📚 Simultaneous Equations | 联立方程组
Simultaneous equations are a set of two or more equations with multiple unknowns that must be solved together. In IGCSE Mathematics, you are expected to solve pairs of linear equations, as well as combinations of one linear and one quadratic equation, using algebraic and graphical methods. Mastering these techniques builds a foundation for more advanced topics like coordinate geometry and calculus.
联立方程组是指含有多个未知数且必须同时求解的两个或多个方程。在IGCSE数学中，你需要用代数方法和图解法求解两个线性方程，以及一个线性方程和一个二次方程的组合。掌握这些方法将为坐标几何和微积分等更高级的内容打下基础。
1. What Are Simultaneous Equations? | 什么是联立方程组？
Simultaneous equations are equations that share the same variables. A solution is a set of values for the variables that satisfies all equations at once. For two linear equations in x and y, this typically corresponds to the point where their graphs intersect on the coordinate plane.
联立方程组是共享相同变量的一组方程。解就是一组能使所有方程同时成立的变量值。对于含有 x 和 y 的两个线性方程，解通常就是它们在坐标平面上图像的交点。
x - y = 2 form a system. The pair x = 3, y = 1 satisfies both: 2(3) + 1 = 7 and 3 - 1 = 2.
例如，方程 2x + y = 7 和 x - y = 2 构成一个联立方程组。x = 3, y = 1 这一对值能同时满足两个方程：2(3) + 1 = 7 且 3 - 1 = 2。
2. Solving Linear Systems by Elimination | 用消元法解线性方程组
The elimination method works by adding or subtracting the equations to cancel out one variable. First, align the equations so that like terms are stacked vertically. Multiply one or both equations if necessary to make the coefficients of one unknown equal in magnitude.
消元法是通过加减方程来抵消一个变量。首先将方程对齐，使同类项上下排列。必要时给一个或两个方程乘以适当的数，使其中一个未知数的系数绝对值相等。
Consider the system:
考虑方程组：
2x + 3y = 12
4x - 3y = 6
Here the y-coefficients are +3 and –3. Adding the two equations eliminates y: (2x + 4x) + (3y – the first equation: 2(3) + 3y = 12 → 6 + 3y = 12 → 3y = 6 → y = 2. The solution is (3, 2).
这里 y 的系数分别是 +3 和 –3。将两个方程相加即可消去 y：(2x + 4x) + (3y – 3y) = 12 + 6，得到 6x = 18，因此 x = 3。代回第一个方程：2(3) + 3y = 12 → 6 + 3y = 12 → 3y = 6 → y = 2。解为 (3, 2)。
3. Solving Linear Systems by Substitution | 用代入法解线性方程组
Substitution is useful when one equation can easily be rearranged to express one variable in terms of the other. Substitute this expression into the other equation to reduce the system to a single equation in one unknown.
如果其中一个方程容易变形，用一个变量表示另一个变量，采用代入法就很方便。将这个表达式代入另一个方程，便可把方程组化为一元方程。
Solve the system:
解方程组：
y = 2x - 5
3x + 4y = 18
Substitute y = 2x – 5 into the second equation: 3x + 4(2x – 5) = 18 → 3x + 8x – 20 = 18 → 11x = 38 → x = 38/11. Then y = 2(38/11) – 5 = 76/11 – 55/11 = 21/11. The exact solution is (38/11, 21/11).
将 y = 2x – 5 代入第二个方程：3x + 4(2x – 5) = 18 → 3x + 8x – 20 = 18 → 11x = 38 → x = 38/11。然后 y = 2(38/11) – 5 = 76/11 – 55/11 = 21/11。精确解为 (38/11, 21/11)。
4. Graphical Interpretation of Linear Systems | 线性方程组的图像解释
Each linear equation describes a straight line. The solution of two simultaneous linear equations is the point where the two lines intersect. Plot both lines on the same axes, and read off the coordinates of the crossing point.
每个线性方程都描述一条直线。两个线性联立方程的解就是两条直线的交点。在同一坐标系中画出这两条直线，在交点处读出坐标即可。
Graphical methods can confirm algebraic solutions, but they may only give approximate values unless the intersection lies exactly on a grid point. Edexcel IGCSE often expects you to draw graphs accurately and estimate solutions to 1 decimal place if needed.
图像法可以验证代数解，但除非交点恰好位于网格点上，否则只能给出近似值。Edexcel IGCSE 通常要求准确绘制图像，并在需要时将解估算至一位小数。
If the lines are parallel, there is no intersection – the system has no solution. If the two equations represent the same line, there are infinitely many points of intersection, giving infinitely many solutions.
如果两条直线平行，就没有交点——方程组无解。如果两个方程代表同一条直线，就有无穷多个交点，即有无穷多组解。
5. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无穷多解
nation yields a true statement like 0 = 0, the equations are dependent (coincident lines).
两个线性方程组成的方程组可能有一组解、无解或无穷多组解。在代数中，如果消元后得出伪命题，例如 0 = 5，说明方程组无解（平行线）。若消元后得到恒等式，例如 0 = 0，说明两个方程是相依方程（重合直线）。
e is a solution. However, 2x + y = 3 and 4x + 2y = 10 are parallel, so no solution exists.
例如，2x + y = 3 和 4x + 2y = 6 代表同一条直线：将第一个方程乘以2就得到第二个。该直线上任意一点都是解。而 2x + y = 3 和 4x + 2y = 10 平行，因此没有解。
6. Solving Linear and Quadratic Simultaneous Equations | 解一次与二次联立方程组
Edexcel IGCSE also requires solving a system where one equation is linear and the other is quadratic. The method of substitution is usually the best approach: rearrange the linear equation to express y in terms of x (or x in terms of y), then substitute into the quadratic to obtain a quadratic equation in one variable.
