📚 IGCSE Maths: Solving Simultaneous Equations | IGCSE数学：解联立方程
Simultaneous equations appear in almost every IGCSE Mathematics paper. They ask you to find the values of two unknowns that satisfy two equations at the same time. You need to master algebraic methods such as elimination and substitution, and you should also be able to interpret solutions graphically. This revision guide covers the key concepts, worked examples, common mistakes, and exam tips.
联立方程几乎出现在每一份 IGCSE 数学试卷中。题目要求你找出同时满足两个方程的两个未知数的值。你需要熟练掌握消元法和代入法等代数方法，同时也应能够从图像上解释解的含义。本复习指南涵盖核心概念、例题解析、常见错误以及考试技巧。
1. What Are Simultaneous Equations? | 什么是联立方程？
A simultaneous equation system consists of two or more equations with two or more unknowns. The solution is the pair of values that makes all equations true at the same time. On a graph, the solution is the point where the lines cross.
联立方程组由两个或更多含有两个或更多未知数的方程组成。解就是能够同时使所有方程成立的一组数值。在图像上，解就是两条直线相交的点。
y = 2 because these values satisfy both equations. Checking by substitution is essential.
例如，x + y = 5 和 x − y = 1 组成一个二元一次方程组。解是 x = 3 和 y = 2，因为这组值同时满足两个方程。代入检验非常重要。
- Unknown: a letter representing an unknown number | 未知数：表示未知数的字母
- Solution: the value pair that satisfies every equation | 解：满足每个方程的一组值
- Linear equation: an equation whose graph is a straight line | 一次方程：图像为一条直线的方程
2. The Elimination Method | 消元法
The elimination method works by adding or subtracting the equations to remove one variable. This is often the fastest method when the coefficients of one variable are the same or easily made the same.
消元法通过将两个方程相加或相减来消去一个变量。当某个变量的系数相同或容易化成相同时，这通常是最快的方法。
m 3x + 2y = 12 and 3x − y = 3. Subtract the second equation from the first to eliminate x:
考虑方程组 3x + 2y = 12 和 3x − y = 3。用第一个方程减去第二个方程以消去 x：
(3x + 2y) − (3x − y) = 12 − 3
3y = 9
y = 3
tute y = 3 into 3x − y = 3 to find x = 2. Always write the final solution as x = 2, y = 3.
现在将 y = 3 代入 3x − y = 3，求得 x = 2。最后一定要写出解 x = 2, y = 3。
x + 2y = 8, multiply the first by 3 and the second by 2 to make the x-coefficients both 6.
如果系数不相同，先给一个或两个方程乘上合适的数。例如对于 2x + 3y = 7 和 3x + 2y = 8，将第一个方程乘以 3，第二个方程乘以 2，使 x 的系数都变成 6。
3. The Substitution Method | 代入法
The substitution method is useful when one equation already has a variable as the subject. Rearrange one equation to make x or y the subject, then substitute that expression into the other equation.
当其中一个方程已经把某个变量写成主项时，代入法非常有用。将其中一个方程变形为 x 或 y 作为主项，然后将该表达式代入另一个方程。
For the system y = 2x + 1 and x + 3y = 14, substitute y = 2x + 1 into the second equation:
对于方程组 y = 2x + 1 和 x + 3y = 14，将 y = 2x + 1 代入第二个方程：
x + 3(2x + 1) = 14
x + 6x + 3 = 14
7x = 11
x = 11/7
y = 29/7. Fractions are acceptable in IGCSE answers unless the question asks for decimals.
然后将 x = 11/7 代回 y = 2x + 1，得到 y = 29/7。除非题目要求用小数，否则 IGCSE 答案中可以使用分数。
Substitution is also the standard method for non-linear simultaneous equations, such as one linear and one quadratic equation.
