📚 Solving Simultaneous Equations | 解联立方程
Simultaneous equations are a core topic in IGCSE Mathematics. They test your ability to handle multiple algebraic relationships at once and appear frequently in both non-calculator and calculator papers. Mastering them opens the door to solving real-world problems involving two or more unknown quantities.
联立方程是IGCSE数学的核心内容。它考察你同时处理多个代数关系的能力，并且在非计算器和计算器试卷中都很常见。掌握联立方程，意味着你能够解决涉及两个或多个未知量的实际问题。
1. What Are Simultaneous Equations? | 什么是联立方程？
Simultaneous equations are a set of equations that share the same variables. A solution to the system is a set of values that satisfies every equation in the system at the same time.
联立方程是共享相同变量的一组方程。该方程组的解是一组同时满足其中每个方程的数值。
For two linear equations with two unknowns, there are three possible outcomes:
对于含有两个未知数的两个线性方程，有且仅有三种可能的结果：
One unique solution — the lines intersect at a single point.
唯一解——两条直线相交于一点。
No solution — the lines are parallel and never intersect.
无解——两条直线平行，永不相交。
Infinitely many solutions — the lines are coincident (the same line).
无穷多解——两条直线重合（是同一条直线）。
x + y = 8, x − y = 2 → x = 5, y = 3
es x = 5 and y = 3 satisfy both equations, so they form the unique solution of the system.
数值 x = 5 和 y = 3 同时满足两个方程，因此它们构成该方程组的唯一解。
2. The Elimination Method | 消元法
The elimination method involves adding or subtracting the equations to remove one variable. This reduces the system to a single equation in one variable.
消元法通过将两个方程相加或相减来消去一个变量，从而把方程组化为一元一次方程。
Example | 例题:
3x + y = 7 … (1)
2x − y = 3 … (2)
Add equation (1) and equation (2). The y terms cancel because +y and −y sum to zero.
将方程(1)和方程(2)相加，y 项相互抵消，因为 +y 与 −y 的和为零。
3x + 2x = 7 + 3
5x = 10 → x = 2
Now substitute x = 2 into equation (1):
再将 x = 2 代入方程(1)：
3(2) + y = 7 → y = 1
Therefore, the solution is x = 2, y = 1.
因此，解为 x = 2，y = 1。
If the coefficients are not opposites, first multiply one or both equations by a constant to make them match or cancel.
如果对应项的系数不是相反数，可以先将一个或两个方程同乘某个常数，使系数相同或相反，再进行加减消元。
3. The Substitution Method | 代入法
The substitution method is especially useful when one variable already has a coefficient of 1, or when one equation is explicitly solved for a variable.
当某个变量系数为1，或者某个方程已经写成“y = …”的形式时，代入法尤为方便。
Example | 例题:
y = 2x + 1 … (1)
x + y = 10 … (2)
Substitute y = 2x + 1 from (1) into (2):
将(1)中的 y = 2x + 1 代入(2)：
x + (2x + 1) = 10
3x + 1 = 10 → 3x = 9 → x = 3
Then substitute x = 3 back into (1):
然后把 x = 3 代回(1)：
y = 2(3) + 1 = 7
The solution is x = 3, y = 7.
解为 x = 3，y = 7。
Always check your answer by substituting both values into the original equations.
务必把两个值代回原方程进行检验。
4. The Graphical Method | 图象法
Graphically, the solution of two linear equations is the point of intersection of their straight-line graphs.
从图象上看，两个线性方程的解就是两条直线图象的交点坐标。
Example | 例题:
y = x + 1
y = −2x + 7
Plot both lines on the same axes. The point where they cross is the solution.
在同一坐标系中画出两条直线，它们相交的点坐标就是方程组的解。
| x | 0 | 1 | 2 |
| y = x + 1 | 1 | 2 | 3 |
| y = −2x + 7 | 7 | 5 | 3 |
points (2, 3) appear in the tables, so the lines intersect at (2, 3). Hence x = 2, y = 3.
两个表格中都有点 (2, 3)，因此两条直线相交于 (2, 3)，即 x = 2，y = 3。
The graphical method is useful for estimation, but in an exam you should use algebraic methods when an exact answer is required.
图象法适用于估算，但考试中需要精确答案时，应使用代数方法。
5. Linear and Quadratic Systems | 一次方程与二次方程联立
ratic equation, such as y = x² − 5. The substitution method is the standard approach here.
IGCSE考试常将一次方程与二次方程结合，例如 y = x² − 5。代入法是比较标准的方法。
Example | 例题:
y = x + 1 … (1)
y = x² − 3 … (2)
Substitute (1) into (2):
将(1)代入(2)：
x + 1 = x² − 3
x² − x − 4 = 0
This quadratic does not factorise easily, so use the quadratic formula:
该二次方程不易因式分解，因此使用求根公式：
x = [−(−1) ± √((−1)² − 4 × 1 × (−4))] / (2 × 1)
x = (1 ± √17) / 2
So x ≈ 2.56 or x ≈ −1.56. Use y = x + 1 to find the corresponding y-values:
因此 x ≈ 2.56 或 x ≈ −1.56。再由 y = x + 1 求出对应的 y 值：
y ≈ 3.56 or y ≈ −0.56
There are two solutions: (2.56, 3.56) and (−1.56, −0.56).
