📚 Linear Equations and Simultaneous Equations for IGCSE Mathematics | IGCSE 数学：一次方程与联立方程
Linear equations and simultaneous equations are essential algebra skills in IGCSE Mathematics. They appear in Paper 2 and Paper 4, often combined with graphs, word problems and real-life applications. Mastering the methods of elimination, substitution and graphical interpretation will help you solve both straightforward and extended problems with confidence.
一次方程与联立方程是 IGCSE 数学中必不可少的代数技能。它们常出现在试卷二和试卷四中，并与图像、应用题和实际情境相结合。掌握消元法、代入法和图像法，能帮助你自信地解决从基础题到拓展题的各种问题。
1. What Is a Linear Equation? | 什么是一次方程？
where a and b are constants and a ≠ 0. In two variables, the standard form is ax + by = c.
一次方程是指未知数的最高次数仅为 1 的方程，其图像是一条直线。一元一次方程的标准形式为 ax + b = 0，其中 a、b 为常数且 a ≠ 0。二元一次方程的标准形式为 ax + by = c。
ax + b = 0, a ≠ 0
ax + by = c
- Linear equations produce straight-line graphs — 一次方程产生直线图像
- The highest power of the unknown is 1 — 未知数的最高次数为 1
- Coefficients can be positive or negative — 系数可以为正或为负
2. Solving One-Step and Two-Step Equations | 解一步方程与两步方程
y 5 to get x = 4. Always keep the equation balanced by doing the same thing to both sides.
解一步方程时，在等式两边同时进行逆运算。例如，若 x + 3 = 7，两边同时减去 3，得到 x = 4。若 5x = 20，两边同时除以 5，得到 x = 4。一定要在等式两边进行相同运算，保持方程平衡。
x + 3 = 7 → x = 7 − 3 = 4
5x = 20 → x = 20 ÷ 5 = 4
, 2x − 5 = 9 becomes 2x = 14 after adding 5 to both sides, then x = 7 after dividing by 2.
两步方程需要先消去加减项，再消去乘除项。例如，2x − 5 = 9，两边先加 5 得到 2x = 14，再除以 2 得到 x = 7。
2x − 5 = 9 → 2x = 14 → x = 7
3. Equations with Brackets and Fractions | 带括号与分数的方程
dividing both sides by 3 to get x + 2 = 7, so x = 5. This is often faster than expanding.
当方程含有括号时，可以先展开括号，也可以在可能的情况下两边同时除以公因数。例如，3(x + 2) = 21 可通过两边同时除以 3 得到 x + 2 = 7，因此 x = 5。这通常比展开更快捷。
3(x + 2) = 21 → x + 2 = 7 → x = 5
For example, x/3 + 2 = 6 becomes x + 6 = 18 after multiplying through by 3, giving x = 12.
对于含分数的方程，每一项都乘以最小公分母以消去分数。例如，x/3 + 2 = 6 两边同时乘以 3，得到 x + 6 = 18，因此 x = 12。
x/3 + 2 = 6 → x + 6 = 18 → x = 12
- Expand brackets or divide first — 先展开括号或先除以公因数
- Multiply by the lowest common denominator to remove fractions — 乘以最小公分母消去分数
4. Introduction to Simultaneous Equations | 联立方程简介
A simultaneous equation system contains two or more equations that share the same unknowns. In IGCSE Mathematics, you usually solve a pair of linear equations to find the single pair of values for x and y that satisfies both equations at the same time.
联立方程组包含两个或多个含有相同未知数的方程。在 IGCSE 数学中，通常需要求解一对一次方程，找到同时满足两个方程的 x 和 y 的唯一一组值。
x + y = 7
x − y = 3
here are three main solution methods: elimination, substitution and graphical representation.
解 x = 5, y = 2 同时满足两个方程，因为 5 + 2 = 7 且 5 − 2 = 3。三种主要解法是：消元法、代入法和图像法。
x = 5, y = 2
5. Solving by Elimination | 消元法
The elimination method works by adding or subtracting the two equations to remove one variable. This is most efficient when the coefficients of one variable are the same or opposites. After finding one variable, substitute it back into either original equation to find the other variable.
消元法通过将两个方程相加或相减来消去一个未知数。当某个未知数的系数相同或互为相反数时，这种方法最为高效。求出一个未知数后，将其代回任意一个原方程即可求出另一个未知数。
ding the equations gives 2x = 10, so x = 5. Substituting x = 5 into x + y = 7 gives y = 2.
对于方程组 x + y = 7 和 x − y = 3，两式相加得到 2x = 10，因此 x = 5。将 x = 5 代入 x + y = 7 得到 y = 2。
(x + y) + (x − y) = 7 + 3 → 2x = 10 → x = 5
by 3 to get 6x − 3y = 15. Then add: 10x = 27, so x = 2.7. Substitute back to find y = 0.4.
