📚 Solving Simultaneous Equations: Substitution, Elimination and Graphical Methods | 解二元一次方程组：代入法、消元法与图像法
aneous equations. In IGCSE Mathematics, you need to find the single pair of values for x and y that satisfies both equations at the same time. This article reviews three main methods: elimination, substitution and graphical representation. Each method is useful in different situations, so you should practise all of them.
一对线性方程如 2x + y = 7 和 x - y = 2 构成一个二元一次方程组。在 IGCSE 数学中，你需要找到同时满足两个方程的 x 与 y 的唯一一组值。本文复习三种主要方法：消元法、代入法和图像表示法。每种方法在不同情形下各有优势，因此你应当全部练习。
1. What Are Simultaneous Equations? | 什么是二元一次方程组？
lutions because you can choose any value for x and then calculate the corresponding y. A second linear equation adds another condition. Together, the two equations are called simultaneous equations. The solution is the pair of values that makes both equations true.
一个含有两个未知数的线性方程，如 3x + 2y = 8，有无限多组解，因为你可以任意选择 x 的值，然后算出对应的 y。第二个线性方程增加了另一个条件。两个方程合在一起就称为联立方程组。方程组的解就是使两个方程都成立的一组值。
Geometrically, each linear equation represents a straight line. The solution of the simultaneous equations is the point where the two lines cross. If the lines intersect at exactly one point, the system has one unique solution. This is the standard case you will meet most often in IGCSE exams.
从几何角度看，每个线性方程表示一条直线。联立方程组的解就是两条直线相交的点。如果两条直线恰好相交于一点，那么这个方程组就有唯一一组解。这是你在 IGCSE 考试中最常遇到的标准情形。
2. Checking a Solution | 检验解
Before solving, it is useful to understand how to check whether a given pair of values is a solution. Substitute the values into both original equations. If both left-hand sides equal the corresponding right-hand sides, the pair is the solution. This same skill is essential after solving to catch arithmetic mistakes.
在求解之前，理解如何检验一组给定值是否为方程组的解是很有用的。将这对值代入两个原方程。如果两个等式的左边都等于对应的右边，那么这组值就是方程组的解。求解之后，这个技能同样重要，可以帮助你发现计算错误。
example, consider the system below. Check whether x = 3 and y = 1 satisfy both equations.
例如，考虑下面的方程组。检验 x = 3 和 y = 1 是否满足两个方程。
2x + y = 7
x - y = 2
nto the second equation: 3 - 1 = 2, which also matches. Therefore, (3, 1) is the solution.
将 x = 3 和 y = 1 代入第一个方程：2(3) + 1 = 6 + 1 = 7，与右边相等。再代入第二个方程：3 - 1 = 2，也相等。因此，(3, 1) 是方程组的解。
3. The Elimination Method (Same Coefficients) | 消元法（系数相同）
The elimination method works by adding or subtracting the two equations to remove one unknown. The easiest case occurs when the coefficients of one unknown are already the same or opposite in sign. If the coefficients are opposite, add the equations; if they are the same, subtract one equation from the other.
消元法通过将两个方程相加或相减来消去一个未知数。最简单的情况是某个未知数的系数已经相同或互为相反数。如果系数互为相反数，就将两个方程相加；如果系数相同，就用一个方程减去另一个方程。
In the system above, the coefficients of y are +1 and -1. They are opposite in sign, so adding the two equations cancels the y term.
在上面的方程组中，y 的系数分别是 +1 和 -1。它们互为相反数，因此将两个方程相加就可以消去 y 项。
(2x + y) + (x - y) = 7 + 2
3x = 9
x = 3
lution is (3, 1). Always write the solution as a pair in brackets, with the x value first.
求出 x 之后，将 x = 3 代入任意一个原方程。使用 x - y = 2，得到 3 - y = 2，所以 y = 1。方程组的解为 (3, 1)。写解时一定要用括号写成一对，x 值在前。
4. The Elimination Method (Multiplying First) | 消元法（先乘以倍数）
In many systems, the coefficients of neither unknown are the same. You then multiply one or both equations by suitable constants to make the coefficients of one unknown equal. After that, add or subtract as needed to eliminate that unknown.
