Solving Linear Simultaneous Equations | 解二元一次方程组

📚 Solving Linear Simultaneous Equations | 解二元一次方程组

In IGCSE mathematics, simultaneous equations appear frequently in algebra and word problems. They allow you to find a single pair of values for two unknowns that satisfy two conditions at the same time. This article covers the elimination method, substitution, graphical interpretation, special cases, and exam-style questions.

在 IGCSE 数学中，方程组经常出现在代数和文字题中。它们帮助你求出同时满足两个条件的两个未知数的一组值。本文涵盖消元法、代入法、图像解释、特殊情况以及考试型例题。

1. What Are Simultaneous Equations? | 什么是方程组？

Simultaneous equations are a set of equations that share the same variables. A solution is a pair of values (x, y) that makes every equation true at the same time. For two linear equations in two unknowns, the solution gives the exact point where the two lines would meet on a graph.

方程组是一组共享相同变量的方程。解是使每一个方程同时成立的一组值 (x, y)。对于两个未知数的两个一次方程，解给出了两条直线在图像上相交的确切位置。

For example, the pair x + y = 5 and x − y = 1 has the solution x = 3, y = 2 because 3 + 2 = 5 and 3 − 2 = 1. The pair of values satisfies both equations.

例如，方程组 x + y = 5 和 x − y = 1 的解是 x = 3，y = 2，因为 3 + 2 = 5 且 3 − 2 = 1。这组值同时满足两个方程。

Linear simultaneous equations contain only variables raised to the power 1. You will not see x², y³, or xy in this topic. If higher powers appear, the system is nonlinear and different techniques are required.

二元一次方程组只包含次数为 1 的变量。在本主题中不会出现 x²、y³ 或 xy。如果出现更高次数，那就是非线性方程组，需要使用其他方法。

2. The Elimination Method | 消元法

The elimination method works by adding or subtracting equations to remove one variable. First, make the coefficients of one variable the same or opposite. Then add or subtract the equations and solve the remaining equation.

消元法通过将方程相加或相减来消去一个变量。首先，使其中一个变量的系数相同或互为相反数。然后将方程相加或相减，求解剩余的一个方程。

Example 1: Solve the system

例 1：解方程组

2x + y = 7

x − y = 2

Add the two equations: (2x + y) + (x − y) = 7 + 2, so 3x = 9 and x = 3. Substitute x = 3 into x − y = 2 to get 3 − y = 2, so y = 1.

将两个方程相加：(2x + y) + (x − y) = 7 + 2，得到 3x = 9，因此 x = 3。将 x = 3 代入 x − y = 2，得到 3 − y = 2，所以 y = 1。

Sometimes you must multiply one or both equations first. If the coefficients do not match, choose the least common multiple. This makes the coefficients equal before you add or subtract.

有时需要先将一个或两个方程乘以适当的数。如果系数不匹配，可以选择最小公倍数。这样就能在相加或相减之前使系数相等。

Example 2: Solve 3x + 2y = 12 and 2x + 5y = 19.

例 2：解方程组 3x + 2y = 12 和 2x + 5y = 19。

Multiply the first equation by 2: 6x + 4y = 24. Multiply the second equation by 3: 6x + 15y = 57. Subtract the first result from the second: 11y = 33, so y = 3. Substitute y = 3 into 3x + 2y = 12 to get 3x + 6 = 12, so x = 2.

将第一个方程乘以 2：6x + 4y = 24。将第二个方程乘以 3：6x + 15y = 57。用第二个结果减去第一个结果：11y = 33，所以 y = 3。将 y = 3 代入 3x + 2y = 12，得到 3x + 6 = 12，因此 x = 2。

Elimination is usually the fastest method when both equations are in standard form ax + by = c. It reduces the system to one simple linear equation.

当两个方程都是标准形式 ax + by = c 时，消元法通常是最快的方法。它把方程组简化为一个简单的一次方程。

3. The Substitution Method | 代入法

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