📚 Solving Linear Simultaneous Equations | 线性二元一次方程组求解
In IGCSE Mathematics, linear simultaneous equations are a core algebra topic. They involve two unknown variables and two equations, and you are expected to find the single pair of values that satisfies both equations at once. Mastering elimination and substitution gives you reliable algebraic methods for exact answers.
在 IGCSE 数学中，线性联立方程组是代数部分的核心考点。它通常包含两个未知数和两个方程，题目要求你找出一对同时满足两个方程的数值。掌握消元法和代入法可以让你用可靠的代数方法求出精确答案。
1. What Are Linear Simultaneous Equations? | 什么是线性联立方程组
A linear equation in two variables, such as 2x + y = 7, has infinitely many solutions on its own. When two such equations are paired, the solution is the ordered pair (x, y) that makes both equations true.
一个含有两个变量的线性方程，例如 2x + y = 7，本身有无穷多个解。当两个这样的方程组成联立方程组时，解就是使两个方程同时成立的有序数对 (x, y)。
ax + by = c
dx + ey = f
The general form is ax + by = c and dx + ey = f, where a, b, c, d, e and f are constants. IGCSE questions may use integers, decimals, fractions or negative coefficients.
一般形式为 ax + by = c 和 dx + ey = f，其中 a、b、c、d、e 和 f 均为常数。IGCSE 题目可能使用整数、小数、分数或负系数。
Linear means that neither variable is raised to a power higher than 1, and variables are not multiplied together. This keeps the graphs as straight lines.
线性是指每个变量的次数都不超过 1，且变量之间不相互乘积。因此它们的图像都是直线。
2. The Elimination Method | 消元法
Elimination makes the coefficient of one variable equal in both equations, then adds or subtracts the equations to remove that variable. This leaves a simple one-variable equation.
消元法通过使某个变量在两个方程中的系数相同，然后将两个方程相加或相减，从而消去该变量。这样可以得到一个只含一个变量的简单方程。
To eliminate the same variable, multiply one or both equations by suitable constants. Choose the smallest possible multiplier to keep the arithmetic neat.
为了消去同一个变量，可以给一个或两个方程乘上合适的常数。尽量选择最小的乘数，以减少计算量。
After finding the first variable, substitute its value back into either original equation. This gives the value of the second variable. Always check both values in the original system.
求出第一个变量后，将它代回任意一个原方程，即可求出第二个变量的值。最后一定要将两个数值代入原方程组进行检验。
3. Worked Example: Elimination | 消元法典型例题
Solve the simultaneous equations:
解下列联立方程组：
2x + y = 7 … (1)
x − y = 2 … (2)
Add equations (1) and (2): the y terms cancel because +y and −y sum to 0. This gives 3x = 9, so x = 3.
将方程 (1) 与 (2) 相加：由于 +y 与 −y 相加为 0，y 项被消去。得到 3x = 9，因此 x = 3。
Substitute x = 3 into equation (2): 3 − y = 2, so y = 1. The solution is (3, 1).
将 x = 3 代入方程 (2)：3 − y = 2，解得 y = 1。因此解为 (3, 1)。
Check in equation (1): 2(3) + 1 = 7, which is correct. This confirms the solution.
代入方程 (1) 检验：2 × 3 + 1 = 7，结果正确。这验证了解是正确的。
4. The Substitution Method | 代入法
Substitution is useful when one equation already has one variable as the subject, such as y = 2x + 1. Replace this variable in the other equation to produce a one-variable equation.
当其中一个方程已经把某个变量写成主项时，例如 y = 2x + 1，代入法非常有用。将该变量替换到另一个方程中，就可以得到只含一个变量的方程。
If neither equation is in the form y = … or x = …, rearrange one equation first. Substitution often has slightly more steps, but it shows clear logical reasoning in word problems.
