📚 Solving Simultaneous Equations | 解联立方程组
Simultaneous equations are a core topic in IGCSE Mathematics. They test your ability to find values of two or more unknown variables that satisfy multiple equations at the same time. Mastering this topic is essential for Paper 2 and Paper 4, and it also appears in extended problem-solving questions across other subjects.
联立方程组是IGCSE数学的核心考点之一。它考察你能否在多个方程同时成立的条件下求出未知数的值。掌握这个主题对Paper 2和Paper 4至关重要，也常常出现在跨学科的综合应用题中。
1. What Are Simultaneous Equations? | 什么是联立方程组
Simultaneous equations are a set of equations that share the same unknown variables. For example, the two equations 2x + y = 7 and x − y = 2 form a pair of simultaneous equations. To solve them, we must find the values of x and y that make both equations true at the same time.
联立方程组是指一组共享相同未知数的方程。例如，2x + y = 7 和 x − y = 2 这两个方程就组成一个联立方程组。解这个方程组，就需要找出同时使两个方程成立的 x 和 y 的值。
There are two main situations you will encounter: linear-linear systems (both equations are straight lines) and non-linear systems (one equation is linear and the other is quadratic, such as y = x²).
你会遇到两种主要情况：线性-线性系统（两个方程都是直线方程）和非线性系统（一个方程是线性的，另一个是二次方程，如 y = x²）。
2. The Elimination Method | 消元法
The elimination method involves adding or subtracting the equations to remove one variable. First, make sure the coefficients of one variable are the same (or opposites) in both equations.
消元法是通过将两个方程相加或相减来消去一个未知数。首先，需要确保某一个未知数的系数在两个方程中相同（或互为相反数）。
Example: Solve the system
3x + 2y = 12
5x − 2y = 4
示例：解方程组
3x + 2y = 12
5x − 2y = 4
Notice that the coefficients of y are +2 and −2, which are opposites. Add the two equations:
注意 y 的系数是 +2 和 −2，互为相反数。将两个方程相加：
(3x + 2y) + (5x − 2y) = 12 + 4
8x = 16
x = 2
Now substitute x = 2 into the first equation:
3(2) + 2y = 12
6 + 2y = 12
2y = 6
y = 3
然后把 x = 2 代入第一个方程：
3(2) + 2y = 12
6 + 2y = 12
2y = 6
y = 3
So the solution is x = 2 and y = 3. You can check by substituting both values into the second equation: 5(2) − 2(3) = 10 − 6 = 4 ✓
所以解为 x = 2，y = 3。你可以将两个值代入第二个方程验证：5(2) − 2(3) = 10 − 6 = 4 ✓
If the coefficients are not the same, multiply one or both equations by a constant before adding or subtracting.
如果系数不相同，则需要先将一个或两个方程乘以适当的常数，再进行相加或相减。
3. The Substitution Method | 代入法
The substitution method is best used when one equation is already in the form y = … or x = … . You replace the variable in the other equation with the expression given.
代入法最适合于某个方程已经变形为 y = … 或 x = … 形式的情况。你将另一个方程中的未知数替换为这个表达式。
Example: Solve the system
y = 2x + 1
3x + y = 11
示例：解方程组
y = 2x + 1
3x + y = 11
Substitute y = 2x + 1 into the second equation:
3x + (2x + 1) = 11
5x + 1 = 11
5x = 10
x = 2
将 y = 2x + 1 代入第二个方程：
3x + (2x + 1) = 11
5x + 1 = 11
5x = 10
x = 2
Then substitute back: y = 2(2) + 1 = 5. So the solution is x = 2, y = 5.
然后代回求 y：y = 2(2) + 1 = 5。因此解为 x = 2，y = 5。
A useful tip: after finding one variable, always substitute into one of the original equations rather than an equation you have manipulated, to avoid repeating an error.
一个实用技巧：求出一个未知数后，应代入原始方程（而不是你重新整理过的方程），以免将同样的错误重复两次。
4. The Graphical Method | 图像法
Graphically, the solution to a pair of linear simultaneous equations is the coordinates of the point where the two lines intersect. You need to draw both straight lines accurately on the same set of axes.
从图像上看，两个线性方程的解就是两条直线交点的坐标。你需要在同一坐标系中准确画出两条直线。
Example: Draw y = 2x + 1 and y = −x + 4. The intersection point is where x = 1 and y = 3. Verify: 3 = 2(1) + 1 ✓ and 3 = −(1) + 4 ✓
示例：画出 y = 2x + 1 和 y = −x + 4。两条直线的交点就是 x = 1，y = 3。验证：3 = 2(1) + 1 ✓，且 3 = −(1) + 4 ✓
When drawing a line, choose at least three points to avoid mistakes. For example, for y = 2x + 1, you can use x = 0, 1, 2 and plot the points (0,1), (1,3), (2,5).
