📚 Solving Simultaneous Equations: Elimination, Substitution and Graphs | IGCSE数学联立方程：消元、代入与图像法
In IGCSE Mathematics, simultaneous equations often appear in both Paper 1 and Paper 2, and they are a core algebra skill. You need to be confident in solving two equations with two unknowns using elimination, substitution, or graphs, and you must be able to form equations from real-life problems. This revision guide covers the essential methods, special cases, and exam techniques.
在 IGCSE 数学中，联立方程经常出现在 Paper 1 和 Paper 2 中，是代数部分的核心技能。你需要熟练运用消元法、代入法或图像法求解含有两个未知数的两个方程，并能从实际问题中建立方程。本复习指南涵盖基本方法、特殊情形和考试技巧。
1. What Are Simultaneous Equations? | 什么是联立方程
A simultaneous equation system consists of two or more equations that share the same variables. The solution is the pair of values that makes every equation true at the same time. For IGCSE, you will usually work with two equations and two unknowns, such as x and y.
联立方程组由两个或两个以上含有相同未知数的方程组成。解就是能同时使每一个方程都成立的一组未知数值。在 IGCSE 中，你通常会处理两个方程和两个未知数，例如 x 和 y。
For example, the system:
例如，方程组：
x + y = 7
2x − y = 2
oth equations are satisfied when you substitute these values. Understanding this idea helps you check whether an answer is reasonable.
它的解是 x = 3 和 y = 4，因为当你代入这两个值时，两个方程都成立。理解这个概念有助于你判断一个答案是否合理。
2. Solving by Elimination | 消元法
Elimination is often the quickest method when the coefficients of one variable can easily be made the same. You add or subtract the two equations to remove that variable, leaving a simple one-variable equation.
当某个未知数的系数可以方便地变为相同时，消元法通常是最快的方法。你将两个方程相加或相减，消去那个未知数，从而得到一个简单的一元方程。
The standard steps are:
标准步骤如下：
- Write both equations in the form ax + by = c. 将两个方程写成 ax + by = c 的形式。
- Multiply one or both equations so that the coefficients of one variable match in size. 将一个或两个方程乘以适当的数，使某个未知数的系数大小相等。
- Add or subtract the equations to eliminate that variable. 将方程相加或相减，消去该未知数。
- Solve the remaining equation and substitute back to find the other variable. 解剩下的一元方程，并回代求出另一个未知数。
For example, solve:
例如，解以下方程组：
2x + y = 10
x − y = 2
= 4. Substituting x = 4 into x − y = 2 gives 4 − y = 2, so y = 2. The solution is (4, 2).
将两个方程相加，得到 3x = 12，因此 x = 4。将 x = 4 代入 x − y = 2，得到 4 − y = 2，所以 y = 2。解为 (4, 2)。
3. Solving by Substitution | 代入法
. You substitute this expression into the other equation and solve the resulting linear equation.
当一个方程已经整理成某个未知数作为主项的形式时，例如 y = 2x + 1，代入法特别有效。你将这个表达式代入另一个方程，然后解得到的一元方程。
For example, solve:
例如，解：
y = 2x + 1
3x + y = 16
2x + 1 = 16. This simplifies to 5x + 1 = 16, so 5x = 15 and x = 3. Then y = 2(3) + 1 = 7.
将 y = 2x + 1 代入 3x + y = 16，得到 3x + 2x + 1 = 16。化简为 5x + 1 = 16，所以 5x = 15，x = 3。然后 y = 2(3) + 1 = 7。
Substitution is especially important for non-linear simultaneous equations, which appear on the Extended syllabus.
代入法对于非线性联立方程尤其重要，这类题目出现在 Extended 课程大纲中。
4. Graphical Interpretation | 图解法
Each linear equation represents a straight line on the coordinate plane. The solution of two simultaneous equations is the point where the two lines intersect. Graphically, you can estimate the solution by reading the coordinates of this intersection.
每一个线性方程都代表坐标平面上的一条直线。两个联立方程的解就是两条直线相交的点。从图像上看，你可以通过读取这个交点的坐标来估算解。
ns y = x + 1 and y = −x + 5, a small table of values helps you plot both lines accurately.
对于方程 y = x + 1 和 y = −x + 5，一个简单的取值表可以帮助你准确地画出两条直线。
| x | -1 | 0 | 1 | 2 |
| y = x + 1 | 0 | 1 | 2 | 3 |
| y = −x + 5 | 6 | 5 | 4 | 3 |
plotting accurately on graph paper gives a reliable solution, but algebra gives the exact values.
两条直线相交于点 (2, 3)，因此解为 x = 2 和 y = 3。在考试中，在坐标纸上准确绘图可以得到可靠答案，但代数方法能给出精确值。
5. Special Cases: No Solution and Infinite Solutions | 特殊情形：无解与无穷多解
the system has no solution. Algebraically, you will reach a false statement such as 0 = 3.
