📚 IGCSE Maths T-4-1030: Simultaneous Equations | IGCSE 数学 T-4-1030：联立方程
Simultaneous equations are a central topic in IGCSE Mathematics. They require you to find values of unknown variables that satisfy more than one equation at the same time, and they appear on almost every extended and core paper in some form.
联立方程是 IGCSE 数学的一个核心主题。它要求你求出能够同时满足多个方程的未知变量的值，而且它几乎会以各种形式出现在每一份核心卷与拓展卷中。
This revision guide covers the graphical meaning of a solution, the substitution and elimination methods, special cases with no solution or infinitely many solutions, word problems, checking techniques and the mistakes that cost most exam marks.
本复习指南涵盖解的图形意义、代入法与消元法、无解与无穷多解等特殊情形、应用题、检验技巧，以及最容易丢分的常见错误。
1. What Are Simultaneous Equations? | 什么是联立方程
A simultaneous equation is a set of two or more equations that share the same variables. The solution is the set of values that makes every equation true at the same time.
联立方程是一组含有相同变量的两个或两个以上的方程。解就是能同时使每一个方程都成立的那组值。
The word 'simultaneous' simply means 'at the same time'. You are not solving one equation and then the other; you are looking for one pair of values (x, y) that works in both equations at once.
"联立"（simultaneous）的意思就是"同时"。你不是先解一个方程再解另一个，而是寻找一对值 (x, y)，使它在两个方程中同时成立。
rk (4 + 1 = 5) but the second would give 4 − 1 = 3, not 1, so that pair is not a solution.
例如，方程组 x + y = 5 与 x − y = 1 的解是 x = 3、y = 2，因为 3 + 2 = 5 且 3 − 2 = 1。如果你试 x = 4、y = 1，第一个方程成立（4 + 1 = 5），但第二个方程给出 4 − 1 = 3，不等于 1，所以这组值不是解。
At IGCSE level you will usually work with two linear equations in two unknowns. The same logic can be extended to three equations in three unknowns, and to one linear plus one non-linear equation, but the two-variable linear case is the main exam focus.
在 IGCSE 阶段，你通常处理的是含有两个未知数的两个线性方程。同样的逻辑可以推广到三个方程三个未知数，以及"一个线性方程加一个非线性方程"的情形，但两变量的线性方程组才是考试的主要重点。
orms are accepted, but you must give both values - a lone x = 3 earns only half the marks.
答案既可以写成 x = 3、y = 2，也可以写成坐标对 (3, 2)。两种写法都被接受，但你必须同时给出两个值——只写 x = 3 只能拿到一半的分。
2. Graphical Meaning: Intersection of Lines | 图形意义：直线的交点
Every linear equation in x and y represents a straight line on a coordinate grid. Solving a pair of simultaneous equations is therefore the same as finding the point where the two lines cross.
每一个关于 x 和 y 的线性方程在坐标系中都表示一条直线。因此，解一组联立方程等价于找出两条直线的交点。
Because two straight lines can only meet in one of three ways, there are exactly three possible outcomes for a linear system.
由于两条直线只可能有三种相对关系，所以线性方程组恰好有三种可能的结果。
- A single crossing point: one unique solution, for example (3, 2).
相交于一点：唯一解，例如 (3, 2)。 - Parallel lines that never meet: no solution at all.
平行且永不相交：无解。 - Two lines that lie on top of each other: infinitely many solutions.
两条直线完全重合：无穷多解。
To solve graphically, build a table of values for each equation, plot both lines accurately with a ruler, and read off the coordinates of the intersection. This method is excellent for checking an algebraic answer.
用图解法求解时，先为每个方程列出取值表，用直尺准确地画出两条直线，然后读出交点的坐标。这种方法非常适合用来检验代数解。
Remember that a graphical solution is only approximate unless the intersection falls exactly on a grid line. If you read off (2.9, 2.1) when the true answer is (3, 2), you may lose accuracy marks, so always state the algebraic method as your main answer.
记住，除非交点正好落在网格线上，否则图解法的结果只是近似值。如果你读出 (2.9, 2.1) 而真实答案是 (3, 2)，就可能因为精度不够而丢分，所以应把代数方法作为主要解法。
3. The Substitution Method | 代入消元法
Substitution means rewriting one equation so that one variable is expressed in terms of the other, then replacing that variable in the second equation. This reduces the system to a single equation in one unknown.
