📚 Mastering Simultaneous Equations | 掌握联立方程组
Simultaneous equations are one of the most important algebra skills in IGCSE Mathematics. They appear in pure algebra questions, coordinate geometry, word problems and even quadratic topics. This revision guide explains how to solve them using elimination, substitution, graphs and careful checking, with plenty of examples you can follow step by step.
联立方程组是 IGCSE 数学中最重要的代数技能之一。它们出现在纯代数题、坐标几何、文字题甚至二次函数题目中。本复习指南将讲解如何使用消元法、代入法、图像法和仔细检验来求解，并提供大量可以逐步跟进的例子。
1. What Are Simultaneous Equations? | 什么是联立方程组？
A simultaneous equation system is a set of two or more equations that share the same variables. The solution is the value or values of the variables that make every equation true at the same time.
联立方程组是共享相同变量的两个或多个方程组成的系统。解就是使每一个方程同时成立的变量的一个或多个值。
o unknowns, x and y. The solution is x = 3 and y = 4 because both equations are satisfied.
例如，下面的系统有两个方程和两个未知数 x 与 y。解是 x = 3 和 y = 4，因为两个方程都被满足。
x + y = 7
x − y = −1
In IGCSE questions, you may be asked to solve the system algebraically or graphically. You may also need to form your own equations from a real-life situation.
在 IGCSE 题目中，你可能需要代数求解或图像求解。你还可能需要根据现实情境自己建立方程。
2. The Elimination Method | 消元法
x + by = c. The idea is to add or subtract the equations so that one variable cancels out.
当两个方程都是 ax + by = c 的形式时，消元法通常是最快的方法。其核心思想是将方程相加或相减，使其中一个变量抵消。
Follow these steps. First, align the x terms, y terms and constants. Second, make the coefficients of one variable the same size. Third, add or subtract to eliminate that variable. Fourth, solve the remaining one-variable equation. Finally, substitute back to find the other variable.
请遵循以下步骤。首先，对齐 x 项、y 项和常数项。第二，使某一变量的系数大小相同。第三，相加或相减以消去该变量。第四，求解剩下的一元方程。最后，代回求出另一个变量。
Example: solve the system
示例：求解方程组
2x + y = 10
x − y = 2
x = 4. Substitute x = 4 into x − y = 2 to get 4 − y = 2, so y = 2. The solution is (4, 2).
将两个方程相加以消去 y：(2x + y) + (x − y) = 10 + 2，因此 3x = 12，x = 4。将 x = 4 代入 x − y = 2，得 4 − y = 2，所以 y = 2。解为 (4, 2)。
y = 5, multiply the second equation by 2 to make the y coefficients both 2, then subtract.
有时你需要在相加或相减之前将一个或两个方程乘以某个数。例如，在方程组 3x + 2y = 12 和 x + y = 5 中，将第二个方程乘以 2，使两个 y 的系数都变成 2，然后相减。
3. The Substitution Method | 代入法
or when one variable is easy to isolate. You replace that variable in the other equation.
当其中一个方程已经有一个变量单独在一边时，例如 y = 2x + 1，或者某个变量很容易分离时，代入法最有用。你将该变量代入另一个方程中。
Example: solve the system
示例：求解方程组
y = 2x + 1
3x + 2y = 16
4x + 2 = 16. This gives 7x = 14, so x = 2. Then y = 2(2) + 1 = 5. The solution is (2, 5).
将 y = 2x + 1 代入第二个方程：3x + 2(2x + 1) = 16，因此 3x + 4x + 2 = 16。得到 7x = 14，所以 x = 2。然后 y = 2(2) + 1 = 5。解为 (2, 5)。
Always use brackets when substituting an expression. This helps you avoid sign errors, especially when a negative coefficient is involved.
代入表达式时一定要使用括号。这有助于避免符号错误，尤其是在涉及负系数时。
4. Graphical Solutions | 图像解法
Each linear equation in two variables can be drawn as a straight line on the coordinate plane. The solution of a simultaneous system is the point where the two lines intersect.
每个二元一次方程都可以在坐标平面上画成一条直线。联立方程组的解就是两条直线相交的点。
he lines y = x + 1 and y = −x + 5 intersect at (2, 3), so x = 2 and y = 3 is the solution.
要用图像法求解，可为每个方程制作数值表，准确地画出两条直线，并读出交点坐标。例如，直线 y = x + 1 和 y = −x + 5 相交于 (2, 3)，所以 x = 2 和 y = 3 就是解。
Graphical methods are useful for checking, but IGCSE questions often ask for algebraic methods when an exact answer is required. The graph may only give an approximate solution if the intersection is not at integer coordinates.
图像法在检查时很有用，但当题目要求精确答案时，IGCSE 通常会要求使用代数方法。如果交点不在整数坐标上，图像可能只能给出近似解。
5. Checking Your Answers | 检验答案
After finding a solution, always substitute both values into the original equations. This takes a few seconds and can catch most arithmetic or sign errors.
求出解之后，一定要将两个值代入原方程进行检验。这只需几秒钟，却能发现大多数算术或符号错误。
k: 2(4) + 2 = 10, and 4 − 2 = 2. Both equations are satisfied, so the solution is correct.
对于方程组 2x + y = 10 和 x − y = 2 的解 (4, 2)，检验：2(4) + 2 = 10，且 4 − 2 = 2。两个方程都成立，因此解是正确的。
If one equation is not satisfied, do not just change the signs. Go back through your working line by line to find where the error occurred.
如果有一个方程不成立，不要只改符号。要逐行检查你的解题过程，找出错误发生的位置。
6. Word Problems: Setting Up Equations | 文字题：建立方程
Many IGCSE problems describe a real-life situation instead of giving the equations directly. Your first job is to define two variables and translate the sentences into algebra.