在许多方程组中，两个未知数的系数都不相同。此时你需要给一个或两个方程乘以合适的常数，使某个未知数的系数相等。然后再根据需要进行加法或减法以消去该未知数。
Consider the system below.
考虑下面的方程组。
3x + 2y = 12
2x - y = 1
The y coefficients are +2 and -1. If you multiply the second equation by 2, the y term becomes -2y. Then the coefficients of y are +2 and -2, which are opposite in sign. Adding the two equations eliminates y.
y 的系数是 +2 和 -1。如果你把第二个方程乘以 2，y 项就变成 -2y。此时 y 的系数为 +2 和 -2，互为相反数。将两个方程相加即可消去 y。
3x + 2y = 12
4x - 2y = 2
, so x = 2. Substitute x = 2 into 2x - y = 1: 4 - y = 1, so y = 3. The solution is (2, 3).
将两个方程相加：3x + 4x = 7x，2y + (-2y) = 0，12 + 2 = 14。因此 7x = 14，所以 x = 2。将 x = 2 代入 2x - y = 1：4 - y = 1，所以 y = 3。方程组的解为 (2, 3)。
5. The Substitution Method | 代入法
The substitution method is useful when one equation is already solved for one variable, or can be rearranged easily. Rearrange the first equation to express one unknown in terms of the other. Then substitute this expression into the second equation.
当一个方程已经化为一个未知数表示另一个未知数的形式，或者容易变形时，代入法非常有用。把第一个方程变形，用一个未知数表示另一个未知数。然后将这个表达式代入第二个方程。
d x - y = 2, the second equation gives x = y + 2. Substitute this into the first equation.
对于同一个方程组 2x + y = 7 和 x - y = 2，由第二个方程可得 x = y + 2。将其代入第一个方程。
2(y + 2) + y = 7
2y + 4 + y = 7
3y = 3
y = 1
ain (3, 1). Substitution is often easier when one unknown already has coefficient 1 or -1.
然后把 y = 1 代回 x = y + 2，得到 x = 3。解仍然是 (3, 1)。当某个未知数的系数已经是 1 或 -1 时，代入法通常更简单。
6. Graphical Method: Drawing Lines | 图像法：画直线
c. Then draw both straight lines on the same coordinate grid. You can use a table of x and y values to plot accurate points, especially when the equation is not easy to sketch mentally.
要用图像法解联立方程组，首先把每个方程改写成斜截式 y = mx + c。然后在同一坐标系中画出两条直线。你可以使用 x 和 y 值的表格来绘制准确的点，尤其是当方程不容易直接画草图时。
anged equations are y = -2x + 7 and y = x - 2. A table of values helps you plot each line.
对于方程组 2x + y = 7 和 x - y = 2，变形后的方程为 y = -2x + 7 和 y = x - 2。数值表可以帮助你画出每条直线。
- For y = -2x + 7: x = 0 gives y = 7; x = 1 gives y = 5; x = 2 gives y = 3.
- 对于 y = -2x + 7：x = 0 时 y = 7；x = 1 时 y = 5；x = 2 时 y = 3。
- For y = x - 2: x = 0 gives y = -2; x = 1 gives y = -1; x = 2 gives y = 0.
- 对于 y = x - 2：x = 0 时 y = -2；x = 1 时 y = -1；x = 2 时 y = 0。
Plot at least three points for each line so that any mistake in one point is easy to spot. Extend both lines across the grid, and label them clearly.
每条直线至少描出三个点，这样如果某个点出错就很容易发现。将两条直线延伸到整个坐标网格，并清楚地标出对应的方程。
7. Reading the Intersection Point | 读取交点
. In the example above, the lines y = -2x + 7 and y = x - 2 intersect at the point (3, 1).
