📚 Simultaneous Equations | 联立方程
Simultaneous equations are a set of two or more equations that must be satisfied by the same variable values at the same time. In IGCSE Edexcel Mathematics, you will encounter both linear and non‑linear systems, and you will learn to solve them using algebraic methods (elimination and substitution) as well as by interpreting their graphs.
联立方程是一组需要同时被相同变量值满足的两个或多个方程。在IGCSE Edexcel数学中，你会遇到线性和非线性方程组，并将学习使用代数方法（消元法和代入法）以及通过图形解释来求解它们。
1. What Are Simultaneous Equations? | 什么是联立方程？
A simultaneous system is simply a collection of equations that share the same unknowns. For example, the pair 2x + y = 7 and x − y = 2 must both hold for the same x and y. The solution is the ordered pair (x, y) that satisfies every equation in the system. In two variables, this corresponds to the point where the graphs of the equations intersect.
联立方程组就是一组共享相同未知数的方程。例如，2x + y = 7 和 x − y = 2 这一对方程必须对同样的 x 和 y 同时成立。解就是满足系统内每一个方程的有序数对 (x, y)。在双变量情况下，这对应于各方程图形相交的点。
ems where one equation is linear and the other is quadratic (e.g., y = x² and y = 2x + 1).
在IGCSE中，大多数方程组涉及两个变量。你需要能够求解线性方程组，以及一个方程是线性、另一个是二次方程（例如 y = x² 和 y = 2x + 1）的方程组。
2. Solving by Elimination | 消元法解联立方程
Elimination is the most efficient technique when both equations are linear. You add or subtract the equations to remove one variable, then solve for the remaining variable. Before adding or subtracting, you may need to multiply one or both equations by a constant so that the coefficients of the chosen variable are equal in magnitude.
当两个方程都为线性时，消元法是最有效的方法。你将方程相加或相减以消去一个变量，然后解出剩下的变量。在相加或相减之前，你可能需要给一个或两个方程乘以一个常数，使得所选变量的系数大小相等。
For instance, to solve
3x + 2y = 12
5x − 2y = 4
he equations: (3x + 2y) + (5x − 2y) = 12 + 4 → 8x = 16 → x = 2. Substitute back to find y.
例如，求解
3x + 2y = 12
5x − 2y = 4
将两方程相加：(3x + 2y) + (5x − 2y) = 12 + 4 → 8x = 16 → x = 2。再代回求出 y。
When coefficients do not match conveniently, multiply first. For example, with 2x + 3y = 13 and 4x − y = 5 by 3 to align the y-coefficients: 12x − 3y = 15. Then add to the first: 14x = 28 → x = 2.
当系数不能方便地匹配时，先乘以适当的数。例如，对于 2x + 3y = 13 和 4x − y = 5，将第二个方程乘以3以对齐 y 的系数：12x − 3y = 15。然后与第一个方程相加：14x = 28 → x = 2。
3. Solving by Substitution | 代入法解联立方程
Substitution is useful when one equation is already solved for one variable, or when one equation is non‑linear. You rearrange one equation to express one unknown in terms of the other, and then substitute this expression into the second equation.
当一个方程已经解出一个变量，或其中一个方程为非线性方程时，代入法很有用。你将一个方程重新整理，用一个未知数表示另一个未知数，然后将此表达式代入第二个方程。
For example, given y = 2x + 1 and x² + y = 7 st into the second: x² + (2x + 1) = 7 → x² + 2x − 6 = 0. Solve the quadratic, then find y.
例如，给定 y = 2x + 1 和 x² + y = 7，将第一个方程中的 y 代入第二个：x² + (2x + 1) = 7 → x² + 2x − 6 = 0。解出这个二次方程，再求出 y。
Substitution also works for linear systems, e.g., from x = 3y − 4 and 2x + y = 5, replace x: 2(3y − 4) + y = 5 → 6y − 8 + y = 5 → 7y = 13 → y = 13/7.