如果两个方程都不是 y = … 或 x = … 的形式，可以先移项整理其中一个方程。代入法通常步骤稍多，但在应用题中逻辑非常清晰。
After substitution, expand brackets carefully and collect like terms. Then solve for the remaining variable and substitute back to find the other variable.
代入后要认真展开括号并合并同类项。然后解出剩下的那个变量，再代回原式求出另一个变量。
5. Worked Example: Substitution | 代入法典型例题
Solve the simultaneous equations:
解下列联立方程组：
y = 2x + 1 … (1)
3x + 2y = 16 … (2)
Substitute y = 2x + 1 into equation (2). This gives 3x + 2(2x + 1) = 16.
将 y = 2x + 1 代入方程 (2)，得到 3x + 2(2x + 1) = 16。
Expand and simplify: 3x + 4x + 2 = 16, so 7x + 2 = 16. Therefore 7x = 14 and x = 2.
展开并合并同类项：3x + 4x + 2 = 16，所以 7x + 2 = 16。因此 7x = 14，x = 2。
Substitute x = 2 back into y = 2x + 1 to get y = 2 × 2 + 1 = 5. The solution is (2, 5).
将 x = 2 代回 y = 2x + 1，得到 y = 2 × 2 + 1 = 5。解为 (2, 5)。
Check in equation (2): 3(2) + 2(5) = 6 + 10 = 16, so the solution is correct.
代入方程 (2) 检验：3 × 2 + 2 × 5 = 6 + 10 = 16，因此解是正确的。
6. Graphical Interpretation | 图像解法与几何意义
Each linear equation represents a straight line on the coordinate plane. The solution of the simultaneous equations is exactly the point where the two lines intersect.
每个线性方程在坐标平面上都表示一条直线。联立方程组的解正是这两条直线相交的点。
If the two lines intersect at one point, the system has one unique solution. The coordinates of the intersection are the values of x and y.
如果两条直线相交于一个点，那么方程组有唯一解。交点坐标就是 x 和 y 的值。
If the lines are parallel, there is no solution. If the lines are the same line, there are infinitely many solutions because every point on the line satisfies both equations.
如果两条直线平行，则方程组无解；如果两条直线重合，则有无穷多解，因为直线上的每一个点都同时满足两个方程。
7. Special Cases: No Solution and Infinite Solutions | 无解与无穷多解
A system has no solution when the two equations contradict each other, such as x + y = 3 and x + y = 5. Algebraically, elimination gives a false statement like 0 = 2.
当两个方程互相矛盾时，方程组无解，例如 x + y = 3 与 x + y = 5。用代数消元会得到 0 = 2 这样的错误等式。
Graphically, these equations represent two parallel lines. Since parallel lines never meet, there is no point that satisfies both equations.
从图像上看，这两个方程表示两条平行直线。因为平行线永不相交，所以不存在同时满足两个方程的点。
A system has infinitely many solutions when one equation is a multiple of the other, such as 2x + 2y = 6 and x + y = 3. Elimination gives 0 = 0, which is always true.
当一个方程是另一个方程的倍数时，方程组有无穷多解，例如 2x + 2y = 6 与 x + y = 3。消元后得到 0 = 0，这是恒等式。
In this case, both equations describe the same line. Any point on that line is a valid solution.
在这种情况下，两个方程描述的是同一条直线。这条直线上的任意一点都是有效解。
8. Word Problems | 应用题解题步骤
Many IGCSE questions are set in real-life contexts. Define two variables clearly, write two equations from the information, then solve using elimination or substitution.
许多 IGCSE 题目都设置在实际情境中。明确设出两个变量，根据题意列出两个方程，然后用消元法或代入法求解。
Example: Two adult tickets and three child tickets cost $24. Five adult tickets and two child tickets cost $49. Let a be the price of one adult ticket and c be the price of one child ticket.
例题：两张成人票和三张儿童票共 24 美元，五张成人票和两张儿童票共 49 美元。设 a 为一张成人票的价格，c 为一张儿童票的价格。
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