两条已画直线的交点是联立方程组的解。交点的 x 坐标就是 x 的值，y 坐标就是 y 的值。在上面的例子中，直线 y = -2x + 7 和 y = x - 2 相交于点 (3, 1)。
Graphical solutions can be less precise when the intersection does not fall exactly on grid lines. In an exam, you should read the coordinates as accurately as the graph scale allows. If the question asks for an approximate answer, a range of acceptable values is often given.
当交点没有正好落在网格线上时，图像法的解可能不够精确。在考试中，你要根据坐标纸的刻度尽可能准确地读出坐标。如果题目要求近似答案，通常会给出一个可接受的取值范围。
8. Special Cases: Parallel and Coincident Lines | 特殊情况：平行与重合直线
Not every system of two linear equations has exactly one solution. If the two lines are parallel, they never intersect, so there is no solution. Algebraically, this happens when the equations have the same left-hand ratio but different constants.
并不是每一个由两个线性方程构成的方程组都恰好有一组解。如果两条直线平行，它们永不相交，因此方程组无解。从代数上看，当两个方程左边对应系数成比例但常数项不同时，就会出现这种情况。
Subtracting one from the other gives 0 = -3, which is impossible, so there is no solution.
例如，x + y = 2 和 x + y = 5 是平行直线。两式相减得到 0 = -3，这是不可能的，因此方程组无解。
o lines are coincident. They lie on top of each other, so every point on the line is a solution. In this case, the system has infinitely many solutions.
如果一个方程是另一个方程的倍数，例如 x + y = 2 和 2x + 2y = 4，那么两条直线重合。它们完全重叠，因此直线上的每一个点都是方程组的解。此时方程组有无穷多组解。
9. Forming Simultaneous Equations from Word Problems | 根据文字题建立方程组
Many IGCSE questions ask you to translate a real-life situation into simultaneous equations. Start by defining two variables, usually x and y, and state clearly what each variable represents. Then form one equation for each condition in the question.
许多 IGCSE 题目要求你把实际情境转化为联立方程组。首先定义两个变量，通常用 x 和 y，并清楚说明每个变量代表什么。然后根据题目中的每一个条件列出一个方程。
e system x + y = 10 and x - y = 4. Adding the equations gives 2x = 14, so x = 7 and y = 3.
例如，两个数的和是 10，差是 4。设较大的数为 x，较小的数为 y。条件给出方程组 x + y = 10 和 x - y = 4。两式相加得到 2x = 14，所以 x = 7，y = 3。
More complex word problems may involve costs, ages or mixtures. Always check that your solution makes sense in the original context. For example, a negative price or an impossible age means you have set up the equations incorrectly.
更复杂的文字题可能涉及费用、年龄或混合物。始终检查你的解在原情境中是否合理。例如，价格为负数或年龄不可能为负，说明你建立方程时出了问题。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Students often lose marks on simultaneous equations because of avoidable mistakes. The most common errors include forgetting to multiply every term in an equation, substituting back into the wrong equation, and misreading signs when subtracting equations. Always write each step clearly and label the equations.
学生在解联立方程组时常常因为一些可以避免的错误而失分。最常见的错误包括：忘记对方程中的每一项都乘以同一个倍数，代回时选错方程，以及相减方程时符号读错。每一步都要写清楚，并给方程标上编号。
- Check the solution in both original equations, not just one.
- 在代入检验时，要代入两个原方程，而不只是一个。
- If elimination gives a false statement such as 0 = 5, write 'no solution'.
- 如果消元后得到一个假等式，如 0 = 5，要写 '无解'。
- If elimination gives 0 = 0, write 'infinitely many solutions'.
- 如果消元后得到 0 = 0，要写 '无穷多组解'。
- In graphical questions, use graph paper and a sharp pencil for accuracy.
- 在图像法题目中，使用坐标纸和削好的铅笔以保证准确性。
Practise each method until you can choose the most efficient one quickly. In the exam, elimination is often fastest when coefficients can be matched easily, while substitution is best when one variable is already isolated.
练习每一种方法，直到你能快速选择最有效的方法。在考试中，当系数容易匹配时，消元法通常最快；当一个变量已经被单独表示出来时，代入法最好。
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