代入法也适用于线性方程组，例如，由 x = 3y − 4 和 2x + y = 5，替换 x：2(3y − 4) + y = 5 → 6y − 8 + y = 5 → 7y = 13 → y = 13/7。
4. Graphical Interpretation | 图形解释
Each linear equation represents a straight line on the Cartesian plane. The solution of a linear simultaneous system is the coordinates of the point where the lines intersect. If the lines are parallel and distinct, there is no solution (inconsistent system). If the lines coincide, there are infinitely many solutions.
每个线性方程在笛卡尔平面上表示一条直线。线性联立方程组的解就是这些直线交点的坐标。如果直线平行且不重合，则无解（矛盾方程组）。如果两直线重合，则有无穷多解。
For a linear–quadratic system, the linear graph is a straight line and the quadratic graph is a parabola. The solutions correspond to the intersection points, and there can be 0, 1, or 2 solutions.
对于线性–二次方程组，线性图形是一条直线，二次图形是一条抛物线。解对应交点的数量，可以是0个、1个或2个。
You can solve graphically by plotting both equations accurately on graph paper and reading the intersection coordinates. While not as precise as algebra, this method reinforces understanding.
你可以通过在坐标纸上精确绘制两个方程的图形，并读出交点坐标来求解。虽然不如代数方法精确，但这种方法能强化理解。
5. Linear and Non‑Linear Systems | 线性与非线性方程组
IGCSE Edexcel expects you to handle systems where one equation is quadratic. The typical form is y = ax² + bx + c (or x = ay² + by + c) and a linear equation such as y = mx + c. You solve by substitution, which yields a quadratic equation in one variable. This quadratic may give two solutions, one solution (tangent), or no real solution.
IGCSE Edexcel 要求你会处理其中一个方程为二次方程的方程组。典型形式是 y = ax² + bx + c（或 x = ay² + by + c）和一个线性方程如 y = mx + c。通过代入法求解，会得到一个关于一个变量的二次方程。该二次方程可能给出两个解、一个解（相切）或无实数解。
Always check that your solutions satisfy both original equations. Sometimes a solution found algebraically may be extraneous due to squaring, but this is rare in IG‑level systems.
始终检查你的解是否满足原方程组。有时代数求解可能因平方产生增根，不过这在IGCSE中很少见。
6. Forming Simultaneous Equations from Word Problems | 从应用题中建立联立方程
Word problems require you to translate written information into algebraic equations. Identify the two unknowns and assign them letters (usually x and y). Then write two independent equations based on the conditions given. Typical contexts include:
应用题要求你将文字信息转化为代数方程。找出两个未知数并用字母表示（通常是 x 和 y）。然后根据给出的条件列出两个独立的方程。常见情境包括：
- Money and costs: "3 pens and 2 pencils cost £2.30, 5 pens and 3 pencils cost £3.70"
- Age problems: "A father is three times as old as his son; in 10 years he will be twice as old."
- Mixtures: "A solution contains x% acid, another contains y% acid."
- Shape and measures: "The perimeter of a rectangle is 30 cm, and its length is twice its width."
- 金钱和成本：“3支钢笔和2支铅笔共£2.30，5支钢笔和3支铅笔共£3.70”
- 年龄问题：“一位父亲的年龄是他儿子的三倍；10年后他将只有两倍”
- 混合物：“一种溶液含 x% 的酸，另一种含 y% 的酸”
- 形状与测量：“一个矩形的周长是30 cm，长是宽的两倍”
After forming the equations, solve them as usual. Always answer the original question and include units where appropriate.
建立方程后，照常求解。务必回答原始问题，并在适当位置包含单位。
7. Common Mistakes to Avoid | 常见错误及避免方法
Many students lose marks due to avoidable errors. Watch out for:
许多学生因可避免的错误而失分。注意：
- Forgetting to multiply all terms when scaling an equation
- Misaligning signs during elimination (e.g., subtracting instead of adding)
- sing the second solution in a quadratic system (set discriminant ≥ 0 and consider both ±√)
- Not substituting the found value back to get the other variable
- Arithmetic slips with negative numbers
- 在缩放方程时忘记乘以所有项
- 在消元时符号处理错误（例如该加却减）
- 在二次方程组中遗漏第二个解（确保判别式 ≥ 0，并考虑正负平方根）
- 求出某个变量后未代回求另一个变量
- 带有负数的算术失误
To avoid these, write each step clearly, check signs, and always verify by plugging the solution back into both original equations.
