📚 Solving Linear Equations | 解一元一次方程
Linear equations are among the most important tools in KS3 mathematics. They help us find unknown values in problems ranging from simple number puzzles to real-world situations such as sharing costs, calculating speeds, or balancing a budget. This article explains how to solve linear equations step by step, using the balance method, inverse operations, and careful checking. By the end, you will be able to solve one-step, two-step, bracket, and fraction equations confidently.
一元一次方程是 KS3 数学中最重要的工具之一。它们帮助我们求出未知数，问题可以是简单的数字谜题，也可以是分摊费用、计算速度或平衡预算等现实情境。本文逐步讲解如何解一元一次方程，包括天平法、逆运算和认真检验。学完后，你将能够自信地解一步方程、两步方程、含括号方程和含分数方程。
1. What Is a Linear Equation? | 什么是一元一次方程？
A linear equation is a mathematical statement that shows two expressions are equal. In KS3, most linear equations involve one unknown, usually written as x or n. The highest power of the unknown is 1, so you will see terms like x, 2x, or -3x, but never x² or x³.
一元一次方程是表示两个表达式相等的数学语句。在 KS3 中，大多数一元一次方程含有一个未知数，通常写作 x 或 n。未知数的最高次数是 1，因此你会看到 x、2x 或 -3x 这样的项，但不会出现 x² 或 x³。
Linear equations appear throughout KS3. They are called linear because if you draw their graphs, you get a straight line. Here, however, we focus on solving them algebraically.
一元一次方程贯穿 KS3 课程。之所以称为线性，是因为如果画出它们的图像，会得到一条直线。不过这里我们重点学习如何用代数方法求解。
2. The Balance Method | 天平法
Think of an equation as a balance scale. Whatever you do to one side, you must do to the other side to keep it balanced. If you add 4 to the left, you must add 4 to the right. This is the golden rule of equation solving.
把方程想象成一个天平。无论你对一边做什么，都必须对另一边做同样的操作，才能保持平衡。如果你在左边加上 4，就必须在右边也加上 4。这是解方程的黄金法则。
The balance method is a powerful way to visualise equation solving. If a balance is level, removing the same weight from both pans keeps it level. In algebra, adding, subtracting, multiplying, or dividing both sides by the same non-zero number keeps the equation true.
天平法是可视化解方程的强大方法。如果天平是水平的，从两个盘中拿走相同的重量，天平仍然保持水平。在代数中，两边同时加上、减去、乘以或除以同一个非零数，方程仍然成立。
x - 5 = 9
x - 5 + 5 = 9 + 5
x = 14
3. Inverse Operations | 逆运算
To isolate the unknown, use inverse operations. Addition and subtraction are inverse operations, and multiplication and division are inverse operations. For a one-step equation, apply the inverse operation to both sides.
为了把未知数单独留在一边，需要使用逆运算。加法和减法互为逆运算，乘法和除法互为逆运算。对于一步方程，在两边同时进行逆运算。
om both sides. For x/5 = 4, multiply both sides by 5. For 6x = 30, divide both sides by 6.
对于 x + 7 = 15，两边同减 7。对于 x/5 = 4，两边同乘 5。对于 6x = 30，两边同除 6。
x + 7 - 7 = 15 - 7 → x = 8
x/5 × 5 = 4 × 5 → x = 20
4. Solving Two-Step Equations | 解两步方程
A two-step equation requires two inverse operations. Always undo addition or subtraction first, then undo multiplication or division. This order follows the reverse of the order of operations.
两步方程需要两次逆运算。总是先消去加法或减法，再消去乘法或除法。这个顺序与运算顺序相反。
, in 5x - 3 = 12, first add 3 to both sides to get 5x = 15, then divide by 5 to get x = 3.
记住顺序：先处理加减，再处理乘除。例如，在 5x - 3 = 12 中，先给两边加 3 得到 5x = 15，然后两边除以 5 得到 x = 3。
3x + 4 = 19
3x = 19 - 4
3x = 15 → x = 5
5. Equations with Brackets | 含括号的方程
When an equation contains brackets, expand them first using the distributive law. Then solve as usual. For example, 2(x + 3) becomes 2x + 6.
当方程含有括号时，先运用分配律展开括号，然后按常规方法求解。例如，2(x + 3) 展开为 2x + 6。
uch as -2(x - 3). Expand carefully: -2 × x = -2x and -2 × -3 = +6, so -2(x - 3) = -2x + 6.
