📚 Mastering Simultaneous Equations: Elimination and Substitution | 精通二元一次方程组：消元法与代入法
Simultaneous equations are a cornerstone of the IGCSE Mathematics syllabus. They require you to find values for two or more unknowns that satisfy two or more equations at the same time. In this article, we will explore two algebraic methods — elimination and substitution — and connect them to the graphical idea of intersecting lines. You will see worked examples, common pitfalls, and exam-style questions to build both confidence and accuracy.
二元一次方程组是 IGCSE 数学大纲中的核心内容。它要求你同时求出满足两个或多个方程的两个或多个未知数的值。本文将讲解两种代数方法——消元法与代入法——并把它们与直线交点的图形概念联系起来。你将看到典型例题、常见错误和考试型练习，以提升解题信心与准确度。
1. What Are Simultaneous Equations? | 什么是二元一次方程组？
A simultaneous equation system is a set of two or more equations that share the same variables. In IGCSE, you will usually work with two linear equations in two unknowns, often written as x and y. A solution is a pair of values (x, y) that makes both equations true at the same time.
联立方程组是由两个或多个含有相同变量的方程组成的集合。在 IGCSE 中，你通常会遇到两个关于两个未知数（通常写作 x 和 y）的线性方程。方程组的解就是能同时使两个方程都成立的一组值 (x, y)。
system below has the solution x = 3 and y = 1 because these values satisfy both equations.
例如，下列方程组的解为 x = 3，y = 1，因为这两个值能使两个方程同时成立。
2x + y = 7
x - y = 2
Check: 2(3) + 1 = 7 and 3 - 1 = 2
2. The Elimination Method | 消元法
Elimination works by adding or subtracting the equations to remove one variable. The goal is to create a single equation in one unknown, which can then be solved directly. Before adding or subtracting, you may need to multiply one or both equations by a constant so that the coefficients of one variable are equal or opposite.
消元法通过将两个方程相加或相减来消去一个未知数。目标是得到一个只含一个未知数的方程，然后直接求解。在相加或相减之前，你可能需要给一个或两个方程乘以某个常数，使其中一个未知数的系数相等或互为相反数。
- y) = 7 + 2, which gives 3x = 9. Solve to get x = 3, then substitute back to find y = 1.
对于方程组 2x + y = 7 和 x - y = 2，将两个方程相加：(2x + y) + (x - y) = 7 + 2，得到 3x = 9。解得 x = 3，再代回原方程求出 y = 1。
2x + y = 7
x - y = 2
Add: 3x = 9 → x = 3
Substitute: 2(3) + y = 7 → y = 1
3. Worked Example: Elimination with Multiplication | 消元法进阶例题
Add to the first: 13x = 26, so x = 2. Substitute into 3x + 2y = 12: 6 + 2y = 12, so y = 3.
当系数不匹配时，需要给一个或两个方程乘以适当的数。例如方程组 3x + 2y = 12 和 5x - y = 7。把第二个方程乘以 2，得到 10x - 2y = 14。与第一个方程相加：13x = 26，所以 x = 2。再代入 3x + 2y = 12：6 + 2y = 12，因此 y = 3。
3x + 2y = 12
5x - y = 7
Second × 2: 10x - 2y = 14
Add: 13x = 26 → x = 2
Then 3(2) + 2y = 12 → 2y = 6 → y = 3
4. The Substitution Method | 代入法
Substitution is most useful when one equation is already solved for one variable, or can be rearranged easily. Replace that variable in the other equation, then solve the resulting single-variable equation.
当一个方程已经写成某个变量的表达式，或者很容易变形时，代入法最为有用。把该变量代入另一个方程，然后求解得到的一元方程。
ond equation: 3x + 2(2x + 1) = 16. Expand to 7x + 2 = 16, so x = 2. Then y = 2(2) + 1 = 5.
对于 y = 2x + 1 和 3x + 2y = 16，将 y 代入第二个方程：3x + 2(2x + 1) = 16。展开得 7x + 2 = 16，所以 x = 2。然后 y = 2(2) + 1 = 5。
y = 2x + 1
3x + 2y = 16
3x + 2(2x + 1) = 16
7x + 2 = 16 → x = 2
y = 2(2) + 1 = 5
5. Worked Example: Substitution with Rearranging | 代入法变形例题
2y - y = 5, giving -13y = -35, y = 35/13. Then x = 10 - 3(35/13) = (130 - 105)/13 = 25/13.
如果两个方程都没有直接写出某个变量的表达式，需要先变形。对于 x + 3y = 10 和 4x - y = 5，将第一个方程变形为 x = 10 - 3y。代入第二个方程得到 4(10 - 3y) - y = 5，即 40 - 12y - y = 5，得到 -13y = -35，y = 35/13。然后 x = 10 - 3(35/13) = (130 - 105)/13 = 25/13。
x + 3y = 10 → x = 10 - 3y
4(10 - 3y) - y = 5
-13y = -35 → y = 35/13
x = 25/13
6. Graphical Interpretation | 图形解释
lines 2x + y = 7 and x - y = 2 intersect at (3, 1), which matches the algebraic solution.
每个线性方程在坐标平面上表示一条直线。方程组的解就是两条直线相交的点。例如，直线 2x + y = 7 和 x - y = 2 相交于点 (3, 1)，这与代数解一致。
You can verify a solution graphically by plotting both lines and reading the intersection coordinates. This also explains why some systems have no solution or infinitely many solutions — the lines may be parallel or coincident.
你可以通过画出两条直线并读取交点坐标来验证解。这也解释了为什么有些方程组无解或有无穷多解——因为两条直线可能平行或重合。
7. Special Cases: No Solution and Infinite Solutions |
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