为避免这些错误，请清晰地写出每一步，检查符号，并始终将解代回原方程组进行验证。
8. Worked Example 1: Linear System | 典型例题1：线性方程组
Solve the simultaneous equations
4x + 3y = 11
2x − y = 3
bstitute x=2 into (2): 4 − y = 3 → y = 1. Solution: (2, 1). Check in (1): 4(2)+3(1)=8+3=11 ✓.
求解联立方程组
4x + 3y = 11
2x − y = 3
标记为 (1) 和 (2)。将 (2) 乘以 3：6x − 3y = 9。与 (1) 相加：(4x+3y)+(6x−3y)=11+9 → 10x = 20 → x = 2。将 x=2 代入 (2)：4 − y = 3 → y = 1。解为 (2, 1)。代回 (1) 检查：4(2)+3(1)=8+3=11 ✓。
9. Worked Example 2: Linear–Quadratic System | 典型例题2：线性–二次方程组
Solve y = x² + 3x − 2 and y = 2x + 4.
求解 y = x² + 3x − 2 和 y = 2x + 4。
(2)+4=8. Solutions: (−3, −2) and (2, 8). Verify both pairs satisfy the quadratic equation.
令两式相等：x² + 3x − 2 = 2x + 4 → x² + x − 6 = 0 → (x+3)(x−2)=0 → x = −3 或 x = 2。当 x=−3 时，y=2(−3)+4=−2；当 x=2 时，y=2(2)+4=8。解为 (−3, −2) 和 (2, 8)。验证这两组都满足二次方程。
10. Strategy for Choosing Elimination or Substitution | 消元法与代入法的选择策略
rs (x², y², xy = ...), substitution is necessary because elimination cannot cancel the non‑linear terms directly.
如果两个方程都以 ax + by = c 的形式给出，消元法通常更快。如果一个方程明确给出了 y = ... 或 x = ...，代入法更直接。对于涉及幂次（x², y², xy = ...）的方程组，必须使用代入法，因为消元法无法直接消去非线性项。
Also, if coefficients are fractions or decimals, consider clearing denominators first to simplify arithmetic.
此外，如果系数是分数或小数，可考虑先去分母以简化运算。
11. Using Technology & Checking Answers | 使用技术工具及检查答案
While you must show algebraic working in exams, you can use a graphing calculator or online tool to check your solutions. Graph both equations and find the intersection points. This is especially helpful for verifying quadratic systems. However, always rely on algebraic manipulation for the marks.
虽然考试中必须展示代数推导过程，但你可以使用图形计算器或在线工具来检查答案。绘制两个方程的图形并找到交点。这对验证二次方程组特别有用。但要始终依赖代数操作来获取分数。
Routine checking: after finding (x, y), substitute into both original equations. If one doesn’t match, retrace your steps.
常规检查：找到 (x, y) 后，代回原方程。如果有一个不成立，就要回溯步骤。
12. Summary & Key Points | 小结与关键点
Simultaneous equations combine two or more conditions into one system. The solution must satisfy every equation. Use elimination for linear systems where coefficients align easily, and substitution when one variable is isolated or when dealing with non‑linear equations. Graphical methods provide a visual check but are less precise. Practice forming equations from real‑world scenarios, and always check your final answers.
联立方程将两个或多个条件组合成一个系统。解必须满足每个方程。当系数容易对齐时，对线性方程组使用消元法；当一个变量被分离出来或处理非线性方程时，使用代入法。图形方法提供视觉检查但精度较低。练习从实际情境中建立方程，并始终检查最终答案。
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