📚 Simultaneous Equations | 联立方程组
Simultaneous equations are a set of two or more equations that share the same variables. The solution to a simultaneous system is the set of values that satisfy all equations at once. In the Edexcel IGCSE Mathematics syllabus, you are expected to solve linear simultaneous equations, one linear and one quadratic, and interpret solutions graphically. Mastering these techniques is essential, as they appear regularly in both the Foundation and Higher tier papers, often embedded in word problems, geometry, or break‑even analysis.
联立方程组是由两个或多个含有相同变量的方程组成的集合。联立方程组的解是同时满足所有方程的一组数值。在 Edexcel IGCSE 数学大纲中，你需要掌握解线性联立方程组、一个线性一个二次的方程组，以及通过图像理解它们的解。熟练掌握这些方法至关重要，因为无论是在基础卷还是进阶卷中，它们都经常出现在文字题、几何题或盈亏平衡分析题里。
1. What Are Simultaneous Equations? | 什么是联立方程组？
A simultaneous equation system requires you to find values for the unknowns (often x and y) that make every equation true at the same time. In IGCSE, the most common type involves two equations with two unknowns. For example, 2x + y = 7 and x − y = 2 form a pair of linear simultaneous equations because the highest power of any variable is 1.
联立方程组要求你找到未知数（通常是 x 和 y）的值，让每个方程同时成立。在 IGCSE 中，最常见的类型是两个方程、两个未知数。例如，2x + y = 7 和 x − y = 2 组成了一对线性联立方程组，因为任意变量的最高次幂都是 1。
2. The Elimination Method | 消元法
The elimination method works best when both equations are linear. You add or subtract the equations to remove one variable, then solve the resulting single equation. For instance, with 2x + y = 7 and x − y = 2, adding them gives 3x = 9, so x = 3. Substituting back gives y = 1. Always check by plugging the values into the second equation: 3 − 1 = 2, which is correct.
当两个方程都是线性方程时，消元法最为适用。将两个方程相加或相减以消去其中一个未知数，然后求解得到的一元一次方程。例如，对于 2x + y = 7 和 x − y = 2，相加得 3x = 9，所以 x = 3；再代回原式得 y = 1。务必检验：3 − 1 = 2，吻合无误。
3. Balancing Coefficients Before Elimination | 配平系数再消元
Sometimes the coefficients of the variable you want to eliminate do not match. You must multiply one or both equations by suitable constants to make the coefficients equal in magnitude. For example, to solve 3x + 2y = 12 and 2x + 5y = −2, you could multiply the first equation by 2 and the second by 3 to get 6x + 4y = 24 and 6x + 15y = −6. Subtracting gives 11y = −30, so y = −30/11. Then find x by substitution.
有时你想消去的那个未知数的系数并不匹配，就必须用一个合适的常数去乘某一个或两个方程，使系数的绝对值相等。例如，解 3x + 2y = 12 和 2x + 5y = −2 时，你可以把第一个方程乘以 2，第二个方程乘以 3，得到 6x + 4y = 24 和 6x + 15y = −6；相减得 11y = −30，即 y = −30/11，再代入求 x。
4. The Substitution Method | 代入法
The substitution method is useful when one equation gives an explicit expression for one variable, such as y = 2x − 1. You replace that variable in the other equation and solve. For instance, given x + 3y = 10 and y = 2x − 1, substitute to get x + 3(2x − 1) = 10, leading to 7x = 13 and x = 13/7, then y = 19/7. This method is particularly helpful when one equation is non‑linear.
当其中一个方程明确表示出某个未知数的表达式（例如 y = 2x − 1）时，代入法就非常实用。将表达式代入另一个方程并求解。例如，给定 x + 3y = 10 和 y = 2x − 1，代入得 x + 3(2x − 1) = 10，化简得 7x = 13，x = 13/7，进而 y = 19/7。当其中一个方程为非线性方程时，这一方法尤其有效。
5. Interpreting Solutions Graphically | 从图像角度理解解
Graphically, the solution of two simultaneous equations corresponds to the point where their graphs intersect. For two linear equations, the lines intersect at exactly one point if they are neither parallel nor coincident. If the lines are parallel, there is no solution; if they coincide, there are infinitely many solutions. Sketching the lines can help you check the reasonableness of an algebraic solution.
从图像上看，两个联立方程组的解对应于它们图像的交点。对于两个线性方程，如果既不平行也不重合，则两条直线恰有一个交点；若平行则无解；若重合则有无穷多解。画出直线草图有助于检验代数解的合理性。
6. One Linear and One Quadratic Equation | 一个线性方程与一个二次方程
IGCSE Higher tier candidates must also solve a system where one equation is linear and the other is quadratic, e.g. y = x² − 3x + 2 and y = 2x − 1. Substitution is the standard approach: replace y in the quadratic with the linear expression to form a quadratic in x alone, then factorise or use the quadratic formula. After finding x-values, substitute back into the linear equation to get y. Expect two solutions, one solution (tangent), or no real solutions.
