📚 031 Simultaneous Equations | 031 联立方程
Simultaneous equations are a fundamental part of the Edexcel IGCSE Mathematics syllabus, requiring you to find values that satisfy two or more equations at the same time. Mastering both algebraic and graphical methods is essential for success in examinations.
联立方程是 Edexcel IGCSE 数学大纲的核心内容之一，要求你找出同时满足两个或多个方程的值。掌握代数方法和图形方法对应试至关重要。
1. What Are Simultaneous Equations? | 什么是联立方程？
A system of simultaneous equations consists of two or more equations with shared variables. The solution is the set of values that makes all equations true at once. In IGCSE, we usually deal with two equations involving x and y.
联立方程组由两个或更多含有共同变量的方程组成。解就是能使所有方程同时成立的那组数值。在 IGCSE 中，我们通常处理含有 x 和 y 的两个方程。
1 and y = −x + 4 are simultaneous: the solution is the point where their graphs intersect.
例如，方程 y = 2x + 1 和 y = −x + 4 是联立的：解就是它们图像的交点。
2. Solving Linear Simultaneous Equations by Elimination | 用消元法解线性联立方程
Elimination is the most efficient method when both equations are linear. The goal is to add or subtract the equations to cancel out one variable.
当两个方程都是线性方程时，消元法是最有效的方法。目标是通过相加或相减消去其中一个变量。
y = 7 and 3x − y = 8. Adding them gives 5x = 15, so x = 3. Substitute back to find y = 1.
考虑方程组：2x + y = 7 和 3x − y = 8。两式相加得 5x = 15，因此 x = 3。代回原方程求得 y = 1。
multiply the second equation by 2 to get 10x + 2y = 26, then subtract the first equation.
如果系数不匹配，在相加或相减之前，需要将其中一个或两个方程乘以适当的倍数。例如，要从 3x + 2y = 12 和 5x + y = 13 中消去 y，先将第二个方程乘以 2 得到 10x + 2y = 26，然后再减去第一个方程。
3. Solving by Substitution | 用代入法求解
Substitution is particularly useful when one equation is already solved for one variable, or when one variable has a coefficient of 1. Express one variable in terms of the other, then substitute into the second equation.
代入法在其中一个方程已经对某一变量求解，或者某个变量的系数为 1 时尤其有用。将一个变量用另一个变量表示出来，然后代入第二个方程。
the second: x + 4(2x − 3) = 15. This simplifies to 9x − 12 = 15, giving x = 3, then y = 3.
对于方程组 y = 2x − 3 和 x + 4y = 15，将第一个方程代入第二个：x + 4(2x − 3) = 15。化简得 9x − 12 = 15，解得 x = 3，然后 y = 3。
4. Graphical Interpretation | 图形解释
Each linear equation represents a straight line on the Cartesian plane. The solution to a pair of linear simultaneous equations is the coordinates of their point of intersection.
每个线性方程在直角坐标平面上都表示一条直线。一对线性联立方程的解就是它们交点的坐标。
If lines are parallel, there is no solution. If the lines coincide, there are infinitely many solutions. Edexcel questions may ask you to use graphs to estimate solutions when lines are drawn.
如果直线平行，则无解。如果两直线重合，则有无数多解。Edexcel 考题可能要求你利用已绘制的直线图像估算解。
5. Linear and Quadratic Simultaneous Equations | 线性与二次联立方程
In IGCSE you will also meet one linear and one quadratic equation. This system can have up to two solutions, corresponding to the intersection points of a straight line and a parabola.
在 IGCSE 中，你还会遇到一个线性方程和一个二次方程组成的方程组。这种方程组最多有两个解，对应一条直线与一条抛物线的交点。
olving these algebraically often involves substituting the linear equation into the quadratic.
例如：y = x² − 4x + 3 和 y = 2x − 3。用代数方法求解这类方程组通常需要将线性方程代入二次方程。
6. Solving Linear-Quadratic Systems by Substitution | 用代入法解线性-二次方程组
The most reliable approach is substitution. Set the two expressions for y equal, or replace y in the quadratic using the linear expression. This yields a quadratic equation in x alone.
最可靠的方法是代入法。令两个表示 y 的表达式相等，或者用线性表达式替换二次方程中的 y。这样可以得到一个只含 x 的二次方程。
3 = 2x − 3. Rearrange to x² − 6x + 6 = 0. Solve by factorisation or the quadratic formula.