代入法的意思是：把其中一个方程改写，使某个变量用另一个变量表示，再把该表达式代入第二个方程。这样方程组就化归为一个只含一个未知数的方程。
Worked example: solve y = 2x + 1 and 3x + y = 11.
例题：解 y = 2x + 1 与 3x + y = 11。
3x + (2x + 1) = 11 → 5x + 1 = 11 → 5x = 10 → x = 2
n is x = 2, y = 5. Check: 3(2) + 5 = 11 and 5 = 2(2) + 1, so both equations are satisfied.
再代回较简单的方程：y = 2(2) + 1 = 5。所以解是 x = 2、y = 5。检验：3(2) + 5 = 11，且 5 = 2(2) + 1，两个方程都成立。
a coefficient, you must divide every term, for example 4y = 3x + 8 becomes y = 0.75x + 2.
当某个变量已经单独位于等号一侧（y = ... 或 x = ...），或者某个系数是 1 或 −1 时，代入法最快。如果被表示的那个变量带有系数，你必须把每一项都除以该系数，例如 4y = 3x + 8 要变成 y = 0.75x + 2。
Typical slips include dropping a minus sign when substituting a negative bracket, and forgetting to multiply the whole bracket: 3(2x − 1) is 6x − 3, not 6x − 1.
常见的失误包括：代入负括号时丢掉负号，以及忘记乘遍整个括号：3(2x − 1) 等于 6x − 3，而不是 6x − 1。
4. The Elimination Method | 加减消元法
Elimination means adding or subtracting the two equations, after multiplying one or both of them if necessary, so that one variable cancels out. It is usually the fastest method for the standard IGCSE layout.
消元法的意思是：必要时先把其中一个或两个方程乘以某个倍数，然后把两个方程相加或相减，使其中一个变量被消掉。对于标准的 IGCSE 题型，它通常是最快的方法。
Example A - coefficients already matched: solve 2x + 3y = 12 and 4x − 3y = 6.
例 A——系数已经配对：解 2x + 3y = 12 与 4x − 3y = 6。
(2x + 3y) + (4x − 3y) = 12 + 6 → 6x = 18 → x = 3
6 + 3y = 12, so 3y = 6 and y = 2. Check in the second: 4(3) − 3(2) = 12 − 6 = 6. Correct.
把 x = 3 代入第一个方程得 6 + 3y = 12，所以 3y = 6，y = 2。代入第二个方程检验：4(3) − 3(2) = 12 − 6 = 6，正确。
y = 14. Multiply the first by 3 and the second by 2 to make the y coefficients equal at 6.
例 B——需要乘倍数：解 3x + 2y = 16 与 2x + 3y = 14。把第一式乘 3、第二式乘 2，使 y 的系数都变成 6。
bsp; minus 4x + 6y = 28 gives 5x = 20 → x = 4
Then 3(4) + 2y = 16, so 2y = 4 and y = 2. Check: 2(4) + 3(2) = 8 + 6 = 14. Correct.
于是 3(4) + 2y = 16，得 2y = 4，y = 2。检验：2(4) + 3(2) = 8 + 6 = 14，正确。
Rule of thumb: if the two matching coefficients have opposite signs, add the equations; if they have the same sign, subtract them. When you multiply an equation, every single term must be multiplied, including the number on the right-hand side.
经验法则：如果两个对应系数符号相反，就把方程相加；符号相同，就相减。乘倍数时，每一项都必须乘，包括等号右边的常数。
5. Choosing the Best Method | 如何选择最合适的方法
Both algebraic methods give the same answer, so choose the one that creates the least algebra. The table below summarises the decision shown in mark schemes and examiner reports.
两种代数方法得到的答案相同，所以要选择运算量最小的那一种。下表总结了评分方案与考官报告中的选择依据。
| Situation | 情形 | Best method | 最佳方法 | Reason | 原因 |
| One variable already isolated, e.g. y = 3x − 2 某个变量已经单独写出 |
Substitution | 代入法 | Nothing needs rearranging 无需变形 |
| A coefficient is 1 or −1 某系数为 1 或 −1 |
Substitution | 代入法 | Rearranging is quick and clean 变形快且整洁 |
| Coefficients are multiples of each other
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