📚 Simultaneous Equations | 联立方程
Simultaneous equations appear all the time in IGCSE Mathematics, often in the context of two linear equations or one linear and one quadratic. They require you to find values of unknowns that satisfy more than one equation at the same time. Mastering both algebraic and graphical methods is essential, as questions can be asked either way and frequently connect to real‑world problem solving.
联立方程在IGCSE数学中无处不在，经常以两个一次方程或一个一次一个二次方程的形式出现。它们要求你找到同时满足多个方程的未知数的值。掌握代数方法和图像方法都至关重要，因为题目可能以任何一种形式出现，并且常常与实际问题的解决联系在一起。
1. What Are Simultaneous Equations? | 什么是联立方程？
ations because they both involve x and y, and we want the pair (x, y) that works for both.
联立方程组是一组共享相同变量的两个或多个方程。解必须使方程组中的每一个方程同时成立。例如，x + y = 5 和 2x − y = 1 就是联立方程，因为它们都含有 x 和 y，而且我们要找到同时满足这两个方程的 (x, y)。
In IGCSE, you will mainly encounter two types: two linear equations (the most common) and one linear with one quadratic equation. The number of solutions depends on the type: two linear equations usually have exactly one solution (unless they are parallel or identical); a linear and a quadratic can have two, one, or no real solutions.
在IGCSE中，你主要会遇到两种类型：两个一次方程（最常见）以及一个一次方程和一个二次方程的组合。解的数量取决于类型：两个一次方程通常恰有一个解（除非两条直线平行或重合）；一个一次和一个二次方程可能有两个、一个或没有实数解。
2. The Elimination Method | 消元法
The elimination method works by adding or subtracting the equations to remove one variable. First, if necessary, multiply one or both equations to make the coefficients of one variable the same (or equal in magnitude but opposite in sign). Then add or subtract to eliminate that variable, solve for the remaining unknown, and substitute back to find the other.
消元法通过对方程进行加法或减法来消去一个变量。首先，如果需要，将其中一个或两个方程乘以适当的数，使某个变量的系数相同（或绝对值相等但符号相反）。然后相加或相减消去该变量，解出剩下的未知数，再代回求出另一个。
Example: 2. Substituting x = 2 into x − 2y = 0 gives 2 − 2y = 0, so y = 1. The solution is (2, 1).
示例： 解方程组 3x + 2y = 8 和 x − 2y = 0。y 的系数为 +2 和 −2，因此将两式相加得 4x = 8，所以 x = 2。把 x = 2 代入 x − 2y = 0 得 2 − 2y = 0，因此 y = 1。解为 (2, 1)。
+ 6y = 14 and 9x − 6y = 12. Adding gives 13x = 26 → x = 2, then substitute to find y = 1.
当系数不容易匹配时，将两个方程都乘以适当的数。例如：2x + 3y = 7 和 3x − 2y = 4。将第一个方程乘以 2，第二个乘以 3 来匹配 y：4x + 6y = 14 和 9x − 6y = 12。相加得 13x = 26 → x = 2，再代回求得 y = 1。
3. The Substitution Method | 代入法
Substitution is useful when one equation is already solved for a variable, or can be easily rearranged. Take one equation and express one variable in terms of the other, then substitute this expression into the other equation. This reduces the system to a single equation in one unknown.
当其中一个方程已经解出某个变量，或者很容易变形时，代入法就很有用。从一个方程中把一个变量用另一个变量表示出来，然后将这个表达式代入另一个方程。这样就把方程组简化为一个一元方程。
Example: 8x − 12 = 10 → 11x = 22 → x = 2. Then y = 2(2) − 3 = 1. The intersection point is (2, 1).
示例： y = 2x − 3 和 3x + 4y = 10。将 y 代入第二个方程：3x + 4(2x − 3) = 10 → 3x + 8x − 12 = 10 → 11x = 22 → x = 2。然后 y = 2(2) − 3 = 1。交点为 (2, 1)。
This method is also the standard approach for one linear and one quadratic. Solve the linear equation for one variable (usually y) and substitute into the quadratic. You will obtain a quadratic in one variable, which may factorise or require the quadratic formula.
这种方法也是处理一个一次方程和一个二次方程的标准方法。将一次方程中解出一个变量（通常是 y），代入二次方程。你会得到一个关于一个变量的二次方程，它可以因式分解或者需要用求根公式来解。
4. Solving One Linear and One Quadratic | 解一次与二次联立方程
). Substituting the linear expression for y into the quadratic yields a quadratic equation in x. Solve this to find the x‑nates of the intersection points, then use the linear equation to find the corresponding y‑coordinates.
