📚 Solving Linear Equations | 解一元一次方程
Linear equations are one of the most important tools in Key Stage 3 mathematics. They help students model relationships in everyday problems and build a strong foundation for more advanced algebra at IGCSE level. This article explains how to solve one-step, two-step and more complex linear equations step by step.
一元一次方程是 KS3 数学中最重要的工具之一。它们帮助学生在日常问题中建立数学模型，也为 IGCSE 更高级的代数学习打下坚实基础。本文逐步讲解如何解一步、两步及更复杂的一元一次方程。
1. What is a linear equation? | 什么是一元一次方程？
ar equations. The word linear comes from the fact that, when plotted on a coordinate plane, the equation represents a straight line.
一元一次方程是指未知数的最高次数为一的方程。例如 x + 5 = 12、4y − 3 = 17 和 z/2 + 1 = 6 都是一元一次方程。linear 一词来源于这类方程在坐标平面上的图像是一条直线。
In KS3, the unknown is usually represented by a letter such as x, y, z or n. The goal is to find the value of the unknown that makes the equation true. This value is called the solution or the root of the equation.
在 KS3 阶段，未知数通常用 x、y、z 或 n 等字母表示。我们的目标是求出使等式成立的未知数的值。这个值称为方程的解或根。
ax + b = c (a ≠ 0)
nd c are numbers and a is not zero. If a were zero, the equation would no longer contain an unknown term and would not be a linear equation.
一元一次方程的一般形式为 ax + b = c，其中 a、b 和 c 是数且 a 不等于零。如果 a 为零，方程中就没有未知项，也就不再是一元一次方程。
2. The balance method | 天平法
An equation is like a balance scale. The two sides of the equation must always have the same value. Therefore, whatever operation you do to one side, you must also do exactly the same operation to the other side. This is called the balance method.
方程就像一架天平。等号两边必须始终具有相同的值。因此，你对一边做什么运算，就必须对另一边做完全相同的运算。这种方法称为天平法。
. The inverse of addition is subtraction, so subtract 7 from both sides. This gives x = 8.
例如，解方程 x + 7 = 15 时，需要去掉左边的 + 7。加法的逆运算是减法，所以两边同时减去 7，得到 x = 8。
x + 7 − 7 = 15 − 7 → x = 8
Understanding inverse operations is essential. Addition and subtraction are inverses, and multiplication and division are inverses. Using the correct inverse operation keeps the equation balanced while isolating the unknown.
理解逆运算非常重要。加法和减法互为逆运算，乘法和除法互为逆运算。使用正确的逆运算可以在保持等式平衡的同时分离出未知数。
3. Solving one-step equations | 解一步方程
+ a = b, x − a = b, ax = b and x/a = b. Each type requires a different inverse operation.
一步方程只需进行一次逆运算即可求解。基本类型有四种：x + a = b、x − a = b、ax = b 和 x/a = b。每种类型需要不同的逆运算。
- For x + 3 = 10, subtract 3 from both sides: x = 7. | 对于 x + 3 = 10，两边同时减去 3：x = 7。
- For x − 4 = 9, add 4 to both sides: x = 13. | 对于 x − 4 = 9，两边同时加上 4：x = 13。
- For 5x = 35, divide both sides by 5: x = 7. | 对于 5x = 35，两边同时除以 5：x = 7。
- For x/6 = 4, multiply both sides by 6: x = 24. | 对于 x/6 = 4，两边同时乘以 6：x = 24。
Always choose the operation that reverses the one applied to the unknown. Writing the operation clearly on both sides helps avoid mistakes and shows your working in tests.
始终选择能逆转未知数上运算的操作。在两边清楚地写出运算有助于避免错误，也能在考试中展示解题步骤。
4. Solving two-step equations | 解两步方程
nvolves both a multiplication and an addition. You must use two inverse operations in the correct order. Usually, you undo addition or subtraction first, then undo multiplication or division.
像 3x + 7 = 19 这样的两步方程涉及乘法和加法两种运算。你必须按正确顺序使用两次逆运算。通常先处理加法或减法，再处理乘法或除法。
9, subtract 7 from both sides to get 3x = 12. Then divide both sides by 3 to obtain x = 4.
对于 3x + 7 = 19，两边同时减去 7，得到 3x = 12。然后两边同时除以 3，得到 x = 4。
3x + 7 − 7 = 19 − 7 → 3x = 12 → x = 4
16. Then divide by 2: y = 8. Reversing the order correctly is a key skill in KS3 algebra.
另一个例子是 2y − 5 = 11。首先两边同时加上 5，得到 2y = 16。然后除以 2，得到 y = 8。正确反转运算顺序是 KS3 代数中的关键技能。
5. Equations with variables on both sides | 变量在等号两边的方程
step is to collect all the x terms on one side and all the number terms on the other side.