有些方程组没有唯一解。如果两条直线平行，它们永不相交，因此方程组无解。从代数上看，你会得到一个矛盾的等式，例如 0 = 3。
x + y = 2
x + y = 5
These equations are inconsistent because the same left side is claimed to equal both 2 and 5. There is no possible pair of x and y.
这两个方程是矛盾的，因为同一个左边同时等于 2 和 5。不存在任何 x 和 y 满足条件。
d 4x + 2y = 8 describe the same line because the second equation is just double the first.
如果两个方程代表同一条直线，则有无穷多解。从代数上看，消元会得到一个恒等式，例如 0 = 0。例如 2x + y = 4 和 4x + 2y = 8 描述的是同一条直线，因为第二个方程只是第一个方程的两倍。
6. Word Problems Leading to Simultaneous Equations | 文字题建立联立方程
Many IGCSE questions ask you to translate a real-life situation into two equations. Start by defining variables clearly, then form two independent equations from the given information.
很多 IGCSE 题目要求你将实际问题转化为两个方程。首先清楚地设未知数，然后根据已知信息建立两个独立的方程。
Example: 3 apples and 2 bananas cost £1.60. 2 apples and 5 bananas cost £1.80. Let a be the price of one apple and b be the price of one banana.
例题：3 个苹果和 2 根香蕉共 1.60 英镑。2 个苹果和 5 根香蕉共 1.80 英镑。设 a 为一个苹果的价格，b 为一根香蕉的价格。
3a + 2b = 1.60
2a + 5b = 1.80
0. Subtracting gives 11a = 4.40, so a = 0.40. Substituting gives b = 0.20. One apple costs £0.40 and one banana costs £0.20.
将第一个方程乘以 5，第二个方程乘以 2，使 b 的系数相等：15a + 10b = 8.00 和 4a + 10b = 3.60。相减得到 11a = 4.40，所以 a = 0.40。代回得到 b = 0.20。一个苹果 0.40 英镑，一根香蕉 0.20 英镑。
7. Linear and Non-linear Systems | 线性与非线性方程组
On the IGCSE Extended syllabus, you may need to solve a system where one equation is linear and the other is quadratic. The substitution method is usually required, often leading to a quadratic equation that must be factorised or solved using the quadratic formula.
在 IGCSE Extended 课程中，你可能需要求解一个线性方程和一个二次方程组成的方程组。通常使用代入法，最终会得到一个需要用因式分解或二次公式求解的一元二次方程。
For example, solve:
例如，解：
y = x² − 4
y = 2x − 1
or x = −1. When x = 3, y = 5; when x = −1, y = −3. The solutions are (3, 5) and (−1, −3).
将 y = 2x − 1 代入 y = x² − 4，得到 x² − 4 = 2x − 1。移项得 x² − 2x − 3 = 0。因式分解得 (x − 3)(x + 1) = 0，所以 x = 3 或 x = −1。当 x = 3 时 y = 5；当 x = −1 时 y = −3。解为 (3, 5) 和 (−1, −3)。
Always check both solutions in the original equations because a quadratic system can produce two valid pairs.
一定要把两组解都代回原方程检验，因为二次方程组可能产生两个有效解。
8. Checking Solutions and Common Mistakes | 解的检验与常见错误
After solving, always substitute your values into both original equations, not only the one you used last. This catches sign errors, arithmetic slips, or copying mistakes.
解完之后，一定要把数值代入两个原始方程，而不仅仅是最后使用的那个方程。这样可以发现符号错误、算术失误或抄写错误。
Common mistakes in IGCSE simultaneous equations include:
IGCSE 联立方程中的常见错误包括：
- Forgetting to multiply every term when balancing coefficients. 在平衡系数时忘记对每一项都进行乘法。
- Subtracting incorrectly when the signs are the same or different. 当符号相同或不同时，减法出现错误。
- Losing a negative sign when substituting back. 回代时丢失负号。
- Not checking a graphical solution against the original equations. 没有将图像法得到的解代回原方程检验。
Writing your working clearly with each equation labelled will help you avoid these errors and gain method marks in exams.
清晰地写出每一步，并给每个方程做标注，有助于你避免这些错误，并在考试中获得过程分。
9. Worked Example: Exam-Style Question | 例题：考试风格
A typical IGCSE question might ask you to solve a system and then use your answer in a related context. Here is a complete worked example.
一道典型的 IGCSE 题目可能会要求你求解一个方程组，然后在相关情境中使用答案。以下是一个完整的解题范例。
2x + 3y = 12
x − y = 1
y + 2 = 12. Therefore 5y = 10 and y = 2. Then x = 2 + 1 = 3. The solution is x = 3, y = 2.
由第二个方程可得 x = y + 1。代入第一个方程：2(y + 1) + 3y = 12。化简得 2y + 2 + 3y = 12，即 5y + 2 = 12。因此 5y = 10，y = 2。然后 x = 2 + 1 = 3
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