📚 Solving Simultaneous Equations | 联立方程的解法
A simultaneous equation is a set of two or more equations that share the same unknown variables. When we solve them, we look for a pair (or more) of values that makes every equation true at the same time. This is one of the most examined topics in IGCSE Mathematics, appearing in both Paper 1 and Paper 2.
联立方程是一组共享相同未知数的两个或更多方程。解联立方程时，我们寻找一组能同时满足所有方程的值。这是 IGCSE 数学中最常考的考点之一，在 Paper 1 和 Paper 2 中都会出现。
1. What Are Simultaneous Equations? | 什么是联立方程
For example, consider the system below:
2x + y = 7
x − y = 2
ime: 2(3) + 1 = 7 and 3 − 1 = 2. A solution to a simultaneous equation system must satisfy all equations simultaneously.
方程组的解是 x = 3，y = 1，因为两个方程同时成立：2(3) + 1 = 7，且 3 − 1 = 2。联立方程组的解必须同时满足所有方程。
At IGCSE level you are expected to solve linear systems with two unknowns, and occasionally a system where one equation is linear and the other is quadratic. You should also be able to interpret solutions graphically.
在 IGCSE 阶段，你需要求解含两个未知数的线性方程组，偶尔还会遇到一个线性方程加一个二次方程的方程组。你还应能通过图像解释解的意义。
2. The Substitution Method | 代入消元法
The substitution method works best when one equation is already written with a single variable on one side, or when it can be rearranged easily.
代入消元法适用于其中一个方程已经写成一侧只有一个变量的形式，或可以轻松重排的情况。
- Rearrange one equation to make x or y the subject.
- Substitute this expression into the other equation.
- Solve the resulting one-variable equation.
- Substitute back to find the other variable.
- 重排其中一个方程，用另一个变量表示 x 或 y。
- 将该表达式代入另一个方程。
- 解所得的一元方程。
- 代回求出另一个变量。
Worked Example: Solve the system
y = x + 3
2x + y = 15
Since y is already the subject, substitute y = x + 3 into the second equation:
2x + (x + 3) = 15
3x + 3 = 15
3x = 12
x = 4
Then y = 4 + 3 = 7. The solution is x = 4, y = 7. Always check: 2(4) + 7 = 15 ✓
因为 y 已经是主项，将 y = x + 3 代入第二个方程：2x + (x + 3) = 15，即 3x + 3 = 15，解得 3x = 12，所以 x = 4。于是 y = 4 + 3 = 7。解为 x = 4，y = 7。务必检验：2(4) + 7 = 15 ✓
3. The Elimination Method | 加减消元法
The elimination method isappears. It is often the fastest method when both equations are in the form Ax + By = C.
加减消元法通过将两个方程相加或相减，使其中一个变量消去。当两个方程都写成 Ax + By = C 的形式时，这通常是最快的方法。
- Arrange both equations in the form Ax + By = C.
- Multiply one or both equations so that the coefficients of x or y match in size.
- Add or subtract to eliminate that variable.
- Solve for the remaining variable, then substitute back.
- 将两个方程整理成 Ax + By = C 的形式。
- 将其中一个或两个方程乘以适当的数，使 x 或 y 的系数绝对值相等。
- 相加或相减以消去该变量。
- 解出剩余变量，再代回求另一个变量。
Worked Example: Solve
3x + 2y = 12
2x + 2y = 8
Both equations have the same y coefficient, 2. Subtract the second equation from the first:
(3x + 2y) − (2x + 2y) = 12 − 8
x = 4
x = 4 into 2x + 2y = 8: 8 + 2y = 8, so 2y = 0, hence y = 0. The solution is x = 4, y = 0.
两个方程的 y 系数相同，都是 2。用第一个方程减去第二个方程：x = 4。将 x = 4 代入 2x + 2y = 8：8 + 2y = 8，故 2y = 0，即 y = 0。解为 x = 4，y = 0。
5 and x − 2y = 0, multiply the second equation by 2 to obtain 2x − 4y = 0, then subtract.
当系数不同时，先乘以适当的数。例如，要从 2x + y = 5 和 x − 2y = 0 中消去 x，可将第二个方程乘以 2，得到 2x − 4y = 0，然后相减。
4. Graphical Interpretation | 图像解释
Each linear equation represents a straight line on the x-y plane. The simultaneous solution is the point where the two lines intersect, i.e. the point whose coordinates satisfy both equations.
每个线性方程在 x-y 平面上代表一条直线。联立方程组的解就是两条直线的交点，即其坐标同时满足这两个方程的点。
cept and y-intercept. For example, the line 2x + y = 7 passes through (0, 7) and (3.5, 0).
要快速画出直线，可将方程重排为 y = mx + c 的形式，或者求出 x 截距和 y 截距。例如，直线 2x + y = 7 经过 (0, 7) 和 (3.5, 0) 两点。
When solving by graph, draw both lines accurately using the same scale, then read the intersection point. The coordinates of the intersection give x and y.
用图像法求解时，需用同一比例尺准确画出两条直线，然后读出交点坐标。交点的横纵坐标就是 x 和 y 的值。
5. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无穷多解
No solution: If the two lines are parallel, they never meet, so the system has no solution. This happens when the equations have the same gradient but different y-intercepts.
无解：如果两条直线平行，它们永不相交，所以方程组无解。当两个方程的斜率相同但 y 截距不同时，就会发生这种情况。
x + y = 5
x + y = 8
at a false statement such as 0 = 3 means the lines are parallel and there is no solution.