Edexcel IGCSE 还要求求解一个线性方程和一个二次方程组成的方程组。通常最适合的方法是代入法：将线性方程变形，用 x 表示 y（或用 y 表示 x），然后代入二次方程中，得到一个一元二次方程。
Solve the system:
解方程组：
y = x + 1
x² + y² = 25
Substitute y = x + 1: x² + (x + 1)² = 25 → x² + x² + 2x + 1 = 25 → 2x² + 2x – 24 = 0 → x² + x – 12 = 0 → (x + 4)(x – 3) = 0. So x = –4 or x = 3. Then y = –3 or y = 4. The solutions are (–4, –3) and (3, 4).
代入 y = x + 1：x² + (x + 1)² = 25 → x² + x² + 2x + 1 = 25 → 2x² + 2x – 24 = 0 → x² + x – 12 = 0 → (x + 4)(x – 3) = 0。因此 x = –4 或 x = 3。然后 y = –3 或 y = 4。解为 (–4, –3) 和 (3, 4)。
7. Using Substitution for Non-linear Systems | 代入法解非线性方程组
When the quadratic is not in standard circle form, the same substitution principle applies. Always substitute the linear expression into the non-linear equation carefully, expanding brackets and simplifying. Be mindful that you might get two, one or zero real solutions depending on the discriminant of the resulting quadratic.
当二次方程不是标准圆的形式时，同样的代入原则依然适用。将一次表达式仔细代入非线性方程，展开括号并化简。注意，根据所得二次方程的判别式，可能会得到两个、一个或零个实数解。
For example, y = 2x – 3 and y = x² – x – 1 lead to 2x – 3 = x² – x – 1 → x² – 3x + 2 = 0 → (x – 1)(x – 2) = 0, giving x = 1, y = –1 and x = 2, y = 1.
例如，y = 2x – 3 和 y = x² – x – 1 得出 2x – 3 = x² – x – 1 → x² – 3x + 2 = 0 → (x – 1)(x – 2) = 0，解得 x = 1, y = –1 和 x = 2, y = 1。
8. Real-World Applications | 实际应用
Simultaneous equations model many practical situations, such as mixing solutions, comparing mobile phone tariffs, or break-even analysis in business. You may need to define variables, form equations from a word problem, and then solve them.
联立方程组可用于模拟许多实际情景，比如混合溶液、比较手机资费方案或商业中的盈亏平衡分析。你可能需要定义变量，根据文字题建立方程组，然后求解。
Example: 'A cinema sells adult tickets for $12 and child tickets for $7. On a day when 200 tickets were sold, the total revenue was $1880. Find the number of adult and child tickets sold.'
例题：“一家电影院成人票每张12美元，儿童票每张7美元。某日售出200张票，总收入1880美元。求售出的成人票和儿童票数量。”
Subtracting 7 times the first equation from the second gives 5a = 480, so a = 96, c = 104.
设 a 为成人票数量，c 为儿童票数量。则有 a + c = 200 和 12a + 7c = 1880。将第二个方程减去第一个方程的7倍，得到 5a = 480，因此 a = 96，c = 104。
9. Common Mistakes to Avoid | 常见错误
Sign errors: When subtracting equations, distribute the minus sign to every term. A classic mistake is subtracting only the first term of the second equation and forgetting the others.
符号错误：方程相减时，必须将减号分配给每一项。典型错误是只对第二个方程的第一项取负号，而忽略了其余项。
Not checking solutions: Always substitute your solution back into both original equations. This catches arithmetic slips and helps you verify that the pair works.
未验算：始终将解代回原方程组，这可以发现计算失误，并验证解是否正确。
Forgetting the quadratic can give two solutions: When a linear and a quadratic are solved, there may be two distinct (x, y) pairs. Make sure to find the corresponding y for each x.
忘记二次方程可能给出两组解：求解一次和二次方程时，可能有两组不同的 (x, y) 解。务必找到每个 x 对应的 y。
Misinterpreting the discriminant: If the resulting quadratic has no real roots (negative discriminant), the system has no real solution. This corresponds to the line not touching the quadratic curve.
误判判别式：若化简后的一元二次方程没有实数根（判别式为负），则方程组无实数解，这对应于直线与二次曲线不相交。
10. Exam Tips & Practice | 考试技巧与练习
In Edexcel IGCSE, simultaneous equations appear frequently on both paper 1 and paper 2. Show all steps clearly – even if you use a calculator for arithmetic, the method marks are awarded for correct substitution and simplification.
在 Edexcel IGCSE 中，联立方程组经常出现在试卷一和试卷二中。要清晰地写出所有步骤——即使使用计算器进行运算，方法分依然会根据正确的代入和化简给出。
For graphical questions, label axes, use a sharp pencil, and plot points carefully. If a question asks you to solve by drawing, you must not use algebra alone; present the graph and state the intersection coordinates. The acceptable tolerance is usually half a small square.
对于图像题，要标记坐标轴，使用尖铅笔，仔细地描点。如果题目要求通过作图求解，就不能只使用代数方法；必须画出图像并标出交点坐标。通常允许的误差为半个小格。
Practice a variety of problems, including ones where both equations need rearranging before elimination. Be comfortable with handling fractional coefficients and with forming equations from word problems. Regular practice builds the confidence to tackle any system the exam may present.
练习各种题型，包括需要先整理再消元的方程组。要熟练处理分数系数以及从文字题中建立方程。经常练习有助于树立信心，从容应对考试中可能出现的任何方程组。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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