📚 Simultaneous Equations | 联立方程
Simultaneous equations are one of the core algebra skills in the IGCSE Mathematics syllabus. You need to be able to solve two equations in two unknowns both algebraically and graphically, and to interpret word problems that lead to a pair of equations. This topic also extends to one linear and one quadratic equation, which can produce more than one solution.
联立方程是 IGCSE 数学大纲中的核心代数技能之一。你需要能够用代数方法和图像方法求解含有两个未知数的两个方程，并理解由文字题建立的一对方程。这一主题还延伸到一个一次方程与一个二次方程的情形，这时可能会得到不止一组解。
1. What Are Simultaneous Equations? | 什么是联立方程
A simultaneous equation problem asks you to find values of unknowns that satisfy two or more equations at the same time. In IGCSE, you will usually meet two equations in two unknowns, often written as x and y. The solution is an ordered pair (x, y) that makes both equations true.
联立方程问题要求你找出同时满足两个或多个方程的未知数值。在 IGCSE 中，通常会遇到两个未知数 x 和 y 的两个方程。解是一个有序数对 (x, y)，它使两个方程都成立。
quations at the same time. You can check this by substitution: 3 + 4 = 7 and 2(3) − 4 = 2.
例如，x + y = 7 和 2x − y = 2 构成一组联立方程。只有一对值 x = 3 和 y = 4 同时满足两个方程。你可以用代入来检验：3 + 4 = 7，并且 2(3) − 4 = 2。
x + y = 7
2x − y = 2
Solution: x = 3, y = 4
There are three main algebraic methods: elimination, substitution, and graphical solution. For harder IGCSE questions, you may also need to solve a system where one equation is quadratic.
有三种主要的代数方法：消元法、代入法和图像解法。对于较难的 IGCSE 题目，你可能还需要求解一个方程为二次方程的方程组。
2. The Elimination Method | 消元法
Elimination works by adding or subtracting the equations to remove one unknown. Choose the unknown whose coefficients are equal or opposite in sign. If the coefficients are not equal, multiply one or both equations by a suitable number first.
消元法通过将方程相加或相减来消去一个未知数。选择系数相等或互为相反数的未知数。如果系数不相等，先给一个或两个方程乘以适当的数。
11 and x − y = 4. Since the y coefficients are +1 and −1, adding the equations removes y.
例：解 2x + y = 11 和 x − y = 4。由于 y 的系数为 +1 和 −1，将两个方程相加即可消去 y。
2x + y = 11
x − y = 4
Add: 3x = 15 → x = 5
y = 4, so y = 1. The solution is x = 5, y = 1. Always write the solution as a clear pair.
将 x = 5 代入 x − y = 4，得到 5 − y = 4，所以 y = 1。解为 x = 5，y = 1。始终把解写成清晰的数对。
n by 3 and the second by 2 to make the y coefficients both 6. Then subtract the equations.
如果系数不相等，要先相乘。对于 3x + 2y = 12 和 5x + 3y = 19，将第一个方程乘以 3，第二个方程乘以 2，使 y 的系数都变成 6。然后将两个方程相减。
9x + 6y = 36
10x + 6y = 38
Subtract: x = 2
Then substitute x = 2 into 3x + 2y = 12 to find y = 3. The solution is (2, 3).
然后将 x = 2 代入 3x + 2y = 12，求得 y = 3。解为 (2, 3)。
3. The Substitution Method | 代入法
Substitution is useful when one equation already has a single unknown as the subject, or when one coefficient is 1. You replace that unknown in the other equation and solve the resulting single equation.
当一个方程已经以某个未知数为主语，或者某个未知数的系数为 1 时，代入法非常有用。你把这个未知数替换到另一个方程中，然后求解得到的一元方程。
+ 2y = 16. Since y is already the subject, substitute y = 2x + 1 into the second equation.
例：解 y = 2x + 1 和 3x + 2y = 16。由于 y 已经是主语，将 y = 2x + 1 代入第二个方程。
3x + 2(2x + 1) = 16
3x + 4x + 2 = 16
7x = 14 → x = 2
Then use y = 2x + 1 to find y = 2(2) + 1 = 5. The solution is x = 2, y = 5.