相减得到 0 = 3，这是不可能的。在加减消元法中，如果得出 0 = 3 这样的假命题，就说明两直线平行，方程组无解。
Infinite solutions: If the two equations represent the same line, every point on that line is a solution.
无穷多解：如果两个方程代表同一条直线，那么该直线上的每一个点都是解。
x + y = 5
2x + 2y = 10
e the first. Subtracting gives 0 = 0, a true statement, meaning infinitely many solutions.
这里第二个方程正好是第一个方程的两倍。相减得到 0 = 0，这是真命题，意味着有无穷多组解。
6. Solving Word Problems | 解应用题
Exam questions often present a real-world scenario. The key skill is translating words into algebraic equations.
考试题常给出实际情境。关键技能是将文字转化为代数方程。
Worked Example: The sum of two numbers is 25 and their difference is 3. Find the two numbers.
例题：两个数之和为 25，之差为 3。求这两个数。
Let the two numbers be x and y. Then:
x + y = 25
x − y = 3
d the two equations: 2x = 28, so x = 14. Then y = 25 − 14 = 11. The numbers are 14 and 11.
设两个数为 x 和 y，则 x + y = 25，x − y = 3。两式相加：2x = 28，所以 x = 14。则 y = 25 − 14 = 11。这两个数是 14 和 11。
Always define your variables clearly in the exam. This gains method marks even if you make a small arithmetic slip later.
考试中务必清楚地定义变量。即使后面出现小的计算失误，也能获得方法分。
7. Non-Linear Simultaneous Equations | 非线性联立方程组
In higher-tier IGCSE papers, you may meet one linear equation and one quadratic equation. Use the substitution method: substitute the linear expression into the quadratic equation.
在 IGCSE 高等卷（Higher Tier）中，你可能会遇到一个线性方程加一个二次方程的情况。使用代入法：将线性表达式代入二次方程。
Worked Example: Solve
y = x + 2
y = x²
Substitute y = x + 2 into y = x²:
x + 2 = x²
x² − x − 2 = 0
(x − 2)(x + 1) = 0
2 or x = −1. When x = 2, y = 4; when x = −1, y = 1. The solutions are (2, 4) and (−1, 1).
将 y = x + 2 代入 y = x²：x + 2 = x²，即 x² − x − 2 = 0，因式分解得 (x − 2)(x + 1) = 0。所以 x = 2 或 x = −1。当 x = 2 时，y = 4；当 x = −1 时，y = 1。解为 (2, 4) 和 (−1, 1)。
Graphically, the line cuts the curve at two points, giving two solutions. This is why a quadratic system can have up to two pairs of solutions.
从图像上看，直线与曲线在两个点相交，因此得到两组解。这就是为什么二次方程组最多可以有两组解。
8. Common Exam Mistakes | 常见考试错误
- Sign errors when subtracting: When subtracting one equation from another, every term must change sign, not just the first term.
- Forgot to substitute back: Finding one variable is only half the solution; you must substitute back to find the second variable.
- No final check: Always check your answer in both original equations.
- Incorrect multiplication: When multiplying an equation, multiply every term on both sides by the same number.
- 相减时符号错误：一个方程减去另一个方程时，每一项都要变号，而不只是第一项。
- 忘记代回：求出一个变量只是完成一半；必须代回求出第二个变量。
- 未进行最终检验：始终将答案代回两个原方程中检验。
- 乘法错误：方程两边乘以一个数时，必须每一项都乘以该数。
A good habit is to write your two values clearly and substitute them into both equations before moving to the next question.
一个好习惯是清晰写出两个值，并在做下一题之前将它们代入两个方程进行检验。
9. Practice Questions | 练习题
Try these questions yourself, then check the answers in the table below.
请自行尝试以下题目，然后对照下面表格中的答案检查。
- Solve by elimination: 5x + y = 17 and 3x − y = 7.
- Solve by substitution: y = 2x − 1 and 3x + 2y = 12.
- Solve the system: 2x + 3y = 13 and 5x + 3y = 28.
- ate whether this system has no solution or infinite solutions: y = 3x + 1 and 2y = 6x + 2.
- 用加减消元法解：5x + y = 17 和 3x − y = 7。
- 用代入消元法解：y = 2x − 1 和 3x + 2y = 12。
- 解方程组：2x + 3y = 13 和 5x + 3y = 28。
- 判断下列方程组无解还是有无穷多解：y = 3x + 1 和 2y = 6x + 2。
| Question | 题目 | Answer | 答案 |
| 1 | x = 3, y = 2 |
| 2 | x = 2, y = 3 |
| 3 | x = 5, y = 1 |
| 4 | Infinite solutions | 无穷多解 |
10. Summary | 总结
Choose the method that suits the question. Use elimination when coefficients already match or can match easily; use substitution when one variable is already isolated or when one equation is quadratic.
选择适合题目的方法。当系数已经相同或容易配成相同时用加减消元法；当某个变量已经被分离，或有一个方程是二次方程时，用代入消元法。
| Method | 方法 | Best Used When | 适用时机 | Key Skill | 关键技能 |
| Elimination | 加减消元 | Both equations are linear, coefficients match easily | Careful addition and subtraction of equations |
| Substitution | 代入消元 | One variable is already isolated, or a quadratic is involved | Substituting and solving quadratics |
| Graphical | 图像法 | Approximate answer is acceptable | Drawing straight lines accurately |
Remember the golden rule: always check your answer by substituting it back into the original equations. This simple step saves many marks.
请记住黄金法则：始终将答案代回原方程检验。这个简单的步骤能帮你避免失分。
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