有时括号外是负号，例如 -2(x - 3)。展开要仔细：-2 × x = -2x，-2 × -3 = +6，所以 -2(x - 3) = -2x + 6。
2(x + 3) = 14
2x + 6 = 14
2x = 8 → x = 4
6. Equations with Variables on Both Sides | 两边含变量的方程
If variables appear on both sides, collect them on one side first. You can add or subtract the same variable term from both sides. Then solve the resulting two-step equation.
如果方程两边都含有变量，先把变量合并到一边。你可以在两边同时加上或减去相同的变量项，然后解得到的两步方程。
xample, in 8x - 3 = 3x + 12, subtract 3x from both sides to keep the coefficient positive.
合并变量项时，通常把较小的变量项移到较大的一边。例如，在 8x - 3 = 3x + 12 中，两边同减 3x，使系数保持为正。
5x - 2 = 2x + 7
5x - 2x = 7 + 2
3x = 9 → x = 3
7. Equations with Fractions | 含分数的方程
To clear fractions, multiply every term by the lowest common denominator. This changes the equation into an equivalent one without fractions. Then solve using the methods above.
为了去分母，将每一项都乘以最小公分母。这样可以把方程化为不含分母的等价方程，然后用上面的方法求解。
of 2 and 3 is 6, so multiply every term by 6 to get 3x + 2x = 30, then 5x = 30 and x = 6.
如果方程含有多个分数，乘以所有分母的最小公倍数。对于 x/2 + x/3 = 5，2 和 3 的最小公倍数是 6，所以每一项乘 6 得到 3x + 2x = 30，然后 5x = 30，x = 6。
x/4 + 1 = 3
x + 4 = 12
x = 8
8. Checking Your Solution | 检验解
Always substitute your answer back into the original equation. If the left side equals the right side, your solution is correct. Checking helps you catch arithmetic mistakes.
始终把你的答案代回原方程。如果左边等于右边，你的解就是正确的。检验可以帮助你发现计算错误。
eck: left side = 2(7) - 4 = 14 - 4 = 10, which equals the right side. So x = 7 is correct.
对于 2x - 4 = 10，解得 x = 7。检验：左边 = 2(7) - 4 = 14 - 4 = 10，与右边相等。所以 x = 7 正确。
9. Word Problems Leading to Linear Equations | 列方程解应用题
Many real-life problems can be solved by forming a linear equation. Identify the unknown, write an equation from the words, solve it, and then check the answer in the context of the problem.
许多现实问题都可以通过列一元一次方程来解决。找出未知数，根据题意列出方程，解方程，然后在题目情境中检验答案。
ngth = 2w + 3; perimeter: 2(w + 2w + 3) = 26 → 6w + 6 = 26 → 6w = 20 → w = 10/3 ≈ 3.33 cm.
例子：一个矩形的长比宽的两倍多 3 cm。如果周长是 26 cm，求宽。设 w 为宽；长 = 2w + 3；周长：2(w + 2w + 3) = 26 → 6w + 6 = 26 → 6w = 20 → w = 10/3 ≈ 3.33 cm。
10. Common Mistakes | 常见错误
Common mistakes include forgetting to apply an operation to both sides, mishandling negative signs, and using the wrong inverse operation. Work slowly and line up each step.
常见错误包括忘记在方程两边同时进行运算、处理负号出错、使用了错误的逆运算。慢慢做，每一步都要对齐。
Another common mistake is not distributing a negative correctly when removing brackets. Write each step clearly, especially when working with negative coefficients.
另一个常见错误是在去括号时没有正确分配负号。每一步都要写清楚，特别是在处理负系数时。
11. Practice Questions | 练习题
(a) x + 9 = 16, (b) 4x - 1 = 11, (c) 3(x - 2) = 12, (d) 7x + 5 = 3x + 21, (e) x/3 - 2 = 4.
尝试以下练习。解每个方程并检验答案：(a) x + 9 = 16，(b) 4x - 1 = 11，(c) 3(x - 2) = 12，(d) 7x + 5 = 3x + 21，(e) x/3 - 2 = 4。
4, (e) x = 18. If you got a different answer, re-check the inverse operation and the sign.
答案：(a) x = 7，(b) x = 3，(c) x = 6，(d) x = 4，(e) x = 18。如果你的答案不同，请重新检查逆运算和符号。
12. Summary | 总结
To solve a linear equation, keep the equation balanced, use inverse operations, simplify brackets
Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com
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