如果系数不相同，需要先将一个或两个方程乘以适当的数。例如，在 4x + 3y = 12 和 2x − y = 5 中，将第二个方程乘以 3 得到 6x − 3y = 15。然后相加：10x = 27，因此 x = 2.7。代回可求得 y = 0.4。
4x + 3y = 12
2x − y = 5 → 6x − 3y = 15
10x = 27 → x = 2.7, y = 0.4
6. Solving by Substitution | 代入法
this expression into the second equation, then solve the resulting one-variable equation.
当一个方程中的某个未知数已经用另一个未知数表示时，例如 y = 2x + 1，代入法非常有用。将这个表达式代入第二个方程，然后求解得到的一元方程。
1) = 10. This simplifies to 3x + 1 = 10, then 3x = 9 and x = 3. Finally, y = 2(3) + 1 = 7.
对于 y = 2x + 1 和 x + y = 10，将 y = 2x + 1 代入 x + y = 10，得到 x + (2x + 1) = 10。化简为 3x + 1 = 10，然后 3x = 9，x = 3。最后，y = 2(3) + 1 = 7。
x + (2x + 1) = 10 → 3x = 9 → x = 3, y = 7
- Substitute the known variable or expression into the other equation — 将已知变量或表达式代入另一个方程
- Always substitute back to find the second unknown — 一定要代回求第二个未知数
7. Graphical Method and Intersection Points | 图像法与交点
Each linear equation can be drawn as a straight line on the coordinate plane. The solution to a pair of simultaneous equations is the point where the two lines intersect. The coordinates of this intersection give the values of x and y that satisfy both equations.
每个一次方程都可以在坐标平面上画成一条直线。联立方程组的解就是两条直线相交的点。这个交点的坐标就是同时满足两个方程的 x 和 y 的值。
ect at (5, 2). You can check this by plotting both lines or by using graph paper in an exam question.
对于 x + y = 7 和 x − y = 3，两条直线相交于点 (5, 2)。你可以通过绘制这两条直线来验证，或在考试题中使用坐标纸完成。
Intersection point: (5, 2)
The graphical method is also useful for showing whether a system has one solution, no solution, or infinitely many solutions.
图像法还可以用来判断方程组有唯一解、无解还是无穷多解。
8. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无穷多解
quation simplifies to 2x + y = 5, which has the same left side but a different right side.
如果两条直线平行，它们永远不会相交，因此方程组无解。当两个方程斜率相同但截距不同时就会出现这种情况。例如，2x + y = 3 和 4x + 2y = 10 是平行的，因为第二个方程化简后为 2x + y = 5，左边相同但右边不同。
2x + y = 3
4x + 2y = 10 → 2x + y = 5
mple, x + y = 2 and 2x + 2y = 4 are equivalent, so every point on that line is a solution.
如果两个方程表示同一条直线，它们完全重合，因此有无穷多解。例如，x + y = 2 和 2x + 2y = 4 等价，因此该直线上的每一个点都是解。
x + y = 2 and 2x + 2y = 4 → infinitely many solutions
9. Word Problems Leading to Simultaneous Equations | 联立方程应用题
Many IGCSE questions ask you to form a pair of simultaneous equations from a real-life situation. Identify the two unknown quantities, assign variables, and translate the given information into two equations. Then solve using elimination or substitution.
许多 IGCSE 题目要求根据实际情境列出一对联立方程。首先确定两个未知量，设出变量，然后将题目信息转化为两个方程。接着使用消元法或代入法求解。
, with x > y. Then x + y = 24 and x − y = 6. Adding gives 2x = 30, so x = 15 and y = 9.
例如，两个数的和为 24，差为 6。设这两个数为 x 和 y，且 x > y。则 x + y = 24 且 x − y = 6。两式相加得到 2x = 30，因此 x = 15，y = 9。
x + y = 24, x − y = 6 → x = 15, y = 9
- Define variables clearly before writing equations — 在列方程前明确设出变量
- Check that the solution makes sense in the original context — 检查解在原情境中是否合理
10. Checking Your Answers and Exam Tips | 检验答案与考试技巧
Always substitute your values back into both original equations to check that they work. This takes a few seconds but can prevent many avoidable errors. In exams, show every step clearly because method marks are often awarded even if a small arithmetic mistake is made.
一定要将得到的值代回两个原方程进行检验。这只需几秒钟，却能避免许多本可避免的错误。在考试中，要清晰地展示每一步，因为即使出现小的计算错误，也常常能获得方法分。
When using elimination, align the equations neatly and label them if necessary. If the numbers are messy, consider using substitution or a graph to estimate the answer first. Time management is important: do not spend too long on one simultaneous equation question.
使用消元法时，要把方程整齐排列，必要时进行编号。如果数字比较复杂，可以考虑使用代入法或先画图估计答案。时间管理很重要：不要在一道联立方程题上花费过长时间。
| Common mistake | How to avoid it | 中文提示 |
| Adding instead of subtracting | Check signs carefully | 仔细检查符号 |
| Forgetting to substitute back | Always find both variables | 一定要求出两个未知数 |
| Not checking the final answer | Substitute into both original equations | 代回两个原方程验证 |
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