📚 Solving Simultaneous Equations | 解联立方程组
Simultaneous equations are a cornerstone of IGCSE Mathematics. They appear in nearly every exam paper, either as a direct algebraic question or as part of a word problem. Mastering them will earn you reliable marks across multiple topics.
联立方程是 IGCSE 数学的基石，几乎每张考卷都会出现，既可能直接考查代数解法，也可能出现在应用题中。掌握联立方程，能帮助你在多个专题中稳定拿分。
1. What Are Simultaneous Equations? | 什么是联立方程？
A simultaneous equation system involves two variables, usually x and y, and two separate equations. A solution is an ordered pair (x, y) that makes both equations true at the same time.
联立方程组含有两个未知数（通常为 x 和 y）以及两个独立方程。解就是一组有序数对 (x, y)，它能同时使两个方程成立。
For example, consider the system:
例如，考虑以下方程组：
2x + y = 7
x - y = 2
3 and y = 1 into both equations confirms the solution, because 2(3) + 1 = 7 and 3 - 1 = 2.
将 x = 3、y = 1 代入两个方程即可验证：2(3) + 1 = 7，且 3 - 1 = 2。
2. The Substitution Method | 代入消元法
Substitution is ideal when one variable has a coefficient of 1. Follow these steps:
当某个未知数的系数为 1 时，代入消元法尤为方便。步骤如下：
- Rearrange one equation to make x or y the subject.
- 将其中一个方程变形，用另一个未知数表示 x 或 y。
- Substitute this expression into the other equation.
- 把该表达式代入另一个方程。
- Solve the resulting single-variable linear equation.
- 解出这个一元一次方程。
- Back-substitute to find the second variable.
- 回代求出另一个未知数。
Worked example: Solve y = 2x - 1 and 3x + 2y = 12.
例：解方程组 y = 2x - 1 与 3x + 2y = 12。
Since y is already the subject, substitute y = 2x - 1 into 3x + 2y = 12:
由于 y 已单独表示，将 y = 2x - 1 代入 3x + 2y = 12：
3x + 2(2x - 1) = 12
3x + 4x - 2 = 12
7x = 14, so x = 2
Then y = 2(2) - 1 = 3. The solution is x = 2, y = 3.
于是 y = 2(2) - 1 = 3，解为 x = 2，y = 3。
3. The Elimination Method | 加减消元法
x + by = c. The goal is to add or subtract the equations so that one variable cancels out.
当两个方程都形如 ax + by = c 时，加减消元法非常有效。核心目标是通过相加或相减，使某个未知数消去。
- Make the coefficients of one variable equal, using multiplication if necessary.
- 必要时通过乘法，使某个未知数的系数相等。
- Add or subtract the equations to eliminate that variable.
- 将两式相加或相减，消去该未知数。
- Solve for the remaining variable, then back-substitute.
- 解出剩下的未知数，再回代求解。
Worked example: Solve 2x + 3y = 13 and 5x + 3y = 22.
例：解方程组 2x + 3y = 13 与 5x + 3y = 22。
Both equations have the same 3y coefficient, so subtract the first equation from the second:
两式中 3y 的系数相同，因此用第二式减去第一式：
(5x - 2x) + (3y - 3y) = 22 - 13
3x = 9, so x = 3
Substitute x = 3 into 2x + 3y = 13: 6 + 3y = 13, hence 3y = 7 and y = 7/3.
将 x = 3 代入 2x + 3y = 13：6 + 3y = 13，故 3y = 7，y = 7/3。
4. Choosing the Right Method | 如何选择合适的方法
Both methods always work, but certain question types favour one over the other.
两种方法都普遍适用，但不同题型各有偏好。
| Scenario 题型 | Recommended Method 推荐方法 | Reason 理由 |
| One equation has y = or x = 某个方程已写成 y = 或 x = | Substitution 代入法 | Avoids extra rearranging 省去多余变形 |
| Coefficients already match 同类项系数已相同 | Elimination 加减消元法 | One step gives the answer 一步即可得解 |
| Fractions or decimals appear 含分数或小数 | Clear denominators first 先化为整数系数 | Simplifies all arithmetic 简化全部计算 |
Whichever method you choose, show full working. Method marks are often awarded even for a small arithmetic slip.
无论选择哪种方法，都要写出完整过程。即使最终计算有误，步骤分也常常可以拿到。
5. Word Problems | 联
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