使用前面的例子 y = x² − 4x + 3 和 y = 2x − 3：代入得到 x² − 4x + 3 = 2x − 3。整理得 x² − 6x + 6 = 0。通过因式分解或求根公式求解。
7. Using the Discriminant to Determine Number of Solutions | 利用判别式判断解的个数
After substituting, the resulting quadratic may have two, one or zero real roots. The discriminant b² − 4ac tells you the nature of the solutions before solving completely. This is a powerful checking tool.
代入后得到的二次方程可能有两个、一个或零个实根。判别式 b² − 4ac 能在完全求解之前就告诉你解的性质。这是一个强大的检验工具。
olution. If the line does not meet the curve, the discriminant is negative, and the system has no real solutions.
如果直线与曲线相切，二次方程会有等根（判别式 = 0），方程组恰有一个解。如果直线与曲线不相交，判别式为负，方程组无实数解。
8. Solving Simultaneous Equations from Word Problems | 解文字题中的联立方程
Many IGCSE problems describe a real-life scenario such as buying items, ages, or mixtures. Define two variables clearly and form two independent equations from the information given.
许多 IGCSE 题目会描述现实场景，如购物、年龄或混合物问题。要清楚地定义两个变量，并根据所给的信息列出两个独立的方程。
Example: '3 apples and 2 bananas cost 1.80, while 5 apples and 1 banana cost 2.30.' a banana. Form 3a + 2b = 1.80 and 5a + b = 2.30. Solve using elimination or substitution.
例如：“3 个苹果和 2 根香蕉共 1.80 英镑，5 个苹果和 1 根香蕉共 2.30 英镑。”设苹果单价为 a，香蕉单价为 b。列出方程 3a + 2b = 1.80 和 5a + b = 2.30。用消元法或代入法求解。
9. Common Mistakes and How to Avoid Them | 常见错误及如何避免
One frequent error is forgetting to multiply both sides of an equation entirely when preparing for elimination. Always apply the multiplier to every term, including the constant on the right.
一个常见错误是在准备消元时忘记将方程两边完全相乘。务必将乘数作用到每一项，包括右边的常数项。
Another is incorrect substitution back into the original equations. Always check your solution by plugging x and y into both original equations. This verifies both values simultaneously.
另一个错误是代回原方程时计算不正确。始终通过将 x 和 y 同时代入两个原方程来检验你的解。这可以同时验证两个数值。
With linear-quadratic systems, students often lose a solution by assuming only one intersection point. Remember the quadratic step could give two x-values, each yielding a corresponding y.
在处理线性-二次方程组时，学生经常因假设只有一个交点而漏掉一个解。记住，二次步骤可能产生两个 x 值，每个都对应一个 y 值。
10. Exam Tips for Edexcel IGCSE | Edexcel IGCSE 考试技巧
Show clear logical steps. Even if you make a calculation slip, you can earn method marks. Always write the final coordinates as a pair (x, y), and if there are two sets, present both clearly.
展示清晰的逻辑步骤。即使计算有误，你也能拿到方法分。始终将最终坐标写成数对 (x, y) 的形式，如果有两组解，要清楚地呈现两者。
When a question asks for an estimate from a graph, draw a line accurately and read values to the accuracy the grid allows. State your answer in the context of the question if it is a word problem.
当问题要求根据图形估算时，要精确地画线，并根据网格的精度读取数值。如果是文字题，还要在题目的上下文中陈述答案。
Practice a variety of problems, including those where you must decide whether elimination or substitution is quicker. The more you practise, the more confident you become.
练习各种问题，包括那些需要你自己判断消元法还是代入法更快的题目。练习得越多，你越有信心。
11. Summary of Key Methods | 关键方法总结
- Elimination: Add/subtract equations after scaling to cancel one variable.
消元法：乘以适当倍数后相加/相减，消去一个变量。 - Substitution: Replace one variable with an expression from the other equation.
代入法：用来自另一个方程的表达式替换一个变量。 - Graphical: Draw lines/curves and identify intersection points.
图形法：绘制直线/曲线并找出交点。 - For linear-quadratic: Substitute linear into quadratic, rearrage, and solve the quadratic equation.
对于线性-二次方程组：将线性方程代入二次方程，整理，并解二次方程。
12. Checking Your Understanding | 检验你的理解
d y = 2x − 3. Find all solutions and sketch the graphs to confirm your algebraic findings.
尝试解方程组 4x − y = 6 和 2x + 3y = 10。然后挑战一个混合方程组：y = x² − 2x + 1 和 y = 2x − 3。找出所有解并画出示意图，验证你的代数结果。
Simultaneous equations form a bridge between algebra and geometry. With systematic practice, you can approach any IGCSE problem with confidence.
联立方程是代数与几何之间的桥梁。通过系统的练习，你可以自信地应对任何 IGCSE 问题。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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