代入法也是求解非线性联立方程的标准方法，例如一个一次方程和一个二次方程组成的方程组。
4. Solving Graphically | 图像法求解
To solve simultaneous equations graphically, draw both lines on the same coordinate grid. The point of intersection gives the solution because its coordinates satisfy both equations.
要用图像法解联立方程，需在同一坐标系中画出两条直线。交点就是解，因为交点的坐标同时满足两个方程。
both lines accurately. The intersection is at (3, 3), so the solution is x = 3 and y = 3.
对于 y = 2x − 3 和 y = −x + 6，准确画出两条直线。交点为 (3, 3)，因此解为 x = 3, y = 3。
Graphical solutions are useful for checking algebraic answers. However, graph reading may not give exact values if the intersection is not at integer coordinates.
图像法可用于检验代数答案。但如果交点不在整数坐标上，读图可能无法给出精确值。
| Method | 方法 | Best for | 最适合 | Accuracy | 精确度 |
|---|---|---|
| Elimination | 消元法 | Coefficients easily matched | 系数容易匹配 | Exact | 精确 |
| Substitution | 代入法 | One variable isolated | 一个变量已单独分离 | Exact | 精确 |
| Graphical | 图像法 | Checking solutions | 检验解 | Approximate | 近似 |
5. Word Problems Involving Simultaneous Equations | 涉及联立方程的应用题
Many IGCSE questions are set in real-life contexts. You must define two variables clearly, form two equations from the information, and then solve them using a reliable method.
许多 IGCSE 题目以现实生活为背景。你必须明确定义两个变量，根据信息列出两个方程，然后用可靠的方法求解。
Example: Two apples and three bananas cost $1.90. Four apples and one banana cost $2.30. Find the cost of one apple and one banana.
例题：两个苹果和三个香蕉花费 $1.90。四个苹果和一个香蕉花费 $2.30。求一个苹果和一个香蕉的价格。
Let a be the cost of one apple and b be the cost of one banana. The equations are:
设 a 为一个苹果的价格，b 为一个香蕉的价格。方程为：
2a + 3b = 1.90
4a + b = 2.30
subtract to eliminate b. You get a = 0.50 and b = 0.30. Write the final answer with units.
将第二个方程乘以 3，然后相减以消去 b。得到 a = 0.50 和 b = 0.30。写出最终答案时要带上单位。
Always check that your values make sense in the original context. A negative price would indicate an error.
务必检查所得值在原始情境中是否合理。负价格通常说明计算有误。
6. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无穷多解
Not every pair of linear equations has one unique solution. If the lines are parallel, there is no solution. If the equations represent the same line, there are infinitely many solutions.
并非每一对一次方程都有唯一解。如果两条直线平行，则无解。如果两个方程表示同一条直线，则有无穷多解。
the left sides are identical but the constants differ. The lines are parallel, so there is no solution.
考虑 2x + 3y = 6 和 4x + 6y = 18。将第二个方程除以 2 得到 2x + 3y = 9，因此左边相同但常数不同。两条直线平行，所以无解。
12 are equivalent. They describe the same line, so every point on that line is a solution.
另一方面，2x + 3y = 6 和 4x + 6y = 12 是等价的。它们描述同一条直线，因此该直线上的每一个点都是解。
= 5, the system has no solution. If it leads to 0 = 0, the system has infinite solutions.
考试中，如果消元导致 0 = 5 这样的假等式，则方程组无解。如果得到 0 = 0，则方程组有无穷多解。
7. Quadratic-Linear Simultaneous Equations | 二次与线性联立方程
IGCSE papers often include one linear equation and one quadratic equation. Use the linear equation to express y in terms of x, then substitute into the quadratic to form a single quadratic equation in x.
