📚 Solving Simultaneous Equations | 联立方程组的解法
Simultaneous equations are a core topic in IGCSE Mathematics. They require you to find values for two or more unknowns that satisfy two or more equations at the same time. This guide explains the main methods: elimination, substitution, and graphical solutions, with worked examples, special cases, quadratic systems, word problems, and exam tips.
联立方程组是 IGCSE 数学的核心考点。它要求你求出同时满足两个或更多方程的两个或更多未知数的值。本文讲解主要方法：消元法、代入法、图像法，并提供例题、特殊情况、二次方程组、应用题与考试技巧。
1. What Are Simultaneous Equations? | 什么是联立方程组？
A set of simultaneous equations is two or more equations that contain the same variables. A solution is a set of values that makes every equation true at the same time.
联立方程组由两个或更多方程组成，它们含有相同的变量。一个解就是一组能同时满足每一个方程的值。
, the equations 2x + y = 7 and x − y = 2 must both be true for the same values of x and y.
例如，方程 2x + y = 7 和 x − y = 2 必须对相同的 x 和 y 值同时成立。
The solution is x = 3, y = 1 because 2(3) + 1 = 7 and 3 − 1 = 2.
解为 x = 3，y = 1，因为 2(3) + 1 = 7 且 3 − 1 = 2。
For linear simultaneous equations, there is usually exactly one solution, unless the lines are parallel or identical. The solution can be found algebraically or graphically.
对于线性联立方程组，通常恰好有一个解，除非两条直线平行或重合。解可以通过代数方法或图像法求出。
s a pair, such as (x, y) = (3, 1), because an exam mark may depend on clear communication.
最终答案要写成数对，例如 (x, y) = (3, 1)，因为考试评分可能取决于表达是否清楚。
2. Standard Form and Setting Up | 标准形式与列式
inear form ax + by = c. This makes coefficients easy to compare and reduces sign mistakes.
解之前，把每个方程改写成线性标准形式 ax + by = c。这样更容易比较系数，也能减少符号错误。
Identify the unknowns clearly. In word problems, let x and y stand for the two quantities you need to find, and include units if they are given.
清楚确定未知数。在应用题中，令 x 和 y 表示需要求出的两个量；如果有单位，要写出来。
Check whether the equations are linear or whether one is quadratic. A linear equation has variables to the first power only; a quadratic equation contains a squared term such as x².
检查方程是线性的，还是其中一个是二次的。线性方程中变量只出现一次幂；二次方程含有平方项，例如 x²。
Label the equations as (1) and (2). This helps you keep track of which equation you multiply, add, or substitute into.
把方程标为 (1) 和 (2)。这有助于追踪你乘的是哪个方程、加的是哪个方程、代入的是哪个方程。
3. The Elimination Method | 消元法
Elimination is usually the fastest method when both equations are linear and the coefficients of one variable can be matched easily. The idea is to add or subtract the equations to remove one variable.
当两个方程都是线性方程，并且某个变量的系数容易配平时，消元法通常是最快的方法。其思路是通过相加或相减来消去一个变量。
Step 1: Write both equations in the form ax + by = c.
第一步：把两个方程写成 ax + by = c 的形式。
Step 2: Multiply one or both equations so that the coefficients of one variable are equal in size.
第二步：给一个或两个方程乘以适当的数，使某个变量的系数大小相等。
Step 3: Add or subtract the equations to eliminate that variable. If the signs are opposite, add. If the signs are the same, subtract.
第三步：将方程相加或相减，消去该变量。符号相反时相加；符号相同时相减。
Step 4: Solve the resulting one-variable equation.
第四步：解得到的一元方程。
Step 5: Substitute back into one original equation to find the other variable.
第五步：代回原方程之一，求另一个变量。
Worked example: solve 2x + y = 7 and x − y = 2.
例题：解 2x + y = 7 和 x − y = 2。
Add the two equations to eliminate y:
将两个方程相加，消去 y：
(2x + y) + (x − y) = 7 + 2 → 3x = 9 → x = 3
Substitute x = 3 into x − y = 2:
将 x = 3 代入 x − y = 2：
3 − y = 2 → y = 1
The solution is x = 3, y = 1.
