📚 Solving Simultaneous Equations | 联立方程组
A linear equation in two unknowns, such as 2x + 3y = 12, has infinitely many solutions. For example, x = 3, y = 2 works, but so does x = 0, y = 4. A system of simultaneous equations combines two or more equations, and we aim to find the unique pair (or set) of values that satisfies every equation at the same time.
一个含有两个未知数的线性方程，例如 2x + 3y = 12，具有无穷多组解。例如，x = 3，y = 2 成立，但 x = 0，y = 4 也成立。联立方程组将两个或更多方程组合在一起，我们的目标是找到同时满足方程组中每一个方程的唯一一组（或多组）值。
In IGCSE Mathematics, you will be expected to solve these systems algebraically and graphically, and also to form equations from real-life contexts. A solid grasp of simultaneous equations strengthens many other areas, including coordinate geometry, curve sketching and problem-solving under time pressure.
在 IGCSE 数学中，你不仅需要用代数方法求解方程组，还要会用图像法求解，并能从实际生活情境中建立方程。扎实掌握联立方程组，对坐标几何、曲线作图以及限时解题等许多方面都有重要帮助。
1. What Are Simultaneous Equations | 什么是联立方程组
When two linear equations share two unknowns, they form a simultaneous system. We normally write them with one equation above the other and label them (1) and (2).
当两个线性方程含有两个共同的未知数时，它们就构成了一个联立方程组。我们通常把一个方程写在另一个上方，并标号为 (1) 和 (2)。
2x + 3y = 12 … (1)
x − y = 1 … (2)
The solution is the pair (x, y) that makes both equations true. Many students mistakenly solve each equation separately and then guess; instead, you must use elimination, substitution or graphing to consider both equations together.
解就是能使两个方程同时成立的那一组 (x, y)。许多同学会错误地分别解每个方程然后猜测；正确做法是用消元法、代入法或图像法，把两个方程放在一起考虑。
You should also know the special cases: if the lines are parallel, the system has no solution; if the two equations represent the same line, there are infinitely many solutions.
你还需要了解特殊情况：如果两条直线平行，方程组无解；如果两个方程表示同一条直线，则有无穷多组解。
2. Forming Equations from Worded Problems | 从文字题建立方程
The first step in many examination questions is translating English into algebra. Look for keywords: “sum” suggests addition, “difference” suggests subtraction, “twice” suggests multiplication by 2, and “is” usually becomes the equals sign.
解决许多应用题的第一步是把文字翻译成代数。注意关键词：”和”提示加法，”差”提示减法，”两倍”提示乘以 2，而”是”通常变成等号。
For example: “The sum of two numbers is 15 and their difference is 3.” Let the two numbers be x and y. Then we write:
例如：”两个数的和是 15，它们的差是 3。”设两个数分别为 x 和 y，则可得：
x + y = 15
x − y = 3
Adding the equations gives 2x = 18, so x = 9. Substituting back gives y = 6. Always state your answer clearly: the numbers are 9 and 6.
两式相加得 2x = 18，所以 x = 9。代回原式得 y = 6。务必清楚地写出答案：这两个数是 9 和 6。
3. The Elimination Method | 消元法
The elimination method is the most widely used technique for linear systems. Its goal is to cancel one unknown by adding or subtracting suitably aligned multiples of the equations.
消元法是线性方程组中最常用的方法，其核心目标是通过给方程乘以适当的倍数后相加或相减，消去一个未知数。
Let us solve the system:
我们来解下面的方程组：
2x + 3y = 12 … (1)
x − y = 1 … (2)
Choose to eliminate y. Multiply equation (2) by 3 to get 3x − 3y = 3. Now the coefficient of y in (1) is +3 and in the new equation it is −3. Add the two equations column by column:
我们选择消去 y。将方程 (2) 两边乘以 3，得到 3x − 3y = 3。此时方程 (1) 中 y 的系数为 +3，而新方程中为 −3。将两方程逐列相加：
(2x + 3y) + (3x − 3y) = 12 + 3
5x = 15
x = 3
Now substitute x = 3 into either original equation. Using x − y = 1 gives 3 − y = 1, so y = 2. The solution is x = 3, y = 2.
然后把 x = 3 代入任意一个原方程。用 x − y = 1 得 3 − y = 1，所以 y = 2。解为 x = 3，y = 2。
Always check your answer in both equations: 2(3) + 3(2) = 12 ✓ and 3 − 2 = 1 ✓.
记得把答案代入两个方程检验：2(3) + 3(2) = 12 ✓，3 − 2 = 1 ✓。
4. The Substitution Method | 代入法
Substitution is especially useful when one equation is already in the form y = … or x = …. We replace that unknown in the other equation, turning a two-variable system into a single-variable equation.
当一个方程已经是 “y = …” 或 “x = …” 的形式时，代入法尤为方便。我们将该未知数替换到另一个方程中，从而把含两个未知数的方程组化简为一个一元方程。
Solve the system:
解下列方程组：
y = 2x − 1 … (1)
x + 3y = 8 … (2)
Substitute (1) into (2):
将 (1) 代入 (2)：
x + 3(2x − 1) = 8
x + 6x − 3 = 8
7x = 11
x = 11/7
Then y = 2(11/7) − 1 = 22/7 − 7/7 = 15/7. So the solution is x = 11/7, y = 15/7.
再代入得 y = 2(11/7) − 1 = 22/7 − 7/7 = 15/7。因此解为 x = 11/7，y = 15/7。
When the answer is a fraction, leave it as a fraction in its simplest form; do not round unless the question asks for an approximation.
当答案是分数时，保留最简分数形式；除非题目要求近似值，否则不要四舍五入。
5. Clearing Fractions and Decimals First | 先处理分数与小数
Exam equations sometimes contain fractions, such as x/2 + y/3 = 3. The safest approach is to multiply every term by the lowest common multiple of the denominators before applying elimination or substitution.
考试中的方程有时含有分数，例如 x/2 + y/3 = 3。最稳妥的做法是先给每一项乘以分母的最小公倍数，把分数化为整数，再进行消元或代入。
Solve the system:
解下列方程组：
x/2 + y/3 = 3 … (1)
x/3 + y/2 = 2 … (2)
Multiply (1) by 6 and (2) by 6:
将 (1) 和 (2) 分别乘以 6：
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