📚 Simultaneous Equations | 联立方程详解
Simultaneous equations are a fundamental topic in IGCSE Mathematics. They involve finding the values of two or more unknown variables that satisfy two or more equations at the same time. Mastering this skill is essential for solving problems in algebra, geometry, and even real-life applications.
联立方程是IGCSE数学中的基础考点。它要求我们找到同时满足两个或两个以上方程的未知数值。掌握这一技能对于解决代数、几何乃至实际生活中的问题都至关重要。
1. What Are Simultaneous Equations? | 什么是联立方程
When two or more equations share the same unknown variables, we call them simultaneous equations. The solution is the set of values that makes every equation true at the same time. For example, in the system 2x + y = 5 and x – y = 1, the solution is x = 2 and y = 1, because both equations hold when these values are substituted.
当两个或多个方程含有相同的未知数时，我们称之为联立方程。解联立方程就是找到一组能同时使所有方程成立的数值。例如，在方程组 2x + y = 5 和 x – y = 1 中，解是 x = 2、y = 1，因为代入后两个方程都成立。
In the IGCSE syllabus, you will encounter linear simultaneous equations (where both equations are straight lines) and sometimes non-linear systems (where one equation is quadratic). The number of equations must equal the number of unknown variables for a unique solution to exist.
在IGCSE考纲中，你会遇到线性联立方程（两个方程都对应直线），有时也会遇到非线性方程组（其中一个方程是二次的）。方程的数量必须等于未知数的数量，才可能存在唯一解。
2. The Substitution Method | 代入法
The substitution method is especially useful when one equation has a single variable isolated or can be easily rearranged. You express one variable in terms of the other, then substitute that expression into the second equation. This reduces the system to a single equation in one variable.
代入法特别适合其中一个方程中某个变量的系数为1或容易变形的情况。你先将一个变量用另一个变量表示出来，然后将这个表达式代入第二个方程，从而把方程组转化为只含一个变量的一元方程。
Let’s solve the system y = 2x + 3 and 3x + 2y = 11 using substitution.
我们用代入法解方程组 y = 2x + 3 和 3x + 2y = 11。
y = 2x + 3 → 3x + 2(2x + 3) = 11
Simplify the second equation: 3x + 4x + 6 = 11, so 7x + 6 = 11, thus 7x = 5, giving x = 5/7. Now substitute back into y = 2x + 3:
化简第二个方程：3x + 4x + 6 = 11，即 7x + 6 = 11，所以 7x = 5，得到 x = 5/7。然后代回 y = 2x + 3：
y = 2 × (5/7) + 3 = 10/7 + 21/7 = 31/7
So the solution is x = 5/7 and y = 31/7. Always check by substituting both values into the original equations.
因此解为 x = 5/7，y = 31/7。记得将两个值代回原方程进行检验。
3. The Elimination Method | 消元法
The elimination method is often the fastest for linear systems. The goal is to add or subtract the equations so that one variable cancels out. You may first need to multiply one or both equations by a constant so that the coefficients of one variable are opposites or equal.
消元法对于线性方程组通常最高效。我们的目标是通过相加或相减使一个变量被消去。你可能需要先将一个或两个方程乘以适当的常数，使某个变量的系数互为相反数或相等。
Solve the system 2x + 3y = 8 and 3x – 2y = -1 by elimination.
用消元法解方程组 2x + 3y = 8 和 3x – 2y = -1。
Multiply the first equation by 2 and the second equation by 3, so the y-coefficients become 6 and -6:
将第一个方程乘以2，第二个方程乘以3，使y的系数变为6和-6：
4x + 6y = 16
9x – 6y = -3
Now add the two equations: 4x + 9x + 6y – 6y = 16 – 3, which gives 13x = 13, so x = 1. Substitute x = 1 into 2x + 3y = 8:
现在将两式相加：4x + 9x + 6y – 6y = 16 – 3，得到 13x = 13，所以 x = 1。将 x = 1 代入 2x + 3y = 8：
2(1) + 3y = 8 → 2 + 3y = 8 → 3y = 6 → y = 2
The solution is therefore x = 1, y = 2. In your exam, clearly show which equation you multiply and by what factor, because method marks are awarded.
因此解为 x = 1，y = 2。在考试中，请清楚写出你乘以了哪个方程以及乘以什么系数，因为步骤分是按过程给分的。
4. The Graphical Method | 图像法
In the graphical method, you plot both equations on the same set of axes. The coordinates of the intersection point represent the solution. This method is useful for visualising the meaning of the solution, but it relies on accurate drawing and reading, which may lead to approximation errors.
