因此长度相等就意味着数量相等，矩形更长就意味着数量更大。
If one part of the bar is x and the other part is 7, then the whole bar has length x + 7. If the whole is known to be 15, the picture hands you the equation directly.
若条形的一部分是 x，另一部分是 7，那么整条的长度就是 x + 7。若已知整体为 15，图形就"直接给出"方程。
x + 7 = 15
The balance idea is the same whether the numbers are whole numbers, decimals or fractions.
一旦方程与图形一致，就按"保持两侧平衡"来解：两边同时减 7，得到 x = 8。无论数字是整数、小数还是分数，平衡的思想都完全一样。
The picture is not part of the final answer, but it is the bridge between the words and the algebra. Examiners award marks for the correct equation, so this bridge is worth building every time.
图形不是最终答案的一部分，但它是文字与代数之间的桥梁。评分时方程列对就有分，所以每次解题都值得搭好这座桥。
3. Key Word Clues in Word Problems | 应用题中的关键词线索
Certain words signal addition, subtraction, multiplication or division. Use them to decide how to build the equation, but always read the whole sentence before you commit to an operation.
某些关键词表示加、减、乘、除。利用它们来决定如何建立方程，但在确定运算之前一定要把整句话读完。
| Word / Phrase | 中文提示 | Operation | 运算 |
|---|---|---|---|
| sum, total, altogether, in all | 总和、共计、一共 | + | 加法 |
| more than, increased by, exceeds | 比……多、增加、超过 | + | 加法 |
| difference, less than, decreased by | 差、比……少、减少 | − | 减法 |
| product, times, twice, double, of | 乘积、倍、两倍、"的" | × | 乘法 |
| quotient, divided by, per, shared equally | 商、除以、每、平均分 | ÷ | 除法 |
| is, was, will be, gives, results in | 是、将要是、等于、得到 | = | 等号 |
Watch the order in subtraction phrases. "5 less than a number" means x − 5, not 5 − x, and "the number decreased by 8" means x − 8. Getting the order wrong is one of the most common lost marks in this topic.
注意减法短语的语序。"比某数少 5"表示 x − 5，而不是 5 − x；"某数减少 8"表示 x − 8。把顺序弄反是这一考点最常见的失分原因之一。
4. Define the Unknown Clearly | 明确定义未知数
Always begin the algebra with a sentence that starts "Let x be ...". Write down the unit as well: "Let x be the number of pens" is much safer than "Let x be pens", because the unit tells you what the final number means.
列式前先用一句话开头："设 x 为……"。最好连单位一起写清楚："设 x 为铅笔的支数"比"设 x 为铅笔"安全得多，因为单位决定了最终答案的含义。
If two quantities are linked, express both in terms of the same letter. If a notebook costs £3 more than a pen and a pen costs £x, then a notebook costs £(x + 3), and a total of 4 pens and 2 notebooks costs 4x + 2(x + 3).
若两个量相互关联，就用同一个字母表示它们。若笔记本比钢笔贵 3 英镑，而钢笔为 x 英镑，则笔记本为 (x + 3) 英镑，4 支钢笔与 2 本笔记本的总价为 4x + 2(x + 3)。
Never use x for two different things in the same question, and never change the meaning of x halfway through the solution. If the question asks for two numbers, find x first and then calculate the second one from your definition.
同一道题中绝不能用 x 表示两个不同的量，也不能在解题途中改变 x 的含义。若题目要求两个数，先求出 x，再根据你的定义算出第二个数。
Write the final equation as a complete sentence in symbols, then read it back in words. "Three times the number plus five equals twenty-six" is a quick mental check that your equation really says what the question says.
把最终方程写成一句完整的符号句，再把它读回文字。"某数的三倍加五等于二十六"，这一句口头回读能快速检验方程是否真的与题意一致。
3x + 5 = 26
5. The Five-Step Method | 五步解题法
A fixed routine keeps you calm under exam pressure and stops you from skipping the most valuable marks. Use the same five steps for every linear word problem you meet.
固定的解题流程能让你在考场压力下保持镇定，也能避免漏掉最容易得分的步骤。遇到任何一元一次方程应用题，都按这五步走。
- Step 1 - Read twice: once for the story, once for the numbers and conditions. | 第一步 读两遍：第一遍读情境，第二遍读数字与条件。
- Step 2 - Define: write "Let x be ..." with a unit. | 第二步 设元：写出"设 x 为……"，并带上单位。
- Step 3 - Translate: sketch a bar model and write the equation. | 第三步 翻译：画条形模型并写出方程。
- Step 4 - Solve: use the balance method, showing every line. | 第四步 求解：用平衡法解方程，每一步都写出来。
- Step 5 - Check and answer: substitute back and write a sentence. | 第五步 检验作答：代回检验，并用一句话作答。
For example, "The sum of three times a number and 5 is 26. Find the number." 5 to get 3x = 21, then divide by 3 to get x = 7. Check: 3 × 7 + 5 = 26, which is correct.
例如："某数的三倍与 5 的和为 26，求这个数。"设这个数为 x，则 3x + 5 = 26。两边减 5 得 3x = 21，再除以 3 得 x = 7。检验：3 × 7 + 5 = 26，正确。
6. Age Problems | 年龄问题
In every age problem, the key fact is that everyone grows older by the same number of years. "In 12 years' time" means add 12 to every person's age; "5 years ago" means subtract 5 from every person's age.
所有年龄问题的关键事实是：每个人增加的年数相同。"12 年后"意味着每个人的年龄都加 12；"5 年前"意味着每个人的年龄都减 5。
Example: A father is three times as old as his son. In 12 years' time he will be twice as old as his son. Find the son's present age.
例题：父亲现在的年龄是儿子的 3 倍。12 年后，父亲的年龄将是儿子的 2 倍。求儿子现在的年龄。
Let the son's present age be x years, so the father's present age is 3x years. In 12 years the son will be x + 12 and the father will be 3x + 12.
设儿子现在 x 岁，则父亲现在 3x 岁。12 年后儿子为 x + 12 岁，父亲为 3x + 12 岁。
3x + 12 = 2(x + 12)
om both sides to get x + 12 = 24, so x = 12. The son is 12 years old and the father is 36.
展开括号：3x + 12 = 2x + 24。两边同减 2x 得 x + 12 = 24，所以 x = 12。儿子 12 岁，父亲 36 岁。
ill be 24 and 48, and 48 = 2 × 24. Notice that the bracket is essential - doubling the son's age means doubling the whole of x + 12, not just x.
按原题文字检验：12 年后两人分别为 24 岁和 48 岁，而 48 = 2 × 24。注意括号必不可少——"儿子的 2 倍"是对 (x + 12) 整体取 2 倍，而不是只对 x 取 2 倍。
7. Consecutive Numbers and Digit Problems | 连续整数与数字问题
Consecutive integers differ by 1, so three consecutive integers can be written as n, n + 1 and n + 2. Using one letter for all three is what makes the equation simple.
连续整数彼此相差 1，因此三个连续整数可写成 n、n + 1、n + 2。用同一个字母表示三个数，正是让方程变简单的关键。
ich simplifies to 3n + 3 = 51, so 3n = 48 and n = 16. The three numbers are 16, 17 and 18.
若它们的和是 51，则 n + (n + 1) + (n + 2) = 51，化简得 3n + 3 = 51，所以 3n = 48，n = 16。三个数为 16、17、18。
2 and n + 4. For example, n + (n + 2) + (n + 4) = 72 gives 3n + 6 = 72, so n = 22 and the
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