方程组有两组解：(2.56, 3.56) 和 (−1.56, −0.56)。
Remember: a line and a quadratic curve can intersect at 0, 1, or 2 points.
注意：一条直线与一条抛物线可能没有交点、有一个交点或两个交点。
6. Special Cases | 特殊情况
Not every system has a single unique solution. Recognising special cases quickly will save you time.
并非每个方程组都有唯一解。快速识别特殊情况可以为你节省时间。
Parallel lines: efficient ratio but different constant terms, e.g. y = 2x + 3 and y = 2x − 5. No solution.
平行线：两个方程中变量系数之比相同但常项不同，例如 y = 2x + 3 与 y = 2x − 5。无解。
Identical lines: nstant multiple of the other, e.g. 2x + y = 5 and 4x + 2y = 10. Infinitely many solutions.
重合直线：一个方程是另一个方程的常数倍，例如 2x + y = 5 与 4x + 2y = 10。有无穷多解。
Inconsistent system: elimination leads to a false statement such as 0 = 5. No solution.
矛盾方程组：消元后得到诸如 0 = 5 的错误等式。无解。
In exam questions, if you get a false statement, double-check your working before concluding there is no solution.
在考试中，如果得到错误等式，先检查运算过程，再下“无解”的结论。
7. Word Problems | 应用题
Simultaneous equations are powerful tools for modelling real situations. You need to translate words into mathematical equations.
联立方程是解决现实问题的有力工具。你需要把文字描述转化为数学方程。
Example | 例题:
The sum of two numbers is 15, and their difference is 3. Find the numbers.
两数之和为15，之差为3。求这两个数。
Let the numbers be x and y, where x > y:
设两数为 x 与 y，其中 x > y：
x + y = 15 … (1)
x − y = 3 … (2)
Add the equations:
两式相加：
2x = 18 → x = 9
Then 9 + y = 15, so y = 6.
于是 9 + y = 15，所以 y = 6。
The two numbers are 9 and 6.
这两个数是9和6。
Key steps: define variables clearly, write two independent equations, solve, and give your final answer as a sentence.
关键步骤：清晰设变量，写出两个相互独立的方程，求解，并用完整句子写出最终答案。
8. Common Mistakes | 常见错误
Many students lose marks on this topic due to careless errors. Watch out for the following:
许多学生在这一专题上因粗心而失分。请注意以下几点：
Forgetting to multiply both sides of an equation when using elimination.
用消元法时忘记两边同乘。
Making sign errors when substituting negative values, e.g. x − (−3) written as x − 3.
代入负数时符号出错，例如把 x − (−3) 写成 x − 3。
Not checking the solution in both original equations.
没有把解代回原方程进行验证。
When solving quadratic systems, including only one of the two possible solutions.
解二次方程组时，只写出两个解中的一个。
Writing coordinates in the wrong order, (y, x) instead of (x, y).
坐标顺序写反，写成 (y, x) 而不是 (x, y)。
Developing a checking habit is the single most effective way to avoid these errors.
养成检验的习惯是避免这些错误最有效的方法。
9. Exam Tips | 考试技巧
Use the following strategies to maximise your marks:
以下策略可以帮助你在这类题目中拿到更高的分数：
Choose the method that best fits the question: elimination for matching coefficients, substitution for a variable isolated or for quadratic systems.
选择合适的方法：系数匹配时用消元法；变量已单独写出或含二次方程时用代入法。
Show every step. An incorrect final answer still earns method marks if your working is clear.
写出每一步。即使最终答案错误，清晰的步骤也可获得方法分。
If time allows, substitute your answer back into the equations honestly.
时间允许时，务必把答案代回方程进行验证。
For word problems, state what your variables represent before writing equations.
对于应用题，先声明变量表示什么，再写方程。
Round to a sensible degree of accuracy only when the question asks, e.g. 3 significant figures.
只有在题目要求时才按指定精度取近似值，例如保留3位有效数字。
10. Practice Questions | 练习
Try these questions on your own before checking the answers.
先独立思考完成下面几题，再核对答案。
Question 1 | 题1: Solve x + 2y = 8 and 3x − 2y = 12.
Solution: x = 5, y = 1.5
Question 2 | 题2: Solve y = 3x − 2 and y = 2x² − 5x + 1.
Solution: x = 3, y = 7 and x = 0.5, y = −0.5
Question 3 | 题3: Two pens and three pencils cost $4. Four pens and one pencil cost $6. Find the price of one pen and one pencil.
Solution: pen = $1.40, pencil = $0.40
If you got all three correct, you have mastered the topic. If not, review the relevant section and try again.
三题全对，说明你已经掌握了这一专题。若有不擅长的，请回顾对应章节后再次尝试。
11. Summary | 总结
Simultaneous equations require two key skills: forming equations from information, and solving them accurately. Master elimination, substitution, and the graphical method, and practise enough past-paper questions to build speed.
联立方程需要两种关键能力：从已知信息中构建方程，以及准确求解。掌握消元法、代入法和图象法，并勤练真题以提升速度。
Remember that every method should give the same solution if applied correctly. Use one method to solve and another to check.
记住：方法只要应用正确，结果必然相同。可以用一种方法求解，用另一种方法检验。
With consistent practice, simultaneous equations will become one of the most reliable scoring areas in your IGCSE Mathematics exam.
只要坚持练习，联立方程将成为你IGCSE数学考试中最稳定的得分点之一。
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