图像法是将两个方程画在同一个坐标系中。交点的坐标就是方程组的解。这种方法有助于直观理解解的含义，但依赖于绘图的准确性，读图时可能产生近似误差。
- Step 1: Rearrange each equation into the form y = mx + c (if they are straight lines). | 第一步：将每个方程化为 y = mx + c 的形式（如果它们是直线）。
- Step 2: Draw both lines on the same grid. | 第二步：在同一坐标网格中画出这两条直线。
- Step 3: Read the coordinates of the point where the lines intersect. | 第三步：读出两条直线交点的坐标。
- Step 4: Substitute the point into both equations to check. | 第四步：将交点坐标代入两个方程进行检验。
For example, the lines y = x + 1 and y = -x + 5 intersect at the point (2, 3). This means x = 2 and y = 3 is the solution to the system.
例如，直线 y = x + 1 和 y = -x + 5 的交点是 (2, 3)。这意味着 x = 2，y = 3 是该方程组的解。
In the IGCSE exam, graphical questions often ask you to use a given grid and draw both lines. Make sure you label the axes and the intersection point clearly.
在IGCSE考试中，图像类题目通常会给出网格并要求你画出两条直线。请务必标清坐标轴和交点。
5. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无穷多解
Not every pair of simultaneous equations has a unique solution. If the two lines are parallel (same gradient but different y-intercept), they never meet, so there is no solution. If the two equations represent the same line (one is a multiple of the other), every point on the line is a solution, so there are infinitely many solutions.
并不是每个联立方程组都有唯一解。如果两条直线平行（斜率相同但有不同截距），它们永远不会相交，因此无解。如果两个方程表示的是同一条直线（一个方程是另一个的倍数），则直线上每一点都是解，因此有无穷多解。
For example, the system y = 2x + 1 and y = 2x – 3 has no solution, because both lines have gradient 2 but different intercepts.
例如，方程组 y = 2x + 1 和 y = 2x – 3 无解，因为两条直线斜率都为2，但截距不同。
On the other hand, the system 2x + 4y = 8 and x + 2y = 4 has infinitely many solutions, because if you multiply the second equation by 2, you get the first equation exactly.
相反，方程组 2x + 4y = 8 和 x + 2y = 4 有无穷多解，因为将第二个方程乘以2就得到第一个方程。
In algebra, you can detect these cases during elimination. If a false statement like 0 = 5 appears, there is no solution. If a true statement like 0 = 0 appears, there are infinitely many solutions.
在代数求解时，你可以在消元过程中发现这些情况。如果出现 0 = 5 这样的矛盾式，则无解；如果出现 0 = 0 这样的恒等式，则有无穷多解。
6. Solving Word Problems | 应用题解法
Simultaneous equations are powerful tools for solving word problems. The key is to define the unknown variables clearly, then translate the conditions in the problem into two equations. Read the problem carefully and identify the relationships between quantities.
联立方程是解决应用题的有力工具。关键在于清晰定义未知变量，然后将题目中的条件转化为两个方程。请仔细阅读题目，识别数量之间的关系。
Example: “A pen and a notebook cost $5 together. The notebook costs $3 more than the pen. Find the cost of each item.”
例如：“一支笔和一个笔记本共5美元。笔记本比笔贵3美元。求每件物品的价格。”
Let x be the price of the pen and y be the price of the notebook. Then:
设 x 为笔的价格，y 为笔记本的价格。那么：
x + y = 5
y = x + 3
Substitute y = x + 3 into the first equation: x + (x + 3) = 5, so 2x + 3 = 5, giving 2x = 2 and x = 1. Then y = 1 + 3 = 4. The pen costs $1 and the notebook costs $4.
将 y = x + 3 代入第一个方程：x + (x + 3) = 5，即 2x + 3 = 5，所以 2x = 2，x = 1。于是 y = 1 + 3 = 4。笔的价格为1美元，笔记本为4美元。
Always state your final answer in a sentence, including the correct units. This improves clarity and earns communication marks.
始终用完整句子写出最终答案，并包含正确单位。这样能提高清晰度并获得表达分。
7. Non-linear Simultaneous Equations | 非线性联立方程
In the Extended IGCSE paper, you may need to solve a system where one equation is linear and the other is quadratic, such as y = x² + 1 and y = 3x – 1. These systems can have up to two solutions, because a line and a curve can intersect at most twice.
