📚 Solving Simultaneous Equations | 解联立方程
A system of simultaneous equations is a set of two or more equations that share the same unknown variables. In IGCSE Mathematics, solving simultaneous equations is an essential skill that appears both in algebra questions and in examination problem-solving contexts.
联立方程是含有相同未知数的多个方程所构成的方程组。在 IGCSE 数学中，解联立方程是代数部分的核心技能，也常常出现在应用题和综合题中。
1. Understanding Linear Simultaneous Equations | 理解线性联立方程
Linear simultaneous equations have the general form ax + by = c and dx + ey = f, where x and y are the unknown variables. The solution to such a system is an ordered pair (x, y) that satisfies both equations at the same time.
线性联立方程的一般形式为 ax + by = c 和 dx + ey = f，其中 x 与 y 是未知数。方程组的解是一对有序数 (x, y)，它同时满足两个方程。
For example, in the system:
3x + y = 11
2x − y = 4
the value x = 3 and y = 2 satisfies both equations, so (3, 2) is the solution.
例如，方程组：
3x + y = 11
2x − y = 4
当 x = 3, y = 2 时两个方程同时成立，因此 (3, 2) 是方程组的解。
2. The Elimination Method | 消元法
Elimination is a systematic way to remove one variable by adding or subtracting the equations. The key is to make the coefficient of one variable the same in both equations, then add or subtract to eliminate it.
消元法是通过相加或相减两个方程，使其中一个未知数的系数相同，从而消去该变量的方法。
Solve the following system using elimination:
3x + y = 11
2x − y = 4
Step 1: Add the equations to remove y because the coefficients of y are +1 and −1. This gives 5x = 15, so x = 3.
第一步：将两个方程相加，因为 y 的系数为 +1 和 −1，相加可消去 y，得到 5x = 15，因此 x = 3。
Step 2: Substitute x = 3 into the first equation: 3(3) + y = 11, so y = 2.
第二步：将 x = 3 代入第一个方程：3(3) + y = 11，所以 y = 2。
Step 3: Check the solution in the second equation: 2(3) − 2 = 4, which is correct.
第三步：将解代入第二个方程验证：2(3) − 2 = 4，成立。
3. The Substitution Method | 代入法
Substitution is especially useful when one equation contains a variable expressed directly, such as y = 2x − 1. You replace that variable in the other equation with the given expression.
当一个方程已经直接表示出某个变量，例如 y = 2x − 1 时，代入法特别方便。你只需把另一个方程中的该变量替换成这个表达式。
Example: Solve
y = 2x − 1
x + 2y = 11
Substitute y = 2x − 1 into x + 2y = 11:
将 y = 2x − 1 代入 x + 2y = 11：
x + 2(2x − 1) = 11
Then x + 4x − 2 = 11, so 5x = 13, giving x = 2.6. Next substitute back: y = 2(2.6) − 1 = 4.2. Therefore the solution is x = 2.6, y = 4.2.
化简得 x + 4x − 2 = 11，所以 5x = 13，x = 2.6。回代入 y = 2(2.6) − 1 = 4.2。因此解为 x = 2.6, y = 4.2。
4. The Graphical Method | 图像法
Graphically, each linear equation represents a straight line. The point where the two lines intersect gives the solution to the simultaneous equations.
从图像上看，每个线性方程都对应一条直线。两条直线的交点就是联立方程组的解。
For example, plot y = 2x + 1 and y = −x + 4 on the same axes.
例如，在同一坐标平面上画出 y = 2x + 1 和 y = −x + 4。
At x = 1, the first line gives y = 3, and the second line also gives y = 3. The lines intersect at (1, 3), so the solution is x = 1, y = 3.
当 x = 1 时，第一条直线得到 y = 3，第二条直线也得到 y = 3。两条直线交于点 (1, 3)，因此解为 x = 1, y = 3。
The graphical method is useful for estimation, but it can be inaccurate if the intersection is not at exact integer coordinates. Always solve algebraically for exact values in an exam.
图像法适合用于估算，但如果交点不是整数坐标，精度就会不足。考试中应使用代数方法求出精确解。
5. Solving Word Problems with Simultaneous Equations | 用联立方程解应用题
Many real-life problems can be translated into simultaneous equations. The first step is to define the unknown variables clearly, then form two equations from the given conditions.
许多实际问题都可以转化为联立方程。第一步是清楚定义未知数，然后根据题目条件列出两个方程。
Example: A total of 500 tickets were sold for a concert. Adult tickets cost $12 and student tickets cost $8. The total revenue was $5200. Find the number of adult tickets and student tickets sold.
例题：一场音乐会共售出 500 张门票。成人票每张 $12，学生票每张 $8，总收入为 $5200。求成人票和学生票各售出多少张。
Let a be the number of adult tickets and s be the number of student tickets.
设 a 为成人票数量，s 为学生票数量。
From the total tickets: a + s = 500.
From the revenue: 12a + 8s = 5200.
由总票数：a + s = 500。
由总收入：12a + 8s = 5200。
Using substitution or elimination gives a = 300 and s = 200. So 300 adult tickets and 200 student tickets were sold.
用代入法或消元法解得 a = 300, s = 200。因此售出成人票 300 张，学生票 200 张。
6. Equations with Decimals and Fractions | 含小数和分数的方程
When simultaneous equations contain fractions or decimals, you can clear them first by multiplying each equation by an appropriate factor. This often makes elimination simpler.