进阶卷的考生还需求解一个线性、一个二次的联立方程组，例如 y = x² − 3x + 2 和 y = 2x − 1。标准方法是代入：用线性表达式替换二次方程中的 y，得到仅含 x 的一元二次方程，然后因式分解或用求根公式求解。求出 x 的值后，再代回线性方程求 y。可能出现两个解、一个解（相切）或无实数解。
7. Solving Word Problems with Simultaneous Equations | 用联立方程组解文字题
Many IGCSE problems describe real‑life situations: buying items with given totals, ages, or mixing solutions. Define two variables, write two equations from the information, and solve. For example: ‘Three pens and two notebooks cost £8; two pens and five notebooks cost £13.’ Let p be the price of a pen and n the price of a notebook. Form 3p + 2n = 8 and 2p + 5n = 13, then solve by elimination or substitution.
许多 IGCSE 题目描述现实情境：给定总价的购物问题、年龄问题或溶液混合问题。设两个变量，根据信息写出两个方程并求解。例如：“三支笔和两个笔记本共 8 英镑；两支笔和五个笔记本共 13 英镑。”设笔的单价为 p，笔记本的单价为 n，列出 3p + 2n = 8 和 2p + 5n = 13，然后用消元法或代入法求解。
8. Breaking Even: Cost and Revenue | 盈亏平衡：成本与收入
A typical application involves cost and revenue functions. The break‑even point occurs when total cost equals total revenue. If cost C = 50 + 3x and revenue R = 8x, where x is the number of units, solve 50 + 3x = 8x to get x = 10. The simultaneous equations are C = 50 + 3x and R = 8x, with the condition C = R. This is a simple linear‑linear system.
一个典型的应用涉及成本函数和收入函数。盈亏平衡点出现在总成本等于总收入时。若成本 C = 50 + 3x，收入 R = 8x，其中 x 为商品数量，解 50 + 3x = 8x 得 x = 10。联立方程组为 C = 50 + 3x 和 R = 8x，并附加条件 C = R。这是一个简单的线性‑线性方程组。
9. Speed, Distance and Time Systems | 速度、距离与时间的方程组
Simultaneous equations can model journeys. For instance, a car travels a total distance of 300 km in 4 hours. Part of the journey is at 80 km/h, the rest at 60 km/h. Let t₁ be the time at 80 km/h and t₂ the time at 60 km/h. Then t₁ + t₂ = 4 and 80t₁ + 60t₂ = 300. Solve to find the time spent at each speed, and then the respective distances.
联立方程组可以建立行程模型。例如，一辆车在 4 小时内总共行驶 300 公里，部分路程以 80 km/h 行驶，其余以 60 km/h 行驶。设以 80 km/h 行驶的时间为 t₁，60 km/h 的时间为 t₂，则 t₁ + t₂ = 4，且 80t₁ + 60t₂ = 300。解出各速度下所用时间，进而得到相应的路程。
10. Common Mistakes and How to Avoid Them | 常见错误与规避方法
One frequent error is forgetting to multiply the whole equation when balancing coefficients, leading to a wrong constant term. Another is misreading the sign when subtracting equations, especially when a negative sign appears. Always double‑check your arithmetic and, crucially, substitute the obtained values back into the original equations to verify they satisfy both. If a quadratic arises, remember that a negative discriminant means no real solutions.
一个常见错误是在配平系数时忘记乘遍整个方程，导致常数项出错。另一个是相减方程时正负号弄错，尤其是遇到负号时。务必仔细检查计算，关键是要将求得的数值代回原方程验证是否都成立。如果出现二次方程，记住判别式为负意味着没有实数解。
11. Using Technology: Graphical Calculators | 使用技术：图形计算器
At IGCSE, graphical calculators or graphing software can be used to visualise solutions. Plot both equations and use the intersect function to find coordinates of crossing points. This is especially helpful for checking algebraic solutions or solving systems with a quadratic. However, you must still be able to solve by hand, as non‑calculator papers demand full written methods.
在 IGCSE 中，可以使用图形计算器或绘图软件直观地观察解。画出两个方程的图像，并用交点功能求得交点坐标。这对检验代数解或求解含二次方程的方程组尤其有用。但你必须仍然能够手算，因为非计算器试卷要求展示完整的书写过程。
12. Summary of Solution Steps | 求解步骤总结
To solve simultaneous equations efficiently, first decide whether elimination or substitution is more suitable. For two linear equations, adjust coefficients and eliminate. For mixed linear‑quadratic systems, substitute the linear expression into the quadratic. Sketch graphs to confirm the number of solutions. Always write your final answer clearly as x = …, y = … and check your work.
要高效解联立方程组，首先判断消元法还是代入法更合适。线性方程组则配平系数后消元；线性‑二次方程组则将线性表达式代入二次方程。画出图像草图以确认解的数量。答案应清晰地写成 x = …、y = … 的形式并验算。
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