有些一元一次方程的未知数出现在等号两边。例如 5x + 2 = 2x + 14。第一步是把所有含 x 的项移到一边，把所有常数项移到另一边。
sides: 3x + 2 = 14. Then subtract 2 from both sides: 3x = 12. Finally divide by 3: x = 4.
两边同时减去 2x，得到 3x + 2 = 14。然后两边同时减去 2，得到 3x = 12。最后除以 3，得到 x = 4。
5x + 2 − 2x = 2x + 14 − 2x → 3x + 2 = 14 → x = 4
When moving a term from one side to the other, its sign changes. This is a short way of thinking about adding or subtracting the same term on both sides. Be careful with negative terms.
把一项从一边移到另一边时，它的符号会改变。这是对两边同时加上或减去同一项的一种简便理解方式。处理负项时要格外小心。
6. Equations with brackets | 含括号的方程
When a linear equation contains brackets, you can either expand the bracket first or divide both sides by the coefficient outside the bracket first. Both methods give the same answer if performed correctly.
当一元一次方程含有括号时，既可以先展开括号，也可以先用括号外的系数去除两边。如果操作正确，两种方法得到的结果相同。
= 14, expanding gives 2x + 6 = 14. Subtract 6 from both sides: 2x = 8. Divide by 2: x = 4.
对于 2(x + 3) = 14，展开得 2x + 6 = 14。两边同时减去 6：2x = 8。除以 2：x = 4。
2(x + 3) = 14 → 2x + 6 = 14 → x = 4
3 to get x = 4. This is often quicker when the right side is divisible by the coefficient.
也可以先两边同时除以 2，得到 x + 3 = 7，然后减去 3 得到 x = 4。当等号右边能被系数整除时，这种方法通常更快。
o both sides: 6x = 24. Divide by 6: x = 4. Dividing first also gives 2x − 1 = 7, so x = 4.
对于 3(2x − 1) = 21，先展开得 6x − 3 = 21。两边同时加上 3：6x = 24。除以 6：x = 4。先除以 3 也能得到 2x − 1 = 7，所以 x = 4。
7. Equations with fractions | 含分数的方程
Equations with fractions are common in KS3 and can be solved by clearing the fraction or by using inverse operations step by step. The key is to remember that a fraction bar means division.
含分数的方程在 KS3 中很常见，可以通过去分母或逐步使用逆运算来求解。关键是要记住分数线表示除法。
+ 2 = 5, first subtract 2 from both sides: x/3 = 3. Then multiply both sides by 3: x = 9.
对于 x/3 + 2 = 5，先两边同时减去 2：x/3 = 3。然后两边同时乘以 3：x = 9。
x/3 + 2 = 5 → x/3 = 3 → x = 9
, add 1 to both sides: (2x)/5 = 4. Multiply both sides by 5: 2x = 20. Divide by 2: x = 10.
对于 (2x)/5 − 1 = 3，两边同时加上 1：(2x)/5 = 4。两边同时乘以 5：2x = 20。除以 2：x = 10。
When fractions have different denominators, you can first multiply every term by the lowest common multiple of the denominators. This clears the fractions and leads to an equation without denominators.
当方程中的分数分母不同时，可以先用所有分母的最小公倍数乘以每一项。这样可以去掉分母，得到一个不含分数的方程。
8. Checking your solution | 检验方程的解
After finding a solution, you should always check it by substituting the value back into the original equation. If the left side equals the right side, your solution is correct. This check is especially useful in tests.
求出解后，应始终把该值代回原方程进行检验。如果左边等于右边，说明解是正确的。这种检验在考试中特别有用。
ft side: 3(4) + 7 = 12 + 7 = 19. The left side equals the right side, so x = 4 is correct.
对于 3x + 7 = 19，我们得到 x = 4。将 x = 4 代入左边：3(4) + 7 = 12 + 7 = 19。左边等于右边，所以 x = 4 是正确的。
3(4) + 7 = 19 → 12 + 7 = 19 → true
If the two sides are not equal after substitution, you have made a mistake in solving. Go back and check your inverse operations, signs and arithmetic carefully.
如果代入后两边不相等，说明解题过程中出现了错误。回去仔细检查逆运算、符号和算术。
9. Common mistakes and how to avoid
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