在IGCSE扩展试卷中，你可能会遇到一个一次方程和一个二次方程组成的方程组，例如 y = x² + 1 和 y = 3x – 1。这类方程组最多可能有两个解，因为直线与曲线最多相交于两点。
To solve such a system, use the substitution method: replace y in one equation with its expression from the other equation.
解这种方程组时通常使用代入法：将其中一个方程中的 y 用另一个方程的表达式替换。
x² + 1 = 3x – 1 → x² – 3x + 2 = 0
Factorise: (x – 1)(x – 2) = 0, so x = 1 or x = 2. Now find the corresponding y-values by substituting into the linear equation:
因式分解：(x – 1)(x – 2) = 0，所以 x = 1 或 x = 2。然后代入线性方程求对应的 y 值：
When x = 1, y = 3(1) – 1 = 2. When x = 2, y = 3(2) – 1 = 5. So the solutions are (1, 2) and (2, 5).
当 x = 1 时，y = 3(1) – 1 = 2。当 x = 2 时，y = 3(2) – 1 = 5。因此解为 (1, 2) 和 (2, 5)。
In your working, always show the substitution step clearly, and remember to solve the resulting quadratic by factorisation, completing the square, or the quadratic formula.
解题时要清楚展示代入过程，并记得用因式分解、配方法或求根公式来解所得的一元二次方程。
8. Common Mistakes and Exam Tips | 常见错误与考试技巧
Many students lose marks in simultaneous equation questions because of small errors. Being aware of common pitfalls will help you avoid them in the exam.
许多学生在联立方程题中因细小错误而丢分。了解常见陷阱有助于你在考试中避开它们。
- Mistake 1: Forgetting to multiply every term when multiplying an entire equation. | 错误一：将一个方程整体乘以常数时，忘记乘每一项。
- Mistake 2: Sign errors when subtracting equations. Change all signs carefully. | 错误二：相减时符号出错。请仔细变号。
- Mistake 3: Substituting the value back into the wrong equation. | 错误三：代回时代入了错误的方程。
- Mistake 4: Not writing the final answer as an ordered pair (x, y). | 错误四：没有将最终答案写成有序数对 (x, y)。
Exam tip: Always check your solution by substituting both values into both original equations. This takes only a few seconds and guarantees full accuracy.
考试技巧：务必把两个值代回原方程检验。这只需几秒钟，却能确保完全正确。
Also, read the question to see what form it asks for — exact fractions are often required, rather than decimal approximations.
此外，注意题目要求的答案形式——通常要求精确分数，而不是近似小数。
9. Practice Questions | 练习题
Try these questions yourself before checking the answers. They cover the main types of simultaneous equations you will face in the IGCSE exam.
请先独立尝试下列题目，再对照答案。它们覆盖了IGCSE考试中联立方程的主要题型。
| Question | 题目 | Answer | 答案 |
| 1. Solve: 3x + y = 7 and x – y = 1 | 1. 解方程组：3x + y = 7，x – y = 1 | x = 2, y = 1 |
| 2. Solve: 2x + 5y = 12 and 4x – 3y = -2 | 2. 解方程组：2x + 5y = 12，4x – 3y = -2 | x = 1, y = 2 |
| 3. Solve: y = x² and y = 2x + 3 | 3. 解方程组：y = x²，y = 2x + 3 | x = -1, y = 1 and x = 3, y = 9 |
| 4. Two numbers have a sum of 15 and a difference of 3. Find the numbers. | 4. 两个数之和为15，差为3，求这两个数。 | 9 and 6 |
If you found Q3 difficult, review Section 7 on non-linear simultaneous equations. For Q4, set up two equations using x + y = 15 and x – y = 3, then solve by elimination.
如果你觉得第3题困难，请回顾第7节关于非线性联立方程的内容。对于第4题，列出方程 x + y = 15 和 x – y = 3，然后用消元法求解。
10. Conclusion | 总结
Simultaneous equations are a powerful and versatile skill in IGCSE Mathematics. Whether you use substitution, elimination, or graphs, the key is to understand the structure of the problem and to practise consistently. Mastery of this topic will not only earn you marks but also strengthen your overall algebraic ability.
联立方程是IGCSE数学中一项强大而通用的技能。无论你使用代入法、消元法还是图像法，关键是理解问题的结构并坚持练习。掌握这一主题不仅能帮你获得分数，还能增强你整体的代数能力。
Remember to check your answers, show clear workings, and read each question carefully. With regular practice, simultaneous equations will become one of the most reliable topics in your exam.
记住要检查答案、清晰展示计算过程，并仔细审题。通过规律练习，联立方程将成为你考试中最稳定的得分点之一。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导