📚 Simultaneous Equations: Linear and Quadratic Systems | 联立方程：线性与二次方程组
In IGCSE Mathematics, simultaneous equations ask you to find values that satisfy two or more equations at the same time. This topic connects algebra, graphs and problem solving, and it appears in both Core and Extended papers.
在 IGCSE 数学中，联立方程要求你求出同时满足两个或更多方程的值。这个主题将代数、图像和问题求解联系起来，在核心卷和扩展卷中都会出现。
1. What Are Simultaneous Equations? | 什么是联立方程？
A simultaneous equation system is a set of equations that share the same variables. A solution must satisfy every equation in the system.
联立方程组是一组共用相同变量的方程。解必须满足方程组中的每一个方程。
ltaneous in x and y. The pair x = 3, y = 2 satisfies both because 3 + 2 = 5 and 3 − 2 = 1.
例如，x + y = 5 和 x − y = 1 是关于 x 和 y 的联立方程。x = 3，y = 2 满足两个方程，因为 3 + 2 = 5 且 3 − 2 = 1。
In IGCSE, you usually meet two main types: two linear equations, and one linear equation with one quadratic equation.
在 IGCSE 中，你通常会遇到两种主要类型：两个线性方程，以及一个线性方程与一个二次方程。
2. The Elimination Method for Linear Systems | 线性方程组的消元法
Elimination is often the quickest method when both equations are linear. The idea is to add or subtract the equations so that one variable disappears.
当两个方程都是线性方程时，消元法通常是最快的方法。其思路是让两个方程相加或相减，使其中一个变量被消去。
Worked example: solve the system below.
例题：解下面的方程组。
2x + 3y = 13
4x − y = 5
First, multiply the second equation by 3 so that the y coefficients are 3 and −3.
首先，将第二个方程乘以 3，使 y 的系数变为 3 和 −3。
2x + 3y = 13
12x − 3y = 15
the two equations: 14x = 28, so x = 2. Substitute x = 2 into the first equation to find y.
现在将两个方程相加：14x = 28，因此 x = 2。将 x = 2 代入第一个方程求出 y。
2(2) + 3y = 13
4 + 3y = 13
3y = 9
y = 3
is x = 2, y = 3. Always check both original equations: 2(2) + 3(3) = 13 and 4(2) − 3 = 5.
解为 x = 2，y = 3。一定要代回两个原方程检验：2(2) + 3(3) = 13，且 4(2) − 3 = 5。
3. The Substitution Method | 代入法
Substitution works well when one equation has a single variable on its own or a variable with coefficient 1. You rearrange one equation and replace that variable in the other equation.
当一个方程中有一个变量单独出现或系数为 1 时，代入法非常有效。你先变形一个方程，再把该变量代入另一个方程。
Worked example: solve x + 2y = 8 and 2x − y = 1.
例题：解 x + 2y = 8 和 2x − y = 1。
From the first equation, x = 8 − 2y. Substitute this into the second equation.
由第一个方程得 x = 8 − 2y。将它代入第二个方程。
2(8 − 2y) − y = 1
16 − 4y − y = 1
−5y = −15
y = 3
substitute y = 3 back into x = 8 − 2y, giving x = 8 − 6 = 2. The solution is x = 2, y = 3.
现在将 y = 3 代回 x = 8 − 2y，得到 x = 8 − 6 = 2。解为 x = 2，y = 3。
4. Graphical Interpretation | 图像解释
Each linear equation in two variables represents a straight line. Solving simultaneous linear equations is the same as finding the point where the two lines intersect.
每个含有两个变量的线性方程都表示一条直线。解线性联立方程就相当于找出两条直线的交点。
If the lines intersect at one point, there is one unique solution.
如果两条直线相交于一点，则有一个唯一解。
If the lines are parallel, there is no solution.
如果两条直线平行，则无解。
If the lines coincide, there are infinitely many solutions.
如果两条直线重合，则有无数个解。
| System | Graph | Number of solutions |
| Different gradients | Intersecting lines | 1 |
| Same gradient, different intercept | Parallel lines | 0 |
| Same gradient, same intercept | Coincident lines | Infinite |
In exam questions, you may be asked to solve a system algebraically and then identify the intersection point on a graph.
在考试题中，你可能需要先用代数方法解方程组，然后在图像上标出交点。
5. One Linear and One Quadratic Equation | 一个线性方程与一个二次方程
When one equation is linear and the other is quadratic, substitution is usually the best approach. Replace the linear expression into the quadratic equation to form a single quadratic in one variable.
当一个方程是线性方程、另一个是二次方程时，代入法通常是最佳方法。将线性表达式代入二次方程，形成一个只含一个变量的二次方程。
Worked example: solve y = x + 1 and x² + y² = 25.
例题：解 y = x + 1 和 x² + y² = 25。
Substitute y = x + 1 into x² + y² = 25.
将 y = x + 1 代入 x² + y² = 25。
x² + (x + 1)² = 25
x² + x² + 2x + 1 = 25
2x² + 2x − 24 = 0
x² + x − 12 = 0
Factorise: (x + 4)(x − 3) = 0, so x = −4 or x = 3. Now find the matching y values.
因式分解：(x + 4)(x − 3) = 0，所以 x = −4 或 x = 3。现在求出对应的 y 值。
x = −4, y = −4 + 1 = −3. For x = 3, y = 3 + 1 = 4. The solutions are (−4, −3) and (3, 4).
当 x = −4 时，y = −4 + 1 = −3。当 x = 3 时，y = 3 + 1 = 4。解为 (−4, −3) 和 (3, 4)。
This type of system can have two solutions, one solution, or no real solutions, depending on how the line meets the curve.
这类方程组可以有两个解、一个解或无实数解，具体取决于直线与曲线的交点情况。
6. Using the Discriminant to Predict Solutions | 用判别式判断解的个数
ic of the form ax² + bx + c = 0. The discriminant tells you how many real solutions exist.
将线性方程代入二次方程后，你会得到一个形如 ax² + bx + c = 0 的二次方程。判别式可以告诉你存在多少个实数解。
D = b² − 4ac
If D > 0, there are two real solutions; the line cuts the curve twice.
如果 D > 0，则有两个实数解；直线与曲线相交两次。
If D = 0, there is one repeated real solution; the line is a tangent.
如果 D = 0，则有一个重复的实数解；直线是曲线的切线。
If D < 0, there are no real solutions; the line does not meet the curve.
如果 D < 0，则没有实数解；直线与曲线不相交。
a = 1, b = 2, c = −1, so D = 2² − 4(1)(−1) = 8. Since D > 0, there are two real solutions.
例如，解 y = x + 2 和 y = x² + 3x + 1 得到 x + 2 = x² + 3x + 1，因此 x² + 2x − 1 = 0。这里 a = 1，b = 2，c = −1，所以 D = 2² − 4(1)(−1) = 8。因为 D > 0，所以有两个实数解。
7. Word Problems Leading to Simultaneous Equations | 联立方程应用题
Many IGCSE questions hide simultaneous equations in real-life contexts. Start by defining clear variables, then translate each sentence into an equation.
许多 IGCSE 题目将联立方程隐藏在现实情境中。先明确设出变量，然后把每一句话转化为一个方程。
Example: 3 adult tickets and 2 child tickets cost $31. 2 adult tickets and 5 child tickets cost $39. Find the cost of one adult ticket and one child ticket.
例题：3 张成人票和 2 张儿童票共 31
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