Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations are one of the most fundamental topics in Key Stage 3 mathematics. They involve finding the unknown value, often represented by a letter such as x, that makes an equation true. Mastering how to solve these equations builds the foundation for algebra, problem-solving, and more advanced topics in secondary school. Whether you are working with one-step, two-step, or multi-step equations, the key is to perform the same operation on both sides to keep the equation balanced. This article covers everything you need to know about solving linear equations for the Cambridge Lower Secondary Checkpoint, with clear examples, common pitfalls, and real-life applications.

一元一次方程是KS3数学中最基础的主题之一。它们要求我们找出使等式成立的未知数，通常用字母如 x 表示。掌握如何解这些方程能为代数、问题解决以及中学更高阶的内容打下坚实基础。无论你面对的是简单的一步方程、两步方程还是更复杂的多步方程，核心原则都是对方程两边同时进行相同的运算以维持平衡。本文涵盖了剑桥初中Checkpoint考试中关于解一元一次方程的所有要点，配合清晰的示例、常见错误分析和实际应用。

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1. Introduction to Linear Equations | 一元一次方程简介

A linear equation is an equation where the highest power of the variable is 1. For example, 3x + 4 = 10 is linear, but x² + 2 = 9 is not. In KS3, you will mainly work with equations that have one variable and can be solved by isolating it on one side of the equals sign. Equations are like a balance scale: whatever you do to one side, you must do to the other to keep it equal.

一元一次方程是指变量的最高次数为1的方程。例如，3x + 4 = 10 是一元一次方程，而 x² + 2 = 9 不是。在KS3阶段，你主要处理的是含有一个变量、且能通过将变量孤立在等号一边来求解的方程。方程就像一架天平：无论你对一边做什么运算，也必须对另一边做同样的运算，以保持平衡。

Common forms you will meet include ax + b = c, where a, b, and c are numbers. Later, you’ll see equations with brackets, fractions, and variables on both sides. The solution to an equation is the value of the variable that makes the statement true. For 2x = 10, the solution is x = 5 because 2 × 5 = 10.

你将遇到的常见形式包括 ax + b = c，其中 a、b 和 c 是数字。之后还会见到带括号、含分数以及两边都有变量的方程。方程的解就是使等式成立的变量值。对于 2x = 10，解为 x = 5，因为 2 × 5 = 10。

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2. Balancing the Equation | 等式平衡原理

The golden rule for solving equations is “do the same to both sides”. If you add, subtract, multiply, or divide one side by a number, you must apply the exact same operation to the other side. This preserves the equality. Imagine a balanced scale: adding 3 kg to the left pan means you must add 3 kg to the right pan to keep it level.

解方程的金科玉律是“两边同时进行相同的运算”。如果你对一边进行加、减、乘或除一个数，就必须对另一边做完全相同的运算。这样才能保持等式成立。想象一架平衡的天平：往左边托盘加3千克，就必须也往右边托盘加3千克，以保持水平。

For example, to solve x + 7 = 12, you subtract 7 from both sides: x + 7 − 7 = 12 − 7, giving x = 5. Always write the operation you are performing next to each side to avoid mistakes. Developing this habit early will help you with longer equations later.

例如，解 x + 7 = 12 时，两边都减去7：x + 7 − 7 = 12 − 7，得到 x = 5。一定要在每一边旁边写出你正在进行的运算，以避免错误。尽早培养这个习惯，将有助于你日后处理更长的方程。

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3. One-Step Equations | 一步方程

One-step equations require only one operation to isolate the variable. They can involve addition, subtraction, multiplication, or division. For addition equations like x + 5 = 9, subtract 5 from both sides. For subtraction equations like y − 3 = 2, add 3 to both sides.

一步方程只需要一次运算就能把变量孤立出来。它们可能涉及加法、减法、乘法或除法。对于加法方程如 x + 5 = 9，两边同时减去5。对于减法方程如 y − 3 = 2，两边同时加上3。

Multiplication equations such as 4p = 20 are solved by dividing both sides by 4: p = 5. Division equations like m ÷ 2 = 7 (written as m/2 = 7) are solved by multiplying both sides by 2: m = 14. Always check your solution by substituting it back into the original equation.