然后用 y = 2x + 1 求得 y = 2(2) + 1 = 5。解为 x = 2，y = 5。
le, from x + 2y = 9 you can write x = 9 − 2y and substitute this into the second equation.
你也可以先把一个方程变形，使某个未知数成为主语，再进行代入。例如，由 x + 2y = 9 可以写成 x = 9 − 2y，并将它代入第二个方程。
4. Setting Up Equations from Word Problems | 从文字题建立方程
IGCSE questions often hide simultaneous equations in everyday contexts such as buying pens and pencils, ages, or ticket prices. Start by defining clear variables, then translate each sentence into an equation.
IGCSE 题目常把联立方程隐藏在购买钢笔和铅笔、年龄或票价等日常情境中。首先要清晰地定义变量，然后把每句话翻译成一个方程。
Example: Three cups of tea and two cakes cost 8.40 dollars. Two cups of tea and five cakes cost 11.00 dollars. Let the cost of one cup of tea be t and one cake be c.
例：三杯茶和两块蛋糕售价 8.40 美元。两杯茶和五块蛋糕售价 11.00 美元。设一杯茶的价格为 t，一块蛋糕的价格为 c。
3t + 2c = 8.40
2t + 5c = 11.00
2.00 and 4t + 10c = 22.00. Subtract to get 11t = 20.00, so t = 1.82 dollars approximately.
将第一个方程乘以 5，第二个方程乘以 2，使 c 的系数都变成 10：15t + 10c = 42.00 和 4t + 10c = 22.00。相减得 11t = 20.00，所以 t 约为 1.82 美元。
46 + 2c = 8.40, so 2c = 2.94 and c = 1.47. State the answer in context with correct units.
代回求 c。由 3(1.82) + 2c = 8.40，得到 5.46 + 2c = 8.40，所以 2c = 2.94，c = 1.47。最后要结合情境陈述答案并注明正确单位。
5. Graphical Interpretation | 图像解释
tion because it lies on both lines. For example, x + y = 7 and 2x − y = 2 cross at (3, 4).
每个线性方程都可以在坐标平面上画成一条直线。两条直线的交点就是解，因为它同时位于两条直线上。例如，x + y = 7 和 2x − y = 2 在点 (3, 4) 相交。
t (0, 7) and (7, 0) and join them with a straight line. Do the same for the second equation.
要画一条直线，先找出两个点。对于 x + y = 7，当 x = 0 时 y = 7；当 y = 0 时 x = 7。标出点 (0, 7) 和 (7, 0)，并用直线连接。对第二个方程也同样操作。
In an exam, you may be asked to solve simultaneous equations graphically. Plot both lines accurately, identify the intersection point, and write the coordinates as your solution.
在考试中，可能会要求你用图像方法解联立方程。准确画出两条直线，标出交点，并把交点的坐标作为解写下来。
6. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无穷多解
Two linear equations do not always have one unique solution. If the lines are parallel, there is no solution because they never meet. If the two equations represent the same line, there are infinitely many solutions because every point on the line works.
两个线性方程并不总是只有唯一解。如果两条直线平行，则没有解，因为它们永不相交。如果两个方程表示同一条直线，则有无穷多解，因为直线上的每一个点都满足条件。
air (x, y) can satisfy both. Algebraically, subtracting gives 0 = −4, which is impossible.
例如，x + y = 5 和 x + y = 9 是平行的。它们的左边相同，但常数不同，因此没有任何数对 (x, y) 能同时满足两个方程。从代数上看，相减得到 0 = −4，这是不可能的。
ter multiplying gives 0 = 0, which is always true, so there are infinitely many solutions.
如果一个方程是另一个方程的倍数，例如 x + y = 5 和 2x + 2y = 10，它们就是同一条直线。相乘后相减得到 0 = 0，这恒成立，因此有无穷多解。
7. One Linear and One Quadratic Equation | 一个一次方程与一个二次方程
A more advanced IGCSE skill is solving a system with one linear and one quadratic equation. Use substitution: make x or y the subject of the linear equation, then substitute into the quadratic equation. You will usually get a quadratic in one unknown, which may have two, one, or zero real solutions.
