📚 Simultaneous Equations: Methods and Applications | 联立方程：方法与应用
Simultaneous equations are a group of equations that share the same unknowns and must all be satisfied at the same time. In IGCSE Mathematics, you will usually meet pairs of equations in two unknowns, x and y, and you need to find the exact values that make both equations true.
联立方程是指含有相同未知数且必须同时成立的一组方程。在 IGCSE 数学中，通常会遇到含有两个未知数 x 和 y 的方程对，需要求出使两个方程同时成立的精确值。
1. What Are Simultaneous Equations? | 什么是联立方程？
nfinitely many pairs of solutions because you can choose any x and calculate a matching y.
像 2x + y = 7 这样的单个方程有无穷多组解，因为可以任选一个 x 计算出对应的 y。
e possible pairs are restricted. The solution is the point where the two conditions agree.
当第二个方程（例如 y = 3x + 1）也必须成立时，可行解就被限制了。这个解就是两个条件同时满足的点。
In IGCSE questions, simultaneous equations often arise in number puzzles, money problems, coordinates, and later in quadratic settings.
在 IGCSE 题目中，联立方程常出现在数字谜题、货币问题、坐标问题以及稍后二次方程的情境中。
2. The Elimination Method | 消元法
Elimination works by adding or subtracting the two equations so that one unknown cancels out. First, make the coefficient of one variable equal in size in both equations.
消元法是将两个方程相加或相减，使其中一个未知数被抵消。首先，使某个变量在两个方程中的系数大小相等。
2, the y coefficients are already +1 and -1. Adding the equations gives 4x = 12, so x = 3.
例如，给定 3x + y = 10 和 x - y = 2，y 的系数已经是 +1 和 -1。将两个方程相加得到 4x = 12，所以 x = 3。
Substitute x = 3 back into either original equation to find y = 1. The solution is (3, 1).
将 x = 3 代回任一原方程可求出 y = 1。解为 (3, 1)。
acting. For 2x + 3y = 12 and 4x - 3y = 6, add directly to get 6x = 18, so x = 3 and y = 2.
如果系数不匹配，可以在相加或相减前先给一个或两个方程乘以适当的数。对于 2x + 3y = 12 和 4x - 3y = 6，直接相加得 6x = 18，所以 x = 3，y = 2。
3. The Substitution Method | 代入法
is already written as y = ... or x = .... You replace that variable in the other equation.
当一个方程已经写成 y = ... 或 x = ... 时，代入法非常有用。你将这个变量替换到另一个方程中。
2x + 1 and 3x + 2y = 16, substitute y into the second equation to get 3x + 2(2x + 1) = 16.
对于 y = 2x + 1 和 3x + 2y = 16，将 y 代入第二个方程得到 3x + 2(2x + 1) = 16。
Simplify to 7x + 2 = 16, giving x = 2, and then y = 5. The solution is (2, 5).
化简得 7x + 2 = 16，因此 x = 2，再得 y = 5。解为 (2, 5)。
uation is written as x = 5 - 2y, substitute into 2x + 3y = 8 to obtain 2(5 - 2y) + 3y = 8.
如果第一个方程写成 x = 5 - 2y，代入 2x + 3y = 8 得到 2(5 - 2y) + 3y = 8。
Expanding gives 10 - 4y + 3y = 8, then -y = -2, so y = 2 and x = 1.
展开得 10 - 4y + 3y = 8，然后 -y = -2，所以 y = 2，x = 1。
Substitution is also the main method for solving one linear and one quadratic equation.
代入法也是求解一个一次方程与一个二次方程的主要方法。
4. Graphical Interpretation | 图像解释
Each linear equation represents a straight line on the coordinate plane. Solving simultaneous linear equations is equivalent to finding the intersection point of the two lines.
每个一次方程在坐标平面上表示一条直线。求解联立一次方程等价于找到两条直线的交点。
If the lines intersect at one point, there is exactly one solution. In IGCSE graph questions, you may read the solution directly from a graph or plot lines accurately.
如果两条直线相交于一点，则有唯一解。在 IGCSE 图像题中，可以直接从图像中读取解，或准确地绘制直线。
= -x + 5, the graph shows intersection at (2, 3), which should agree with algebraic work.
对于方程组 y = 2x - 1 和 y = -x + 5，图像显示交点为 (2, 3)，应与代数计算一致。
Graphical solutions are especially useful for checking that an algebraic solution is reasonable and for understanding what a
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