典型的IGCSE题目会给出 y = mx + c（一次）和 y = ax² + bx + c（二次）。将一次表达式代入二次方程中的 y，得到一个关于 x 的二次方程。解这个方程得到交点的 x 坐标，然后用一次方程求出对应的 y 坐标。
Example: rresponding y values from y = 2x + 1 are y = 5 and y = −1. Solutions: (2, 5) and (−1, −1).
示例： y = 2x + 1 和 y = x² + x − 1。代入得 x² + x − 1 = 2x + 1 → x² − x − 2 = 0。因式分解：(x − 2)(x + 1) = 0，所以 x = 2 或 x = −1。由 y = 2x + 1 求出对应的 y 值为 y = 5 和 y = −1。解为 (2, 5) 和 (−1, −1)。
If the quadratic does not factorise, use the quadratic formula: x = [−b ± √b² − 4ac] / (2a). The discriminant (b² − 4ac) tells you how many real solutions exist: two if positive, one if zero, none if negative.
如果二次方程不能因式分解，就用求根公式：x = [−b ± √b² − 4ac] / (2a)。判别式 (b² − 4ac) 告诉你存在多少个实数解：大于零有两个，等于零有一个，小于零则没有。
5. Graphical Interpretation | 图像解释
Simultaneous equations can be solved by drawing graphs. The solution(s) are the coordinates where the graphs intersect. For two linear equations, the intersection is a single point (unless parallel). For a linear and a quadratic, there can be two intersections, one (tangent), or none.
联立方程可以通过画图来求解。解就是图像交点的坐标。对于两个一次方程，交点通常是一个点（除非平行）。对于一个一次和一个二次方程，可能有两个交点、一个交点（相切）或没有交点。
When solving graphically, draw accurate axes and plot points carefully. The x‑coordinate and y‑coordinate of each intersection point must satisfy both equations. Graphical methods are often used to estimate solutions when algebraic manipulation is difficult, but the IGCSE expects you to be able to read values to a reasonable accuracy.
在图像求解时，要精确画出坐标轴并仔细描点。每个交点的 x 坐标和 y 坐标必须同时满足两个方程。当代数处理困难时，通常用图像方法来估计解，但IGCSE要求你能够以合理的精确度读取数值。
6. Word Problems Leading to Simultaneous Equations | 建立联立方程的应用题
Many IGCSE questions are set in real‑life contexts: buying items, ages, money, or geometric situations. The key is to define variables clearly (e.g., let x be the price of a pen and y be the price of a pencil) and then translate the given information into two equations. Solve the system, and finally answer the question in context.
许多IGCSE题目设定在现实生活场景中：购买物品、年龄、金钱或几何情况。关键是要清晰地定义变量（例如，设 x 为一支钢笔的价格，y 为一支铅笔的价格），然后将已知信息转化为两个方程。解方程组，最后在具体情景中回答该问题。
Example: Two apples and three bananas cost £1.90. Five apples and two bananas cost £ = 30p. Always check units and present the answer clearly (e.g., Apple: 40p, Banana: 30p).
示例： 两个苹果和三个香蕉共 £1.90。五个苹果和两个香蕉共 £2.80。设一个苹果的价格为 a 便士，一个香蕉为 b 便士。方程：2a + 3b = 190，5a + 2b = 280。解出 a = 40 便士，b = 30 便士。务必检查单位并清晰地给出答案（例如：苹果 40 便士，香蕉 30 便士）。
7. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无穷多解
Not all simultaneous equations have a unique answer. If two linear equations represent parallel lines (same slope but different y‑intercepts), there is no solution – the system is inconsistent. If the two equations are essentially the same line (one is a multiple of the other), there are infinitely many solutions because every point on the line satisfies both.
并非所有联立方程组都有唯一解。如果两个一次方程表示平行线（斜率相同但 y 截距不同），则无解——方程组是不相容的。如果两个方程本质上是同一条直线（一个是另一个的倍数），则有无穷多解，因为直线上的每个点都同时满足两个方程。
re infinitely many solutions. IGCSE questions sometimes ask you to describe the geometric relationship between the lines based on these outcomes.
你可以在消元过程中发现这些情况：如果你得出一个矛盾式，如 0 = 5，则无解。如果你得出一个恒等式，如 0 = 0，则有无穷多解。IGCSE题目有时会让你根据这些结果描述两条直线之间的几何关系。
8. Using Simultaneous Equations to Find Intersection Points | 利用联立方程求交点
he graphs are straight lines, a line and a curve, or two curves (rare at IGCSE), set the y‑ line is a tangent to a curve: the resulting quadratic has a double root, discriminant = 0.