解为 x = 3，y = 1。
4. Elimination with Scaling and Signs | 需要倍乘与符号处理的消元法
Many exam equations do not have matching coefficients. You must multiply one or both equations first, and then add or subtract carefully.
许多考试方程没有现成的相等系数。你必须先给一个或两个方程乘以适当的数，然后仔细相加或相减。
Example: solve 3x + 2y = 12 and 2x + 3y = 13.
例题：解 3x + 2y = 12 和 2x + 3y = 13。
Multiply the first equation by 3 and the second equation by 2 so that the y coefficients become 6.
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。
9x + 6y = 36
4x + 6y = 26
Subtract the second scaled equation from the first:
用第一个倍乘后的方程减去第二个倍乘后的方程：
5x = 10 → x = 2
Substitute x = 2 into 3x + 2y = 12:
将 x = 2 代入 3x + 2y = 12：
3(2) + 2y = 12 → 6 + 2y = 12 → 2y = 6 → y = 3
Check: 3(2) + 2(3) = 12 and 2(2) + 3(3) = 13. The solution is (2, 3).
检验：3(2) + 2(3) = 12，2(2) + 3(3) = 13。解为 (2, 3)。
5. The Substitution Method | 代入法
Substitution is useful when one equation is already solved for x or y, or when one variable can be isolated without creating complicated fractions.
当一个方程已经解出 x 或 y，或者某个变量可以容易地分离出来且不产生复杂分数时，代入法很有用。
Example: solve y = 2x + 1 and 3x + y = 11.
例题：解 y = 2x + 1 和 3x + y = 11。
Substitute y = 2x + 1 into the second equation:
将 y = 2x + 1 代入第二个方程：
3x + (2x + 1) = 11 → 5x + 1 = 11 → 5x = 10 → x = 2
Now substitute x = 2 back into y = 2x + 1:
再将 x = 2 代回 y = 2x + 1：
y = 2(2) + 1 = 5
Check: 3(2) + 5 = 11 and y = 2(2) + 1 = 5. The solution is x = 2, y = 5.
检验：3(2) + 5 = 11，且 y = 2(2) + 1 = 5。解为 x = 2，y = 5。
6. Substitution with Rearranging | 需要变形的代入法
If neither equation is solved for a variable, choose the variable with coefficient 1 or −1, because it is easiest to isolate.
如果两个方程都没有解出某个变量，选择系数为 1 或 −1 的变量，因为它最容易分离。
Example: solve x + 2y = 8 and 3x − y = 3.
例题：解 x + 2y = 8 和 3x − y = 3。
From the first equation, make x the subject:
由第一个方程，把 x 表示出来：
x = 8 − 2y
Substitute this into the second equation:
将它代入第二个方程：
3(8 − 2y) − y = 3 → 24 − 6y − y = 3 → −7y = −21 → y = 3
Then find x:
再求 x：
x = 8 − 2(3) = 2
Check: 2 + 2(3) = 8 and 3(2) − 3 = 3. The solution is (2, 3).
检验：2 + 2(3) = 8，3(2) − 3 = 3。解为 (2, 3)。
7. Graphical Solutions | 图像法
A graphical solution means drawing both lines on the same axes. The point where the two lines intersect is the solution to the simultaneous equations.
图像法就是在同一坐标系中画出两条直线。两条直线的交点就是联立方程组的解。
Example: solve y = 2x − 1 and y = −x + 5 graphically, and check algebraically.
例题：用图像法解 y = 2x − 1 和 y = −x + 5，并用代数法检验。
Make a table of values for each line, plot at least three points for each line, and draw straight lines with a ruler.
为每条直线列出取值表，每条直线至少描三个点，并用直尺画出直线。
The intersection can also be found algebraically by setting the expressions equal:
交点也可以通过令两个表达式相等来代数求解：
2x − 1 = −x + 5 → 3x = 6 → x =
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