当联立方程中含有分数或小数时，可以在每个方程两边乘以适当的数，先去分母或小数，这样消元会更简便。
Example: Solve
x/2 + y/3 = 8
x/3 − y/4 = 2
Multiply the first equation by 6 to get 3x + 2y = 48. Multiply the second equation by 12 to get 4x − 3y = 24.
第一个方程两边乘以 6，得 3x + 2y = 48。第二个方程两边乘以 12，得 4x − 3y = 24。
Now solve the new system. Multiplying the first equation by 3 and the second by 2 gives:
现在解新的方程组。把第一个方程乘以 3，第二个方程乘以 2：
9x + 6y = 144
8x − 6y = 48
Add them to get 17x = 192, so x = 192/17. Substitute to find y if needed. This method avoids fractions until the final answer.
相加得 17x = 192，所以 x = 192/17。再代入求出 y。这种方法可以避免中途出现分数。
7. Special Cases: No Solution and Infinite Solutions | 特殊情形：无解与无穷解
Not every pair of linear equations has exactly one solution. If the lines are parallel, they never intersect, so there is no solution. If the equations are actually the same line, they have infinitely many solutions.
并非每对线性方程都有唯一解。如果两条直线平行，它们不相交，因此方程组无解。如果两个方程本质上表示同一条直线，则方程组有无穷多解。
An example of no solution is:
2x + 3y = 6
4x + 6y = 15
Here, the second equation is not a multiple of the first in the constant term, so the lines are parallel and distinct. No pair (x, y) satisfies both equations.
这里第二个方程虽然是第一个方程左边系数的倍数，但常数项不成同一比例，因此两条直线平行且不同。不存在 (x, y) 同时满足两个方程。
An example of infinite solutions is:
x + y = 5
2x + 2y = 10
The second equation is simply the first equation multiplied by 2, so both equations represent the same line.
第二个方程是第一个方程乘以 2，所以两个方程表示同一条直线。
8. Non-linear Simultaneous Equations | 非线性联立方程
IGCSE sometimes requires solving a system where one equation is linear and the other is quadratic, for example y = x² − 3x + 2 and y = 2x − 2. You solve them by substitution and then factorise the resulting quadratic.
IGCSE 偶尔会要求解一个线性方程和一个二次方程组成的方程组，例如 y = x² − 3x + 2 与 y = 2x − 2。解法是代入消元，然后对所得二次方程进行因式分解。
Because both equations are equal to y, set the expressions equal:
因为两个方程都等于 y，所以令两个表达式相等：
x² − 3x + 2 = 2x − 2
Rearrange to get x² − 5x + 4 = 0. Factorise: (x − 1)(x − 4) = 0. Thus x = 1 or x = 4.
整理得 x² − 5x + 4 = 0。因式分解：(x − 1)(x − 4) = 0。所以 x = 1 或 x = 4。
Substitute each x into the linear equation y = 2x − 2: if x = 1, y = 0; if x = 4, y = 6. The solutions are (1, 0) and (4, 6).
将每个 x 代入线性方程 y = 2x − 2：当 x = 1 时，y = 0；当 x = 4 时，y = 6。因此解为 (1, 0) 和 (4, 6)。
This method may produce a quadratic that does not factorise; in that case, you can use the quadratic formula or complete the square.
如果所得二次方程无法因式分解，可以使用求根公式或配方法求解。
9. Common Mistakes to Avoid | 常见错误避免
When solving simultaneous equations, students often make sign errors during elimination or forget to substitute back into the original equation. Good habits can prevent these errors.
在解联立方程时，学生常犯消元过程中的符号错误，或忘记回代到原方程验证。养成好习惯可以避免这些错误。
- Always align like terms when adding or subtracting equations. 在进行方程加减时，务必让同类项对齐。
- When multiplying an equation, multiply every term on both sides. 对方程两边进行乘法时，要乘等号两边的每一项。
- Check your answer in both original equations. 把答案同时代入两个原方程进行检验。
- In substitution, use brackets correctly when replacing a variable. 代入时，要用括号正确替换变量。
- If the solution produces a contradiction such as 0 = 5, the system has no solution. 如果解到 0 = 5 这样的矛盾式，说明方程组无解。
10. Exam-style Practice Questions | 真题风格练习
Try the following questions on your own before checking the answers.
请先独立完成下面的练习，再对照答案。
Question 1: Solve by elimination: 5x + 2y = 17 and 3x − 2y = 7.
题目 1：用消元法解方程组：5x + 2y = 17 和 3x − 2y = 7。
Question 2: Solve by substitution: y = 3x − 4 and 2x + y = 11.
题目 2：用代入法解方程组：y = 3x − 4 和 2x + y = 11。
Question 3: The sum of two numbers is 15 and their difference is 3. Find the numbers.
题目 3：两个数的和为 15，差为 3，求这两个数。
Answers: 1. x = 3, y = 1. 2. x = 3, y = 5. 3. The numbers are 9 and 6.
答案：1. x = 3, y = 1。2. x = 3, y = 5。3. 这两个数是 9 和 6。
Mastering simultaneous equations takes regular practice. Once you are confident with elimination, substitution and the graphical interpretation, you will be ready for both foundation and higher tier IGCSE questions.
掌握联立方程需要持续的练习。一旦你熟练掌握了消元法、代入法和图像法，就能从容应对 IGCSE 中基础和高阶的题目。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply