📚 Solving Simultaneous Equations | 解联立方程组
Simultaneous equations are a cornerstone of IGCSE algebra. They require you to find values of unknown variables that satisfy two or more equations at the same time. This topic links algebraic manipulation, graphical interpretation and real-world modelling, so exam questions often combine methods, sketching and problem solving.
联立方程组是 IGCSE 代数的核心内容。它要求你求出同时满足两个或多个方程的未知变量值。这一主题将代数运算、图像解释和实际建模联系起来，因此考试题常结合方法选择、作图与问题解决进行综合考查。
1. What Are Simultaneous Equations? | 什么是联立方程组？
A simultaneous equation system consists of two or more equations that share the same variables. In IGCSE Mathematics, most systems involve two unknown variables, usually x and y. A valid solution must satisfy every equation in the system; if it works for one equation but not the other, it is not a solution.
联立方程组由两个或多个含有相同变量的方程组成。在 IGCSE 数学中，大多数方程组包含两个未知变量，通常用 x 和 y 表示。有效的解必须同时满足方程组中的每一个方程；若只满足其中一个方程而不是全部，则不是方程组的解。
For example, x + y = 5 and x − y = 1 form a system. The pair x = 3, y = 2 satisfies both equations, so (3, 2) is the solution.
例如，x + y = 5 与 x − y = 1 构成一个方程组。x = 3, y = 2 这一组值同时满足两个方程，因此 (3, 2) 是该方程组的解。
2. Graphical Interpretation | 图像解释
Each linear equation in two variables represents a straight line on the coordinate plane. The solution to a simultaneous linear system is the coordinate point where the two lines intersect. This geometric view helps show why systems can have one solution, no solution or infinitely many solutions.
每个二元一次方程在坐标平面上表示一条直线。线性联立方程组的解就是两条直线交点的坐标。这种几何观点有助于理解为什么方程组可能有一个解、无解或无穷多个解。
If the lines meet at exactly one point, the system has one unique solution. Parallel lines have no intersection, so the system has no solution. Coincident lines lie on top of each other, giving infinitely many solutions.
若两直线恰好交于一点，则方程组有唯一解。平行直线没有交点，因此方程组无解。重合直线完全重叠，因此方程有无穷多个解。
3. Solving by Elimination | 消元法
Elimination is usually the fastest method for solving two linear equations. You add or subtract the equations to remove one variable. This method works best when the coefficients of one variable are the same or can easily be made the same.
消元法通常是解两个线性方程组最快的方法。通过将两个方程相加或相减来消去一个变量。当某一个变量的系数相同或容易化为相同时，消元法最为有效。
Write both equations in the standard form ax + by = c, with like terms aligned. Then multiply one or both equations by suitable numbers so that the coefficients of x or y have the same magnitude. Add or subtract the equations to eliminate that variable, then solve the resulting one-variable equation.
先将两个方程写成标准形式 ax + by = c，并将同类项对齐。然后给一个或两个方程乘以适当的数，使 x 或 y 的系数大小相同。将方程相加或相减以消去该变量，再解得到的一元一次方程。
Finally, substitute the found value back into one original equation to find the other variable. Always check both original equations to confirm the solution.
最后，将求得的值代回其中一个原方程，求出另一个变量。始终代入两个原方程进行检验，以确认解的正确性。
Solve: 2x + y = 10 and x − y = 2 → Add equations → 3x = 12 → x = 4, y = 2
解方程组 2x + y = 10 与 x − y = 2，将两式相加得 3x = 12，所以 x = 4，再得 y = 2。
| Method | 方法 | Best for | 最适合 |
|---|---|
| Elimination | 消元法 | Linear systems with easy coefficient alignment | 系数易于配齐的线性方程组 |
| Substitution | 代入法 | One variable already isolated, or one equation is non-linear | 某一变量已单独表示，或其中一个方程为非线性方程 |
4. Solving by Substitution | 代入法
Substitution is useful when one equation already has a variable isolated, such as y = 2x + 1, or when one equation is not linear. Rearrange one equation to express y in terms of x or x in terms of y, then replace that variable in the other equation.
当一个方程已经单独表示出某个变量，例如 y = 2x + 1，或其中一个方程不是线性方程时，代入法非常有用。先将一个方程变形为用 x 表示 y 或用 y 表示 x，再将该表达式代入另一个方程。
For the system y = 2x + 1 and 3x + 2y = 16, substitute y = 2x + 1 into the second equation to get 3x + 2(2x + 1) = 16. This simplifies to 7x + 2 = 16, then 7x = 14, so x = 2. Substituting back gives y = 5.
对于方程组 y = 2x + 1 与 3x + 2y = 16，将 y = 2x + 1 代入第二个方程，得到 3x + 2(2x + 1) = 16。化简得 7x + 2 = 16，再得 7x = 14，所以 x = 2。代回后得到 y = 5。
y = 2x + 1, 3x + 2y = 16 → x = 2, y = 5
方程组 y = 2x + 1 与 3x + 2y = 16 的解为 x = 2, y = 5。
5. Special Cases: No Solution and Infinite Solutions | 特殊情形：无解与无穷多解
Not every simultaneous system has a unique solution. When the two lines are parallel, they have the same slope but different y-intercepts. Algebraically, using elimination will produce a false statement such as 0 = 5, meaning there is no solution.
并非每个联立方程组都有唯一解。当两条直线平行时，它们的斜率相同但截距不同。在代数运算中，使用消元法会产生如 0 = 5 的假等式，说明方程组无解。
When the two equations represent the same line, the graphs coincide. Elimination will lead to an identity such as 0 = 0, meaning there are infinitely many solutions. You can express the solution set using a parameter or by giving one variable in terms of the other.
当两个方程表示同一条直线时，它们的图像重合。消元后会得到如 0 = 0 的恒等式，说明方程组有无穷多个解。你可以用参数表示解集，或将一个变量用另一个变量表示出来。
6. Simultaneous Linear and Quadratic Equations | 一次与二次联立方程
IGCSE also tests simultaneous equations where one equation is linear and the other is quadratic, such as y = x + 2 and y = x² − 1. Graphically, this is the intersection of a straight line and a parabola, so there may be zero, one or two solutions.
IGCSE 还会考查一个方程是一次方程、另一个方程是二次方程的联立方程组，例如 y = x + 2 与 y = x² − 1。从图像上看，这是直线与抛物线的交点，因此可能有零个、一个或两个解。
To solve, use substitution. Replace y in the quadratic equation with the linear expression. This produces a quadratic equation in one variable, which can be solved by factorising, completing the square or the quadratic formula.
求解方法是使用代入法。将二次方程中的 y 替换为一次方程给出的表达式，从而得到一个关于一个变量的二次方程。该方程可通过因式分解、配方法或二次公式求解。
y = x² − 3x + 4 and y = x + 1 → x² − 4x + 3 = 0 → (x − 1)(x − 3) = 0
对于 y = x² − 3x + 4 与 y = x + 1，令两式相等得到 x² − 4x + 3 = 0，即 (x − 1)(x − 3) = 0。
The resulting x-values are 1 and 3. Substituting back gives the solutions (1, 2) and (3, 4). Always check both solutions in both original equations.
可解得 x = 1 或 x = 3。代回后得到 (1, 2) 和 (3, 4) 两组解。务必在原来的两个方程中检验两组解。
7. Using Graphs to Solve Quadratic Simultaneous Equations | 用图像解二次联立方程
When a question asks for a graphical solution, plot the straight line and the quadratic curve accurately on graph paper. The x-coordinates of the intersection points give the solutions of the simultaneous system. You may then read off the y-coordinates or substitute back to find exact values.
当题目要求用图像求解时，应在坐标纸上准确绘制直线和二次曲线。交点横坐标就是联立方程组的解。然后你可以读出纵坐标，或代回求精确值。
You should also be able to sketch the graphs to determine the number of solutions without calculating them fully. A line cutting a parabola twice gives two solutions, a tangent gives one solution, and a line missing the parabola gives no solution.
你还应学会通过画草图判断解的个数，而不必完整计算。直线与抛物线相交于两点，则有两个解；直线与抛物线相切，则有一个解；直线与抛物线无交点，则无解。
8. Real-Life Applications | 实际应用
Simultaneous equations model many real-world situations, such as costs and revenue, mixtures, speed and time, tickets and pricing, or supply and demand. To set up equations from a word problem, define the variables clearly and write two relationships using the given information.
联立方程组可用于建模许多实际问题，如成本与收入、混合物、速度与时间、票价与定价、供求关系等。解应用题时，要明确定义变量，并根据已知信息写出两个关系式。
For example, if 3 pencils and 2 pens cost £4.80, while 5 pencils and 3 pens cost £7.20, you can write two equations and solve for the individual prices. Always express your final answer in the context of the problem and check whether negative or fractional answers make sense.
例如，若 3 支铅笔和 2 支钢笔花费 £4.80，而 5 支铅笔和 3 支钢笔花费 £7.20，你可以列出两个方程并求解单品价格。最后一定要结合题意表达答案，并检查负数解或分数解是否合理。
9. Common Mistakes and Exam Tips | 常见错误与考试技巧
A common mistake in elimination is a sign error when subtracting equations. For example, subtracting x − y = 2 from 2x + y = 10 incorrectly may give the wrong coefficient for y. Write the subtraction vertically and change signs carefully.
消元法中常见的错误是相减方程时出现符号错误。例如，从 2x + y = 10 中减去 x − y = 2，如果减号处理不当，y 的系数就会出错。应竖直列式并仔细变号。
Another frequent error is multiplying only part of an equation by a coefficient. Every term on both sides must be multiplied. In quadratic simultaneous equations, students sometimes forget to find both solutions or fail to check them in the original linear equation.
另一个常见错误是只给方程的一部分乘以某个系数。等式两边的每一项都必须乘以同一个数。在二次联立方程中，学生有时忘记求出两组解，或没有代入原一次方程进行检验。
Always write your steps clearly, label the equations, and verify the solution by substituting into both original equations. In a word problem, revisit the units and the reasonableness of the answer.
始终清楚书写步骤，给方程编号，并将解代入两个原方程进行验算。在应用题中，还要回看单位和答案的合理性。
10. Worked Examples | 典型例题
Example 1: Solve the linear system 2x + y = 10 and x − y = 2. Adding the two equations gives 3x = 12, so x = 4. Substituting x = 4 into x − y = 2 gives 4 − y = 2, so y = 2. The solution is (4, 2).
例题 1：解线性方程组 2x + y = 10 与 x − y = 2。两式相加得 3x = 12，所以 x = 4。将 x = 4 代入 x − y = 2，得 4 − y = 2，所以 y = 2。解为 (4, 2)。
Example 2: Solve the linear-quadratic system y = x² − 3x + 4 and y = x + 1. Set the two expressions for y equal: x² − 3x + 4 = x + 1. Rearrange to x² − 4x + 3 = 0. Factorise: (x − 1)(x − 3) = 0. Thus x = 1 or x = 3. When x = 1, y = 2; when x = 3, y = 4. The solutions are (1, 2) and (3, 4).
例题 2：解一次与二次联立方程组 y = x² − 3x + 4 与 y = x + 1。令两个关于 y 的表达式相等：x² − 3x + 4 = x + 1。移项得 x² − 4x + 3 = 0。因式分解为 (x − 1)(x − 3) = 0。所以 x = 1 或 x = 3。当 x = 1 时，y = 2；当 x = 3 时，y = 4。解为 (1, 2) 和 (3, 4)。
These worked examples show that a clear, structured method can make simultaneous equations straightforward, even when one equation is quadratic.
这些典型例题表明，清晰、有步骤的方法可以使联立方程组问题变得简单，即使其中一个方程是二次方程也不例外。
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