许多 IGCSE 题目描述的是一个现实情境，而不是直接给出方程。你的首要任务是定义两个变量，并将句子翻译成代数式。
Example: Anna buys 3 apples and 2 bananas for £1.60. Bilal buys 5 apples and 2 bananas for £2.20. Find the cost of one apple and one banana.
示例：Anna 买 3 个苹果和 2 根香蕉花了 £1.60。Bilal 买 5 个苹果和 2 根香蕉花了 £2.20。求一个苹果和一根香蕉的价格。
Let a be the cost of one apple and b be the cost of one banana. The equations are:
设 a 为一个苹果的价格，b 为一根香蕉的价格。方程为：
3a + 2b = 1.60
5a + 2b = 2.20
: 3(0.30) + 2b = 1.60, so 0.90 + 2b = 1.60, giving 2b = 0.70 and b = 0.35. One apple costs £0.30 and one banana costs £0.35.
用第二个方程减去第一个方程：2a = 0.60，所以 a = 0.30。代入 3a + 2b = 1.60：3(0.30) + 2b = 1.60，因此 0.90 + 2b = 1.60，得到 2b = 0.70，b = 0.35。一个苹果价格为 £0.30，一根香蕉价格为 £0.35。
Always state the meaning of your variables clearly and give the final answer in the context of the question, including units where needed.
始终清楚地说明变量的含义，并结合题目情境给出最终答案，包括需要的单位。
7. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无穷多解
Not every system has one unique solution. If the two lines are parallel, the system has no solution. If the two lines are exactly the same, the system has infinitely many solutions.
并非每个方程组都有唯一解。如果两条直线平行，则方程组无解。如果两条直线完全重合，则方程组有无穷多个解。
pair of x and y values can make the same sum equal both 2 and 5, so there is no solution.
平行线示例：x + y = 2 和 x + y = 5。没有任何一对 x 和 y 的值能让同一个和既等于 2 又等于 5，因此无解。
irst, so every point on x + y = 2 is a solution. The system has infinitely many solutions.
重合直线示例：x + y = 2 和 2x + 2y = 4。第二个方程只是第一个方程的两倍，因此 x + y = 2 上的每一个点都是解。该方程组有无穷多个解。
false statement like 0 = 3. Infinite solutions often lead to a true statement like 0 = 0.
在代数中，你可能会注意到当消去变量后得到 0 = 3 这样的假命题时，方程组无解。无穷多解通常会产生 0 = 0 这样的恒真命题。
8. Quadratic-Linear Systems | 二次与一次方程组
IGCSE extended questions sometimes pair a quadratic equation with a linear equation. The system can have zero, one or two real solutions, depending on where the line intersects the curve.
IGCSE 扩展题有时会将一个二次方程与一个一次方程组合。根据直线与曲线的交点情况，该方程组可能有零个、一个或两个实数解。
Example: solve the system
示例：求解方程组
y = x² − 3x + 1
y = x − 2
1 = x − 2. Rearrange to x² − 4x + 3 = 0. Factorise: (x − 1)(x − 3) = 0, so x = 1 or x = 3.
因为两个表达式都等于 y，令它们相等：x² − 3x + 1 = x − 2。移项得 x² − 4x + 3 = 0。因式分解：(x − 1)(x − 3) = 0，所以 x = 1 或 x = 3。
om y = x − 2. When x = 1, y = −1. When x = 3, y = 1. The solutions are (1, −1) and (3, 1).
从 y = x − 2 中求出对应的 y 值。当 x = 1 时，y = −1。当 x = 3 时，y = 1。解为 (1, −1) 和 (3, 1)。
Always substitute back into both original equations to confirm that the pairs really do satisfy the quadratic and the linear equation.
始终将结果代回两个原方程，以确认这些数对确实同时满足二次方程和一次方程。
9. Common Mistakes | 常见错误
ead of 2x = 6 correctly; the main danger is losing a minus sign on the constant or y term.
一个常见错误是在方程相减时忘记改变每一项的符号。例如，用 3x + y = 11 减去 x + y = 5 时，y 项抵消，但主要危险是在常数项或 y 项上丢失负号。
you multiply x + y = 5 by 2, every term and the constant must be multiplied: 2x + 2y = 10.
另一个错误是只将方程的一边乘以某个倍数。如果将 x + y = 5 乘以 2，则每一项和常数都必须相乘：2x + 2y = 10。
Students also sometimes substitute their answer into the equation they have already rearranged, rather than into the original equations. Always check with both original equations.
学生有时会把答案代入他们已经整理过的方程，而不是代入原方程。一定要用两个原方程进行检验。
10. Exam Tips and Practice | 考试技巧与练习
To gain full marks, always show your method clearly. Write the equation numbers or labels, show the multiplication step, show the substitution step, and finish with a clear final answer.
要获得满分，始终清晰地展示你的解题方法。写出方程编号或标签，展示乘法步骤、代入步骤，并写出明确的最终答案。
Use these strategies in the exam:
在考试中运用这些策略：
- Eliminate when both equations are in ax + by = c form. / 当两个方程都是 ax + by = c 形式时使用消元法。
- Substitute when one variable is already isolated. / 当其中一个变量已经被分离时使用代入法。
- Draw graphs only when asked, but use them to check answers. / 只有在题目要求时才画图，但可以用图像检查答案。
- Check your final solution in both original equations. / 将最终解代入两个原方程进行检验。
- Give money answers to two decimal places and state units. / 金额答案保留两位小数并写明单位。
Practice regularly with past paper questions, including word problems and quadratic-linear systems, to build speed and confidence before the exam.
定期练习历年真题，包括文字题和二次与一次方程组，以在考试前提高速度和信心。
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