IGCSE 试卷中常出现一个一次方程和一个二次方程的组合。利用一次方程将 y 用 x 表示，然后代入二次方程，得到一个只含 x 的一元二次方程。
Example: y = 2x + 1 and y = x² + 3x − 1. Set them equal:
例题：y = 2x + 1 和 y = x² + 3x − 1。令两者相等：
2x + 1 = x² + 3x − 1
0 = x² + x − 2
0 = (x + 2)(x − 1)
into y = 2x + 1 to find the corresponding y-values. The solutions are (−2, −3) and (1, 3).
因此 x = −2 或 x = 1。将每个值代回 y = 2x + 1 求出对应的 y 值。解为 (−2, −3) 和 (1, 3)。
Each solution is an ordered pair. Do not mix the x and y values incorrectly.
每个解都是一组有序数对。不要把 x 和 y 的值错误搭配。
8. Common Mistakes and How to Avoid Them | 常见错误与避免方法
ation. For example, multiplying 2x + 3y = 6 by 2 must give 4x + 6y = 12, not 4x + 3y = 12.
常见错误之一是在方程两边同乘一个数时忘记乘每一项。例如，将 2x + 3y = 6 乘以 2 必须得到 4x + 6y = 12，而不是 4x + 3y = 12。
Another frequent mistake is subtracting the equations incorrectly. Always subtract the whole expression, including signs. Use brackets to avoid sign errors.
另一个常见错误是方程相减时出错。一定要减去整个表达式，包括符号。使用括号可以避免符号错误。
Many students also fail to check their answers. Substituting the values back into both original equations takes little time and can catch mistakes quickly.
许多学生也不检验答案。将值代回两个原始方程花费时间很少，却能快速发现错误。
- Always write the final answer as x = ... , y = ... | 最终答案务必写成 x = ... , y = ...
- Keep fractions until the final step | 在最后一步之前保留分数
- Check both equations | 两个方程都要检验
9. Exam Strategy and Tips | 考试策略与技巧
In the IGCSE exam, simultaneous equation questions often carry 3 to 6 marks. Method marks are awarded for correct elimination or substitution even if the final answer has an arithmetic slip.
在 IGCSE 考试中，联立方程题通常占 3 到 6 分。即使最终答案有计算错误，正确的消元或代入步骤也能获得方法分。
Show all working clearly. Examiners can only award method marks if they can see your process. Avoid doing everything mentally.
清晰展示所有解题步骤。考官只有看到你的过程才能给方法分。不要完全心算。
If a question asks for a graphical solution, draw lines with a ruler and label the intersection point. If it asks for an exact solution, use algebra and leave answers in fraction form unless stated otherwise.
如果题目要求图像解，用直尺画线并标出交点。如果要求精确解，使用代数方法，除非另有说明，答案保留分数形式。
Manage your time: elimination and substitution questions are usually quick marks. Practise them until they become automatic.
合理分配时间：消元法和代入法题目通常可以快速得分。反复练习直到熟练自动。
10. Practice Questions with Worked Solutions | 带解析的练习题
Try these questions before looking at the solutions. They cover both linear and quadratic-linear systems.
在查看解析之前先尝试以下题目。它们涵盖线性方程组以及二次与线性方程组。
Question 1: Solve 5x + 2y = 11 and 3x − 2y = 1.
题目 1：解方程组 5x + 2y = 11 和 3x − 2y = 1。
Add the equations: 8x = 12 → x = 1.5
Substitute into 3x − 2y = 1 → 4.5 − 2y = 1 → y = 1.75
Solution: x = 3/2, y = 7/4.
解：x = 3/2, y = 7/4。
Question 2: Solve y = x² + 2x − 3 and y = x + 3.
题目 2：解方程组 y = x² + 2x − 3 和 y = x + 3。
x² + 2x − 3 = x + 3
x² + x − 6 = 0
(x + 3)(x − 2) = 0
x = −3 or x = 2
When x = −3, y = 0. When x = 2, y = 5. The solutions are (−3, 0) and (2, 5).
当 x = −3 时，y = 0。当 x = 2 时，y = 5。解为 (−3, 0) 和 (2, 5)。
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