📚 IGCSE Maths Topic 200: Solving Linear Equations | IGCSE 数学主题 200：解一次方程
Linear equations are one of the most important foundations of IGCSE Mathematics. They appear in almost every paper, either as standalone questions or as part of larger problems in algebra, geometry, and even statistics. A confident grasp of linear equations will help you secure reliable marks and build a strong base for more advanced topics such as simultaneous equations, quadratic equations, and functions.
一次方程是 IGCSE 数学最重要的基础内容之一。它们几乎出现在每一份试卷中，无论是作为独立考题，还是作为代数、几何甚至统计大题的一部分。熟练掌握一次方程可以帮助你稳定得分，也为后续更高级的主题，如联立方程、二次方程和函数，打下扎实基础。
1. What Is a Linear Equation? | 什么是一次方程？
A linear equation is an equation in which the unknown, usually written as x, has an exponent of 1. This means there are no terms such as x², x³, √x xample, 3x + 5 = 20 is linear because x is not squared, cubed, or placed in a denominator.
一次方程是指未知数通常写作 x，且其次数为 1 的方程。这意味着方程中不会出现 x²、x³、√x 或 1/x 这样的项。一次方程通常可以写成 ax + b = c 的形式，其中 a、b、c 是常数。例如，3x + 5 = 20 就是一次方程，因为 x 没有被平方、立方或放在分母中。
Solving a linear equation means finding the value of x that makes both sides of the equation equal. This value is called the solution or the root of the equation. In IGCSE questions, you will often be asked to solve an equation and sometimes to check your answer by substitution.
解一次方程就是找到使等号两边相等的 x 值。这个值称为方程的解或根。在 IGCSE 题目中，你通常会被要求解方程，有时还需要用代入法检验答案。
2. The Balance Method | 天平法
The balance method treats an equation like a pair of balanced scales. Whatever you do to one side, you must do to the other side to keep the equation true. If you add, subtract, multiply, or divide on one side, repeat the exact same operation on the other side. This principle is the golden rule of equation solving.
天平法把方程看作一架平衡的天平。你对一边做什么运算，就必须对另一边做完全相同的运算，才能保持等号成立。如果在一侧加、减、乘或除，就要在另一侧重复完全相同的运算。这个原则是解方程的黄金法则。
+ 5 − 5 = 20 − 5, giving 3x = 15. Then divide both sides by 3: 3x ÷ 3 = 15 ÷ 3, so x = 5.
例如，考虑方程 3x + 5 = 20。首先两边同时减去 5：3x + 5 − 5 = 20 − 5，得到 3x = 15。然后两边同时除以 3：3x ÷ 3 = 15 ÷ 3，所以 x = 5。
3x + 5 = 20 → 3x = 15 → x = 5
3. Solving Two-Step Equations | 解两步方程
Many IGCSE linear equations can be solved in exactly two steps. Start by undoing any addition or subtraction, then undo multiplication or division. Always reverse the order of operations: undo addition or subtraction before multiplication or division. This is because the original expression was built by multiplying first and then adding, so we reverse that order when solving.
许多 IGCSE 一次方程恰好可以用两步解出。首先处理加法或减法，然后处理乘法或除法。记住要逆着运算顺序来：先消去加减，再消去乘除。这是因为原表达式是先乘后加得到的，所以解方程时要反过来处理。
ficient of x is negative, divide carefully. For −3x = 12, divide both sides by −3: x = −4.
例如，解方程 7x − 4 = 24。两边加 4：7x = 28。然后两边除以 7：x = 4。如果 x 的系数是负数，除法时要小心。例如 −3x = 12，两边除以 −3 得 x = −4。
7x − 4 = 24 → 7x = 28 → x = 4
4. Equations with Brackets | 含括号的方程
When an equation contains brackets, expand them first using the distributive law. Multiply each term inside the bracket by the term outside. For example, 2(x − 3) becomes 2x − 6. After expanding, solve the simplified equation using the balance method. This is a very common IGCSE skill because brackets often appear in exam questions.
当方程含有括号时，首先要利用分配律展开。把括号外的项乘以括号内的每一项。例如，2(x − 3) 展开为 2x − 6。展开后，再使用天平法解简化后的方程。这是非常常见的 IGCSE 技能，因为括号经常出现在考题中。
ide the bracket is negative: −2(x − 4) = 10 becomes −2x + 8 = 10, then −2x = 2, so x = −1.
例如，解方程 3(2x + 1) = 27。展开得 6x + 3 = 27。两边减 3：6x = 24。除以 6：x = 4。当括号外的项是负数时要小心：−2(x − 4) = 10 展开为 −2x + 8 = 10，然后 −2x = 2，所以 x = −1。
3(2x + 1) = 27 → 6x + 3 = 27 → 6x = 24 → x = 4
5. Variables on Both Sides | 两边含未知数的方程
If x appears on both sides of the equation, collect the variable terms on one side first. Choose the side with the larger coefficient so that x remains positive. Then collect the constant terms on the other side. This usually creates a two-step equation that you can solve easily.
如果等号两边都含有 x，先把含 x 的项移到同一边。通常选择系数较大的一侧，使 x 保持为正。然后把常数项移到另一边。这样通常会得到一个容易求解的两步方程。
ft side is 5(3) − 3 = 12, and the right side is 2(3) + 6 = 12, so the solution is correct.
例如，解方程 5x − 3 = 2x + 6。两边减去 2x：3x − 3 = 6。两边加 3：3x = 9。除以 3：x = 3。你可以代入 x = 3 检验：左边为 5(3) − 3 = 12，右边为 2(3) + 6 = 12，因此解是正确的。
5x − 3 = 2x + 6 → 3x − 3 = 6 → 3x = 9 → x = 3
6. Equations with Fractions | 含分数的方程
Fraction equations may look harder, but you can clear fractions quickly. If a single fraction equals a number, multiply both sides by the denominator. If there are several terms, multiply every term by the lowest common denominator. This removes the fractions and gives a simpler linear equation.
分数方程可能看起来更难，但你可以快速去分母。如果单个分数等于一个数，两边同时乘以分母。如果有多个项，就把每一项都乘以最小公分母。这样可以消去分数，得到一个更简单的一次方程。
er example is (x + 3)/2 = 5. Multiply both sides by 2: x + 3 = 10. Then subtract 3: x = 7.
例如，解方程 x/4 + 2 = 7。两边减 2：x/4 = 5。两边乘以 4：x = 20。另一个例子是 (x + 3)/2 = 5。两边乘以 2：x + 3 = 10。然后减 3：x = 7。
x/4 + 2 = 7 → x/4 = 5 → x = 20
(x + 3)/2 = 5 → x + 3 = 10 → x = 7
7. Forming Equations from Word Problems | 根据应用题列方程
nslate written language into algebra. Let the unknown quantity be x. Identify key phrases: "more than" means +, "less than" means −, "times" means ×, and "is" means =. Always define the variable clearly before writing the equation.
应用题考查你是否能把文字语言转化为代数式。设未知量为 x。识别关键词：“比……多”表示 +，“比……少”表示 −，“……倍”表示 ×，“是”表示 =。在列方程之前，一定要清楚地定义变量。
racting 3 gives 2n = 16, so n = 8. In IGCSE problems, it is important to state the meaning
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