📚 Solving Simultaneous Equations | 解联立方程组
Simultaneous equations are a set of equations with multiple unknown variables. Solving them means finding values for the variables that satisfy all equations at the same time. In the IGCSE syllabus, you will often solve two linear equations with two unknowns using algebraic methods or graphical methods.
联立方程组是含有多个未知数的方程组。解联立方程组意味着找到同时满足所有方程的一组未知数值。在IGCSE考试大纲中，你通常需要用代数方法或图像法来解含有两个未知数的两个线性方程。
1. What Are Simultaneous Equations? | 什么是联立方程组？
When you have two variables, such as x and y, you need two independent equations to find a unique solution. For example:
当你有两个变量（如x和y）时，需要两个独立的方程才能找到唯一解。例如：
2x + y = 7
x − y = 2
These two equations form a pair of simultaneous equations. The solution (x = 3, y = 1) satisfies both equations because 2×3+1=7 and 3−1=2.
这两个方程构成一对联立方程组。解（x=3，y=1）同时满足两个方程，因为2×3+1=7且3−1=2。
2. Graphical Method | 图像法
Each linear equation can be drawn as a straight line on a coordinate grid. The point where the two lines intersect gives the common solution. For the equations above, line y = 7 − 2x and line y = x − 2 intersect at (3, 1).
每个线性方程都可以在坐标网格上画成一条直线。两条直线的交点坐标就是共同解。对于上面的方程，直线 y = 7 − 2x 和直线 y = x − 2 在点 (3, 1) 处相交。
This method is useful for estimating solutions, but algebra gives exact values. When lines are parallel, there is no solution; when they coincide, there are infinitely many solutions.
这种方法适用于估算解，但代数方法能得到精确值。若两直线平行，则无解；若二者重合，则有无穷多解。
3. Substitution Method | 代入法
Substitution involves rearranging one equation to express one variable in terms of the other, then substituting this expression into the second equation.
代入法的步骤是：先由一个方程解出一个变量关于另一个变量的表达式，再把这个表达式代入第二个方程。
x + 3y = 11 → x = 11 − 3y
Then substitute into 2x − y = 5:
然后代入 2x − y = 5：
2(11 − 3y) − y = 5
22 − 6y − y = 5
−7y = −17
y = 17⁄7
Now find x by substituting y back. This method works well when one equation is already solved for one variable, or when coefficients are small.
再将y代回求x。当一个方程已经解出某个变量，或系数较小时，代入法非常有效。
4. Elimination Method | 消元法
Elimination (also called addition or subtraction method) removes one variable by adding or subtracting the two equations. First make the coefficients of one variable the same in both equations.
消元法（也叫加减消元法）通过将两个方程相加或相减来消去一个变量。首先使两个方程中同一个变量的系数相等。
3x + 2y = 12
5x − 2y = 4
Add the equations to eliminate y:
将两个方程相加以消去y：
8x = 16
x = 2
Substitute x = 2 into one original equation to get y = 3.
将x=2代入原方程之一，得到y=3。
5. When Coefficients Are Different | 当系数不同时
If the coefficients of the variable to eliminate are different, multiply one or both equations by suitable numbers first. For example:
如果要消去的变量系数不同，先给一个或两个方程乘以适当的数。例如：
2x + 3y = 8 (1)
3x + 2y = 7 (2)
Multiply (1) by 2 and (2) by 3 so both have 6x:
将(1)乘以2，(2)乘以3，使两者都含有6x：
4x + 6y = 16
9x + 6y = 21
Subtract (2)−(1): 5x = 5, so x = 1. Then y = 2.
两式相减得5x = 5，所以x = 1，进而y = 2。
6. Special Cases | 特殊情况
Sometimes the equations have no solution (inconsistent system) or infinitely many solutions (dependent system). For example:
有时方程组无解（不相容系统），或有无穷多解（相依系统）。例如：
x + y = 5
2x + 2y = 12
The second equation simplifies to x + y = 6, which contradicts x + y = 5. No solution exists. Graphically, these lines are parallel.
第二个方程化简后为x + y = 6，与x + y = 5矛盾，因此无解。在图像上两条直线平行。
7. Solving Word Problems | 解决应用题
Many real-world problems can be translated into simultaneous equations. Let the unknowns be variables, write equations from the given information, then solve them.
许多实际问题可以转化为联立方程组。把未知量设为变量，根据已知信息列出方程，然后求解。
The sum of two numbers is 15.
Their difference is 3.
Find the numbers.
Let the numbers be a and b:
设两个数为a和b：
a + b = 15
a − b = 3
Adding gives 2a = 18, so a = 9. Subtracting gives 2b = 12, so b = 6. The numbers are 9 and 6.
相加得2a = 18，所以a = 9；相减得2b = 12，所以b = 6。这两个数是9和6。
8. Checking Your Solution | 检验你的解
Always verify the solution by substituting both values into both original equations. If both statements are true, the solution is correct.
一定要把两个值分别代入两个原方程来检验。如果两个等式都成立，解就是正确的。
For example, for x = 3, y = 1 in the system 2x + y = 7 and x − y = 2:
例如，对于方程组2x + y = 7和x − y = 2，代x=3、y=1：
2(3) + 1 = 7 ✓
3 − 1 = 2 ✓
Checking prevents careless mistakes and is essential in exam questions.
检验能避免粗心错误，在考试中非常重要。
9. Practice with Fractions and Decimals | 分数与小数的练习
Equations may contain fractions or decimals. Multiply every term by the common denominator to clear fractions first.
方程中可能出现分数或小数。先给每一项乘以公共分母以去掉分数。
x⁄2 + y⁄3 = 4
x⁄4 − y⁄6 = 1
Multiply the first equation by 6 and the second by 12:
第一个方程乘以6，第二个乘以12：
3x + 2y = 24
3x − 2y = 12
Now add to eliminate y: 6x = 36, so x = 6. Then y = 3.
相加消去y：6x = 36，所以x = 6，然后y = 3。
10. Common Mistakes to Avoid | 常见错误提醒
Students often forget to multiply every term when scaling equations, or misplace negative signs during subtraction. Always align like terms carefully.
学生在对方程放大倍数时常忘记乘以所有项，或在减法中弄错负号。务必仔细对齐同类项。
When subtracting one equation from another, change the signs of the subtracted equation before adding. For example, subtract 2x + 3y = 8 from 4x + 6y = 16 means adding −2x − 3y + 8.
当一个方程减去另一个方程时，应先改变被减方程各项的符号再相加。例如，从4x + 6y = 16中减去2x + 3y = 8，相当于加上−2x − 3y + 8。
Practice with varied examples builds confidence. Use both substitution and elimination to see which you prefer, but master both for flexibility.
通过多样化的例题练习可以建立自信。代入法和消元法都要熟练掌握，以便灵活选择。
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