乘法方程如 4p = 20，通过两边除以4来解：p = 5。除法方程如 m ÷ 2 = 7（写作 m/2 = 7）通过两边乘以2来解：m = 14。务必通过将解代回原方程进行检验。

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4. Two-Step Equations | 两步方程

Two-step equations involve two operations. A classic example is 2x + 3 = 11. To solve it, first undo the constant term using addition or subtraction, then undo the coefficient (the multiplier) using division or multiplication. The order is the reverse of BODMAS: you deal with what is furthest from x first.

两步方程包含两种运算。一个典型的例子是 2x + 3 = 11。解这类方程时，先通过加减消去常数项，再通过乘除消去系数（乘数）。运算顺序与BODMAS相反：先处理离 x 最远的部分。

Start with 2x + 3 = 11. Subtract 3 from both sides: 2x = 8. Then divide both sides by 2: x = 4. For a subtraction example like 5y − 7 = 18, add 7 first: 5y = 25, then divide by 5: y = 5. Writing each step clearly prevents errors.

以 2x + 3 = 11 为例。两边先减去3：2x = 8。然后两边除以2：x = 4。对于带减法的例子如 5y − 7 = 18，先加7：5y = 25，再除以5：y = 5。清晰写出每一步能防止出错。

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5. Equations with Brackets | 带括号的方程

When an equation contains brackets, your first job is to expand them using the distributive law. For example, 3(x + 2) = 15 expands to 3x + 6 = 15. Then solve the resulting two-step equation: subtract 6 to get 3x = 9, and divide by 3 to find x = 3.

当方程含有括号时，第一个步骤是用分配律展开括号。例如，3(x + 2) = 15 展开为 3x + 6 = 15。然后解得到的两步方程：减去6得 3x = 9，再除以3得 x = 3。

Be careful with negative numbers outside brackets. For −2(x − 4) = 10, expand to −2x + 8 = 10 (because −2 × −4 = +8). Then subtract 8 from both sides to get −2x = 2, and divide by −2 to get x = −1. Always double-check sign rules when expanding.

要特别注意括号外有负数的情况。例如 −2(x − 4) = 10，展开得 −2x + 8 = 10（因为 −2 × −4 = +8）。然后两边减去8得 −2x = 2，再除以 −2 得 x = −1。展开时一定要再三检查符号规则。

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6. Equations with Variables on Both Sides | 两边含变量的方程

Sometimes the variable appears on both sides of the equation, such as 5x + 2 = 3x + 10. The goal is to collect all variable terms on one side and all constant terms on the other. Subtract the smaller variable term from both sides to keep coefficients positive.

有时变量出现在方程的两边，例如 5x + 2 = 3x + 10。目标是让所有含变量的项集中到一边，所有常数项集中到另一边。从两边同时减去较小的变量项，以保持系数为正。

For 5x + 2 = 3x + 10, subtract 3x from both sides: 2x + 2 = 10. Then subtract 2: 2x = 8, so x = 4. You can also move the constant first; the order doesn’t matter as long as you keep equality. Practise rearranging so that the variable ends up on the left, but it’s equally valid on the right.

对于 5x + 2 = 3x + 10，两边同时减去 3x：2x + 2 = 10。然后再减2：2x = 8，所以 x = 4。你也可以先移常数项；只要保持等式平衡，顺序无关紧要。通过练习，习惯于把变量移到左边，但留在右边同样有效。

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7. Equations Involving Fractions | 含分数的方程

When an equation contains fractions, a powerful method is to clear the denominators by multiplying every term by the lowest common multiple (LCM). For example, in x/3 + 2 = 5, subtract 2 first to get x/3 = 3, then multiply both sides by 3: x = 9. Alternatively, multiply everything by 3 at the start: x + 6 = 15, then x = 9.

当方程中含有分数时，一个强大的方法是每一项同时乘以最小公倍数（LCM）以清除分母。例如，在 x/3 + 2 = 5 中，先减去2得 x/3 = 3，然后两边乘以3：x = 9。或者，一开始就把所有项乘以3：x + 6 = 15，得出 x = 9。

For equations like (2x)/5 = 4, multiply both sides by 5: 2x = 20, then x = 10. When there are multiple denominators, find the LCM. For x/2 + x/3 = 5, multiply every term by 6: 3x + 2x = 30 → 5x = 30 → x = 6. This technique simplifies messy fractions quickly.

对于像 (2x)/5 = 4 这样的方程，两边乘以5：2x = 20，所以 x = 10。当有多个分母时，找出最小公倍数。例如 x/2 + x/3 = 5，每一项乘以6：3x + 2x = 30 → 5x = 30 → x = 6。这个技巧能迅速简化繁杂的分数。

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8. Checking Solutions by Substitution | 代入检验解

After finding a solution, always verify it by substituting the value back into the original equation. This simple habit can catch arithmetic mistakes and build confidence. If both sides give the same number, your solution is correct. Otherwise, re-check your working.

求出解之后，一定要将值代回原方程进行验证。这个简单的习惯能发现计算错误并建立信心。如果两边得出相同的数值，你的解就是正确的。否则就要重新检查你的解答过程。

For example, after solving 4(y − 2) = 16 and getting y = 6, check: left side 4(6 − 2) = 4 × 4 = 16, right side 16. They match, so y = 6 is correct. In exams, this check can be shown as a final line to gain marks for verification.

例如，解 4(y − 2) = 16 得到 y = 6 后，检验：左边 4(6 − 2) = 4 × 4 = 16，右边为16。两者相等，因此 y = 6 正确。在考试中，可以将这个验证过程作为最后一步展示出来，以获得检验分。

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9. Common Mistakes to Avoid | 常见错误解析

Many students lose marks because of simple but avoidable errors. One common mistake is forgetting to apply an operation to both sides. For instance, when solving 3x = 9, dividing only the left side by 3 gives x = 9, which is wrong. Always write the operation on both sides explicitly.

许多学生因简单但可避免的错误而失分。一个常见错误是忘记对两边同时做运算。例如，解 3x = 9 时，只将左边除以3，得到 x = 9，这是错误的。一定要清晰地在两边写出所做的运算。

Another mistake is mishandling negative signs, especially when expanding brackets. For −2(x − 3), students sometimes write −2x − 6 instead of −2x + 6. Also, when moving terms across the equals sign, be mindful of sign changes. Using systematic steps prevents these slip-ups.

另一个错误是负号处理不当，特别是在展开括号时。对于 −2(x − 3)，学生有时会错写成 −2x − 6，而正确答案是 −2x + 6。此外，当把项移到等号另一边时，要注意符号变化。使用系统化的解题步骤可以避免这些疏忽。

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10. Real-World Applications | 实际应用

Linear equations are not just abstract exercises; they model countless real-life situations. For example, if a taxi charges a flat fee of £3 plus £2 per mile, the cost C for m miles can be written as C = 2m + 3. If you know the total cost is £15, you solve 2m + 3 = 15 to find m = 6 miles.

一元一次方程不仅仅是抽象的练习，它们可用于模拟无数现实情形。例如，如果出租车起步价为3英镑，每英里2英镑，那么行驶m英里的费用C可表示为 C = 2m + 3。如果已知总费用为15英镑，你可以解方程 2m + 3 = 15，得出 m = 6 英里。

Similarly, when sharing costs among friends, working out how long it takes to fill a tank, or balancing a budget, you set up an equation and solve for the unknown. Recognizing that the variable represents a real quantity makes algebra meaningful and memorable.

同样，在与朋友分摊费用、计算注满水箱所需时间或平衡预算时，你可以建立方程并求解未知量。意识到变量代表了一个真实的物理量，会让代数变得既有意义又容易记住。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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