📚 Solving Linear Equations | 解一次方程
Linear equations are the foundation of algebra. In KS3 mathematics, you learn how to solve equations where the unknown value, often represented by a letter like x, appears only to the first power. Mastering these methods will help you tackle everything from simple puzzles to complex scientific problems later on. This article takes you through the steps, from basic principles to equations with fractions and brackets, all explained clearly for Cambridge learners.
一次方程是代数的基础。在KS3阶段的数学中，你要学习如何求解未知数（通常用字母如x表示）只出现一次方的方程。掌握这些方法将帮助你应对从简单谜题到未来复杂的科学问题。本文带你一步步走过从基本原理到含分数和括号的方程的求解过程，为剑桥学习者讲得清清楚楚。
1. What is a Linear Equation? | 什么是一次方程？
A linear equation is a mathematical statement that shows two expressions are equal, and the highest power of the unknown variable is 1. For example, x + 3 = 7 and 4x − 2 = 10 are linear equations. The solution is the value of the variable that makes the equation true.
一次方程是表明两个表达式相等的数学语句，其中未知变量的最高次幂是1。例如，x + 3 = 7 和 4x − 2 = 10 都是一次方程。方程的解就是使等式成立的变量的值。
2. The Balancing Method | 平衡法
Think of an equation as a balanced scale. Whatever you do to one side, you must do to the other to keep it balanced. If you add 5 to the left, you must add 5 to the right. This idea is the golden rule for solving all linear equations.
把方程想象成一台平衡的天平。你对一边做的任何操作，必须对另一边也做同样的操作以保持平衡。如果你在左边加上5，就必须在右边也加上5。这个思想是求解所有一次方程的金科玉律。
3. Solving One‑Step Equations | 解一步方程
If the equation is x + 4 = 13, simply subtract 4 from both sides. On the left, x + 4 − 4 leaves x; on the right, 13 − 4 = 9. So x = 9. For multiplication, e.g., 3x = 21, divide both sides by 3 to get x = 7.
如果方程是 x + 4 = 13，只需从两边同时减去4。左边，x + 4 − 4 剩下 x；右边，13 − 4 = 9。因此 x = 9。对于乘法，例如 3x = 21，两边同除以3得到 x = 7。
4. Solving Two‑Step Equations | 解两步方程
Consider 2x + 5 = 17. First, subtract 5 from both sides: 2x = 12. Then divide both sides by 2: x = 6. Always undo the addition or subtraction first, then the multiplication or division.
考虑 2x + 5 = 17。首先，两边同减5：2x = 12。然后两边同除以2：x = 6。总是先消除加减法，再消除乘除法。
5. Equations with Brackets | 带括号的方程
When you see brackets, expand them first using the distributive law. For 3(x + 2) = 21, multiply 3 by x and by 2 to get 3x + 6 = 21. Then subtract 6 from both sides: 3x = 15, so x = 5.
当你看到括号，先用分配律展开。对于 3(x + 2) = 21，把3分别乘以x和2得到 3x + 6 = 21。然后两边同减6：3x = 15，所以 x = 5。
6. Unknowns on Both Sides | 未知数在等式两边
If the variable appears on both sides, like 5x + 3 = 2x + 12, start by eliminating the smaller x term. Subtract 2x from both sides to get 3x + 3 = 12. Now it’s a two‑step equation: subtract 3, then divide by 3, giving x = 3.
如果变量出现在两边，例如 5x + 3 = 2x + 12，先消除较小的x项。两边同减2x得到 3x + 3 = 12。现在它变成了一个两步方程：减3，然后除以3，得出 x = 3。
7. Equations Involving Fractions | 含分数的方程
For equations like x/4 + 3 = 7, treat the fraction as a division. Subtract 3 first: x/4 = 4. Then multiply both sides by 4: x = 16. If there are multiple fractions, multiply each term by the lowest common denominator to clear them.
对于像 x/4 + 3 = 7 这样的方程，把分数看成一个除法。先减去3：x/4 = 4。然后两边同乘以4：x = 16。如果有多个分数，把每项都乘以最小公分母来消除分母。
8. Checking Your Solution | 检验你的解
Always substitute your answer back into the original equation to verify. If you got x = 6 for 2x − 4 = 8, check: 2(6) − 4 = 12 − 4 = 8, so it’s correct. This habit prevents careless mistakes.
始终把你的答案代回原方程去验证。如果你对 2x − 4 = 8 求出了 x = 6，检验一下：2(6) − 4 = 12 − 4 = 8，所以是对的。这个习惯能防止粗心的错误。
9. Forming Equations from Word Problems | 根据应用题建立方程
Word problems require translating real‑life situations into algebraic language. For example, ‘Three times a number plus 5 equals 20.’ Let the number be n: 3n + 5 = 20. Solve as usual to find n = 5.
应用题要求把实际情境翻译成代数语言。例如，“一个数的三倍加上5等于20”。设这个数为 n：3n + 5 = 20。照常求解得 n = 5。
10. Common Mistakes to Avoid | 要避免的常见错误
Watch out for sign errors when moving terms across the equals sign. Also, remember to apply operations to both sides, not just one. And don’t forget to multiply every term inside a bracket by the coefficient outside.
移项时要注意符号错误。另外，牢记要对两边进行同样的操作，而不是只对一边。并且不要忘了括号外的系数要与括号内的每一项相乘。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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