📚 Simultaneous Equations – Linear and Quadratic | 联立方程组 – 线性与二次
In IGCSE Mathematics, you will often encounter problems that involve finding values that satisfy more than one equation at the same time. These are called simultaneous equations, and they can be linear or involve a quadratic curve. Mastering both elimination and substitution methods is essential for success in the Edexcel IGCSE exam, as questions regularly require you to solve two linear equations or a linear-quadratic pair, both algebraically and graphically.
在 IGCSE 数学中，你经常会遇到需要找到同时满足多个方程的值的问题。这些方程称为联立方程组，可以是线性的，也可以包含二次曲线。掌握消元法和代入法对于在 Edexcel IGCSE 考试中取得成功至关重要，因为题目经常要求你代数求解或通过图形求解两个线性方程或一个线性-二次方程组。
1. What Are Simultaneous Equations? | 什么是联立方程组？
solution must satisfy all of them at the same time. For example, the pair x + y = 5 and x – d true. In the IGCSE context, you will usually work with two unknowns, most often x and y.
联立方程组是由两个或更多方程组成的一组方程，它们共享相同的未知量，且解必须同时满足所有方程。例如，方程组 x + y = 5 和 x – y = 1 的唯一解是 x = 3, y = 2，因为当你代入这些值时，两个方程都成立。在 IGCSE 的背景下，你通常会处理两个未知量，最常见的是 x 和 y。
2. Types of Simultaneous Equations in IGCSE | IGCSE 中的联立方程组类型
us equations, where one equation is linear and the other is quadratic, such as y = x² + 2x – 3. You may also see a circle equation or other non‑ in consistent: use substitution to reduce the system to a single equation in one variable.
Edexcel IGCSE 主要关注两类：线性联立方程组，其中两个方程都是 ax + by = c 的形式；以及线性-二次联立方程组，其中一个方程是线性的，另一个是二次的，例如 y = x² + 2x – 3。你也可能碰到圆方程或其他非线性形式，但求解的代数技巧是一致的：使用代入法将方程组转化为一个单变量的方程。
3. Solving Linear Simultaneous Equations: Elimination Method | 解线性联立方程组：消元法
The elimination method involves adding or subtracting the two equations to remove one variable, leaving a simple equation to solve. For instance, consider the system 2x + y = 7 and 3x – y = 8. By adding the equations, the +y and – efficients do not match, you may need to multiply one or both equations by a constant before adding or subtracting.
消元法是通过将两个方程相加或相减来消去一个变量，从而得到一个简单的方程来求解。例如，考虑方程组 2x + y = 7 和 3x – y = 8。通过将两个方程相加，+y 和 –y 相互抵消，得到 5x = 15，因此 x = 3。然后回代得到 y = 1。如果系数不匹配，你可能需要先将一个或两个方程乘以一个常数，然后再进行相加或相减。
Example: 2x + y = 7
3x – y = 8
(Add) 5x = 15 → x = 3, y = 1
4. Solving Linear Simultaneous Equations: Substitution Method | 解线性联立方程组：代入法
In the substitution method, you rearrange one equation to express one variable in terms of the other, then substitute this expression into the second equation. For the system x + 2y = 8 and 3x – y = 3, you could make x the subject: x = 8 – 2y. Substituting into the second equation gives 3(8 – 2y) – y = 3, which simplifies to 24 – 6y – y = 3 → –7y = –21 → y = 3. Then x = 8 – larly useful when one variable has a coefficient of 1, but it works for any linear system.
在代入法中，你重新整理一个方程，将其中的一个变量用另一个变量表示，然后将这个表达式代入第二个方程。对于方程组 x + 2y = 8 和 3x – y = 3，你可以将 x 表示为主语：x = 8 – 2y。代入第二个方程得到 3(8 – 2y) – y = 3，化简为 24 – 6y – y = 3 → –7y = –21 → y = 3。然后 x = 8 – 6 = 2。当一个变量的系数为 1 时，此方法特别有用，但它适用于任何线性方程组。
5. Graphical Interpretation of Solutions | 解的图形解释
Graphically, the solution to a pair of simultaneous equations corresponds to the point(s) where their graphs intersect. For two linear equations, the intersection is a single point – unless the lines are parallel (no solution) or coincident (infinitely many solutions). For a linear and a quadratic equation, the line may cut the parabola at 0, 1, or 2 points, which correspond to the number of real solutions. Edexcel IGCSE often asks you to interpret such graphs or to solve equations by drawing them.