画直线时至少要选取三个点以避免误差。例如，对于 y = 2x + 1，可取 x = 0、1、2，描点 (0,1)、(1,3)、(2,5)。
If the lines are parallel, the system has no solution. If the two equations represent the same line, there are infinitely many solutions.
如果两条直线平行，则方程组无解；如果两个方程表示同一条直线，则有无穷多组解。
5. Solving Word Problems | 解应用题
For word problems, convert the information into two equations using two variables. Usually, one equation represents a total, and the other represents a difference, ratio, or relationship.
对于应用题，先用两个未知数将题目信息转化为两个方程。通常，一个方程表示总量关系，另一个表示差、比例或其他关系。
Example: The sum of two numbers is 10 and their difference is 4. Find the numbers.
示例：两个数之和是10，差是4。求这两个数。
Let x be the larger number and y the smaller. Then:
x + y = 10
x − y = 4
设较大的数为 x，较小的数为 y。则：
x + y = 10
x − y = 4
Adding gives 2x = 14, so x = 7. Substituting back: 7 + y = 10, so y = 3. The numbers are 7 and 3.
两式相加得 2x = 14，所以 x = 7。代回：7 + y = 10，所以 y = 3。这两个数分别是7和3。
Always define your variables clearly at the start of a word problem, and write a final sentence with the answer in context.
在做应用题时，务必在开头清晰地定义未知数，并在最后用一句话写出符合题意的答案。
6. Non-Linear Simultaneous Equations | 非线性联立方程组
IGCSE Extended candidates must also solve one linear and one quadratic equation simultaneously. The substitution method is almost always used.
IGCSE Extended（拓展课程）考生还需要解一个线性方程与一个二次方程组成的联立方程组。此时几乎总是使用代入法。
Example: Solve
y = x²
y = x + 2
示例：解
y = x²
y = x + 2
Since y = x² and y = x + 2, we can equate the two expressions:
x² = x + 2
x² − x − 2 = 0
因为 y = x² 且 y = x + 2，所以将两个表达式联立相等：
x² = x + 2
x² − x − 2 = 0
Factorise: (x − 2)(x + 1) = 0, so x = 2 or x = −1.
因式分解得：(x − 2)(x + 1) = 0，所以 x = 2 或 x = −1。
Now find the corresponding y-values. When x = 2, y = 2² = 4. When x = −1, y = (−1)² = 1.
再求出对应的 y 值。当 x = 2 时，y = 2² = 4；当 x = −1 时，y = (−1)² = 1。
Therefore the solutions are (2,4) and (−1,1). Graphically, these are the two intersection points of the parabola and the straight line.
因此解为 (2,4) 和 (−1,1)。在图像上，这两个点就是抛物线与直线的两个交点。
If the resulting quadratic has a negative discriminant (b² − 4ac < 0), then there are no real solutions, meaning the line and the curve do not intersect.
如果得到的二次方程判别式为负（b² − 4ac < 0），则没有实数解，说明直线与曲线不相交。
7. The Discriminant and the Number of Solutions | 判别式与解的个数
When solving a linear and a quadratic equation together, you substitute to obtain a quadratic equation. The discriminant Δ = b² − 4ac tells you how many intersection points exist:
当联立一个一次方程与一个二次方程时，代入后会得到一个一元二次方程。判别式 Δ = b² − 4ac 可以告诉你交点的个数：
| Δ = b² − 4ac | Number of real solutions | 图形含义 |
| Δ > 0 | 2 solutions | 直线与曲线相交于两点 |
| Δ = 0 | 1 solution (tangent) | 直线与曲线相切于一点 |
| Δ < 0 | 0 solutions | 直线与曲线无交点 |
8. Common Mistakes and Exam Tips | 常见错误与考试技巧
Here are the most common errors students make in the exam, and how to avoid them.
以下是考生在考试中最常见的错误以及相应的避免方法。
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Sign errors when subtracting equations. If you subtract one equation from another, changing the sign of every term in the second equation is a reliable strategy. | 在相减时出现符号错误。两式相减时，应先将第二个方程中所有项的符号改变，再与第一式相加，这样最可靠。
-
Forgetting to substitute the found value into an original equation. | 忘记将求出的值代入原方程求另一个未知数。
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Not checking the answer. Plug both values into both equations to verify. | 解完不检查。应将两个值分别代入两个方程进行验证。
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