在坐标几何中，求两个图像的交点正是一个联立方程问题。无论图像是两条直线、一条直线和一条曲线，还是两条曲线（IGCSE中少见），都要令 y 的表达式相等或进行代入。这一技巧也用于证明一条直线是一条曲线的切线：所得二次方程有二重根，即判别式 = 0。
Example: ² − 4x + 4 = 0 → (x − 2)² = 0. One repeated root, so the line touches the curve at (2, 4).
示例： 证明 y = 4x − 4 是 y = x² 的切线。令 x² = 4x − 4 → x² − 4x + 4 = 0 → (x − 2)² = 0。有一个重根，因此直线与曲线相切于 (2, 4)。
9. Choosing the Right Method | 选择正确的方法
For two linear equations, elimination is often faster, especially when coefficients are small integers. Substitution is better when one variable is already isolated or when dealing with a non‑linear equation. In the exam, take a moment to decide which method minimises working and the chance of arithmetic errors. Whichever you choose, always check your solution by substituting both values into the original equations.
对于两个一次方程，消元法通常更快，尤其是当系数为较小的整数时。当一个变量已被分离出来，或者处理非线性方程时，代入法更合适。在考试中，花点时间决定哪种方法可以减少步骤和算术错误的机会。无论选择哪种方法，一定要将两个值都代回原方程进行检验。
arranging them into the standard ax + by = c form helps avoid mistakes during elimination.
有时，动笔之前先整理方程会有很大帮助。例如，如果方程以 2y = 3x + 5 这样的形式给出，将它们整理成标准的 ax + by = c 形式有助于避免消元时的错误。
10. Common Mistakes and How to Avoid Them | 常见错误及如何避免
3) requires careful distribution of the minus sign. Write the operation clearly and double‑check. Another common error is forgetting to find both variables – always complete the full solution and state the pair (x, y).
符号错误是头号陷阱，尤其是在方程相减时。记住：从 (5x − y = 3) 中减去 (2x − 3y = 7) 时，需要小心地分配负号。清楚地写出运算过程，并再次检查。另一个常见错误是只求出一个变量就结束了——一定要完成完整解答并写出有序数对 (x, y)。
In quadratic simultaneous equations, students sometimes substitute incorrectly or expand brackets wrongly. Write every step, use brackets when substituting, and simplify carefully. Also, always check that each solution pair works in both original equations; this catches algebraic slips effectively.
在二次联立方程中，学生有时会错误地代入或展开括号。写出每一步，代入时使用括号，并仔细化简。另外，务必检查每一组解都满足两个原方程；这能有效地发现代数错误。
11. Quick Revision Summary | 快速复习要点
- Elimination: make coefficients match, add/subtract, solve, substitute back.
- Substitution: isolate one variable, replace in the other equation, solve.
- Linear + quadratic: substitute linear into quadratic, solve quadratic, find y.
- Graphical: intersection points = solutions; accuracy matters.
- Word problems: define variables, write two equations, solve, answer in context.
- Special cases: parallel lines → no solution; identical lines → infinite solutions.
- Always check your answers in the original equations.
- 消元法： 使系数匹配，相加或相减，解方程，回代。
- 代入法： 分离出一个变量，代入另一个方程，解方程。
- 一次与二次联立： 将一次代入二次，解二次方程，求出 y。
- 图像法： 交点即为解；准确性很重要。
- 应用题： 定义变量，写出两个方程，求解，在情景中作答。
- 特殊情况： 平行线 → 无解；重合直线 → 无穷多解。
- 永远要检验你的答案代入原方程。
12. Practice Questions to Test Yourself | 自我检测练习题
1. Solve the simultaneous equations: 4x + y = 10 and 2x − 3y = −2.
1. 解联立方程：4x + y = 10 和 2x − 3y = −2。
2. Find the intersection points of y = x² − 3x + 2 and y = x − 1.
2. 求 y = x² − 3x + 2 和 y = x − 1 的交点。
3. The sum of two numbers is 26 and their difference is 4. Find the numbers.
3. 两数之和为 26，差为 4。求这两个数。
4. Show whether the lines 3x + 2y = 7 and 6x + 4y = 15 have any intersection.
4. 判断直线 3x + 2y = 7 和 6x + 4y = 15 是否有交点。
Take your time solving these, and apply the methods discussed. Practice builds speed and accuracy for the IGCSE exam.
花时间解这些题，应用所讨论的方法。练习能提高IGCSE考试的速度和准确度。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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