从图形上看，一组联立方程组的解对应于它们图形相交的点。对于两个线性方程，交点只有一个——除非直线平行（无解）或重合（无穷多解）。对于一个线性方程和一个二次方程，直线可能与抛物线交于 0 个、1 个或 2 个点，这对应着实解的个数。Edexcel IGCSE 经常要求你解释这类图形，或者通过绘制图形来求解方程。
6. Introducing Linear and Quadratic Simultaneous Equations | 引入线性与二次联立方程组
have up to two solutions. To solve, you equate the two expressions for y because they are both equal to y, and then solve the resulting quadratic equation. This yields the x‑coordinates, and you find the corresponding y‑ equation. Expect to work with quadratics that factorise or require the quadratic formula.
当其中一个方程为二次方程时，例如 y = x² + 3x + 2，而另一个方程为线性方程，例如 y = 2x + 1，方程组最多可以有两个解。要解这个方程组，你可以将两个 y 的表达式设成相等，因为它们都等于 y，然后求解所得的二次方程。这会给出 x 坐标，随后你使用线性方程求出相应的 y 值。你需要处理的二次方程可能可以通过因式分解求解，或者需要使用二次公式。
7. Solving Linear-Quadratic Systems by Substitution | 通过代入法解线性-二次方程组
The standard technique is substitution: if you have the system y = x² + 2x – 3 and y = x + 3, set x² + 2x – 3 = x + 3. Rearrange to form a quadratic: x² + x – 6 = 0, which factorises as (x + 3)(x – 2) = 0, giving x = –3 or x = 2. Substitute these into the linear equation y = x + 3: when x = –3, y = 0; when x = 2, y = 5. Therefore the solutions are (–, 0) and (2, 5). Always check that both solutions satisfy the original quadratic equation.
标准技巧是代入法：如果你有方程组 y = x² + 2x – 3 和 y = x + 3，令 x² + 2x – 3 = x + 3。重新整理得到一个二次方程：x² + x – 6 = 0，它可以因式分解为 (x + 3)(x – 2) = 0，得出 x = –3 或 x = 2。将这些值代入线性方程 y = x + 3：当 x = –3 时，y = 0；当 x = 2 时，y = 5。因此解为 (–3, 0) 和 (2, 5)。一定要检查这两个解是否都满足原始的二次方程。
Set equal: x² + 2x – 3 = x + 3
→ x² + x – 6 = 0
→ (x + 3)(x – 2) = 0
→ x = –3 or 2, y from linear equation.
8. Using the Quadratic Formula for Non‑Factorising Quadratics | 使用二次公式处理不能因式分解的二次方程
If the resulting quadratic does not factorise neatly, apply the quadratic formula x = [–b ± √b² – 4ac] / 2a. For example, with the system y = x² + x + 1 and y = 2x + 2, setting equal gives x² – x – 1 = 0, which does not factorise. Then x = [1 ± √1 + 4] / 2 = (1 ± √5) / 2. Substitute into y = 2x + 2 to get the y‑ E questions will provide the formula; just remember to bracket negative numbers carefully.
如果得到的二次方程不能整齐地因式分解，则应用二次公式 x = [–b ± √b² – 4ac] / 2a。例如，对于方程组 y = x² + x + 1 和 y = 2x + 2，设定相等得到 x² – x – 1 = 0，它不能因式分解。那么 x = [1 ± √1 + 4] / 2 = (1 ± √5) / 2。代入 y = 2x + 2 求出 y 值。IGCSE 题目会提供公式；只需记住小心地为负数加上括号。
x = [–b ± √b² – 4ac] / 2a
9. Handling Word Problems | 处理文字题
Many IGCSE past papers present simultaneous equations within real‑ and bananas, or mixing metal alloys. The key is to define your variables clearly (e.g., let a be the number of apples, b the number of bananas) and translate the written conditions into equations. For instance, "The sum of two numbers is 20 and their product is 96" h leads to a quadratic system. Solve step by step, and always check that your answers are realistic (e.g., positive quantities).
许多 IGCSE 历年真题在现实生活情境中设置联立方程组，例如购买苹果和香蕉，或混合金属合金。关键是要清晰地定义变量（例如，设 a 为苹果的数量，b 为香蕉的数量），并将书面条件转化为方程。例如，“两个数的和是 20，它们的积是 96”可转化为 x + y = 20 和 xy = 96，这就构成了一个二次方程组。要逐步求解，并始终检查答案是否现实（例如，数量应为正数）。
10. Common Errors and Tips | 常见错误与提示
incorrect subtraction. In substitution, always substitute into the linear equation after finding x to avoid messy algebra. For linear‑ ax² + bx + c = 0 before factorising or using the formula, and be careful with signs. Graph‑ tise with past Edexcel IGCSE papers to become fluent in recognising which method to apply.
在解线性方程组时，一个常见错误是在消元过程中忘记
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