更进一步的 IGCSE 技能是求解一个一次方程与一个二次方程构成的方程组。使用代入法：把一次方程中的 x 或 y 变为主语，再代入二次方程。你通常会得到一个关于单个未知数的二次方程，它可能有两个、一个或零个实数解。
x² − 3x + 1 and y = 2x − 5. Since both expressions equal y, set them equal to each other.
例：解 y = x² − 3x + 1 和 y = 2x − 5。由于两个式子都等于 y，让它们彼此相等。
x² − 3x + 1 = 2x − 5
x² − 5x + 6 = 0
(x − 2)(x − 3) = 0
n y = 2x − 5. When x = 2, y = −1. When x = 3, y = 1. The solutions are (2, −1) and (3, 1).
所以 x = 2 或 x = 3。将每个值代入一次方程 y = 2x − 5。当 x = 2 时，y = −1。当 x = 3 时，y = 1。解为 (2, −1) 和 (3, 1)。
For a pair of linear equations, the graph shows two lines and one intersection. For one linear and one quadratic equation, the graph shows a line and a parabola, which can intersect at two points, one point, or no points.
对于两个一次方程，图像显示两条直线和一个交点。对于一个一次方程和一个二次方程，图像显示一条直线和一条抛物线，它们可能相交于两点、一点或没有交点。
8. Checking Your Answers | 检验答案
Always substitute your answers into both original equations to check that they work. This not only catches arithmetic mistakes but also ensures that you have not lost a solution or introduced an incorrect one when rearranging.
一定要把答案代入两个原始方程进行检验。这不仅能发现算术错误，还能确保你在变形时没有丢失解或引入错误解。
ck: 2(5) + 1 = 11 and 5 − 1 = 4. Both equations are satisfied, so the solution is correct.
例如，如果你对 2x + y = 11 和 x − y = 4 求得 x = 5，y = 1，检验：2(5) + 1 = 11，且 5 − 1 = 4。两个方程都成立，因此解是正确的。
r the quadratic example, verify (2, −1) and (3, 1) in both y = x² − 3x + 1 and y = 2x − 5.
当方程组有两组解时，要分别检验每一对。对于二次方程的例子，检验 (2, −1) 和 (3, 1) 是否都满足 y = x² − 3x + 1 和 y = 2x − 5。
9. Common Mistakes | 常见错误
2y = 7, not 3x = 15. Always write the subtraction carefully and change signs where needed.
一个常见错误是在方程相减时出现符号错误。例如，从 2x + y = 11 中减去 x − y = 4 得到 x + 2y = 7，而不是 3x = 15。务必仔细书写减法，并在需要时改变符号。
st be multiplied. For 3x + 2y = 12, multiplying by 3 gives 9x + 6y = 36, not 9x + 2y = 12.
另一个错误是只乘方程的一部分。当你把一个方程乘以某个数时，两边的每一项都必须相乘。对于 3x + 2y = 12，乘以 3 得到 9x + 6y = 36，而不是 9x + 2y = 12。
u find x, you must substitute it back to find y. A final pair such as x = 5 is incomplete.
有些学生在求出第一个未知数后忘记求第二个未知数。在消元法中，一旦求出 x，你必须把它代回去求 y。只写出 x = 5 这样的结果是不完整的。
10. Exam Tips and Practice | 考试技巧与练习
as x = ... and y = ... . This earns method marks even if a small arithmetic error occurs.
在 IGCSE 考试中，一定要展示清晰的解题过程。写出你使用的方程，展示乘法或代入步骤，并把最终解标注为 x = ... 和 y = ... 。这样即使出现小的算术错误，也能获得方法分。
For word problems, define your variables explicitly at the start. A statement such as 'Let t be the cost of one tea and c be the cost of one cake' shows the examiner what your letters mean and helps you organise the equations.
对于文字题，在开头明确定义变量。像 '设 t 为一杯茶的价格，c 为一块蛋糕的价格' 这样的陈述能让考官知道字母的含义，也有助于你整理方程。
Practice a mixture of straightforward algebraic systems, word problems, graphical questions, and one linear plus one quadratic systems. The more you practise checking solutions, the more confident you will be under exam pressure.
混合练习各种题型：直接的代数方程组、文字题、图像题，以及一个一次方程和一个二次方程的方程组。你越多练习检验答案，考试时就会越有信心。
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