📚 Mastering Simultaneous Equations for IGCSE Mathematics | IGCSE 数学：掌握联立方程
Simultaneous equations are a fundamental tool in IGCSE Mathematics. They allow you to find unknown values when two or more conditions must be satisfied at the same time. This topic appears in both Core and Extended papers and often links to graphs, quadratic equations, and real-life word problems.
联立方程是 IGCSE 数学中的一个基本工具。当两个或多个条件必须同时满足时，它们可以帮助你求出未知数的值。该主题出现在核心和扩展试卷中，并且经常与图像、二次方程以及现实生活应用题相结合。
1. What Are Simultaneous Equations? | 什么是联立方程？
A simultaneous equation system consists of two or more equations that share the same variables. The solution is the set of values that makes every equation true at the same time.
联立方程组由两个或两个以上含有相同变量的方程组成。解是使每个方程同时成立的一组数值。
In IGCSE Mathematics, you will mostly work with two equations in two unknowns, usually x and y. The solution can be written as an ordered pair (x, y).
在 IGCSE 数学中，你主要处理的是两个未知数 x 和 y 的两个方程。解可以写成有序数对 (x, y)。
For example, x = 3 and y = 2 could be the solution of a system where both equations are true only when these values are used. Checking solutions in the original equations is essential.
例如，x = 3 和 y = 2 可能是某个方程组的解，因为只有使用这些值时两个方程才同时成立。在原方程中检验解是至关重要的。
2. The Elimination Method | 消元法
Elimination is often the quickest method when both equations are linear. You add or subtract the equations to cancel one variable.
当两个方程都是线性方程时，消元法通常是最快的方法。你将方程相加或相减，以消去其中一个变量。
For example, solve 2x + y = 7 and 3x – y = 8. Adding gives 5x = 15, so x = 3. Substituting back gives y = 1.
例如，解方程组 2x + y = 7 和 3x – y = 8。两式相加得到 5x = 15，所以 x = 3。代回任一方程得到 y = 1。
The key steps are: make the coefficients of one variable match in size, add or subtract to eliminate that variable, solve the resulting one-variable equation, and substitute back to find the other variable.
关键步骤是：使某个变量的系数大小相等，相加或相减以消去该变量，求解得到的一元方程，再代回求出另一个变量。
When using elimination, watch the signs carefully. Adding -y and +y cancels, but adding -y and -y gives -2y.
使用消元法时，要特别注意符号。-y 与 +y 相加会抵消，但 -y 与 -y 相加会得到 -2y。
3. The Substitution Method | 代入法
Substitution is useful when one equation has a variable already isolated, such as y = 2x – 3. You replace that variable in the other equation.
当一个方程中某个变量已经被单独表示出来时，如 y = 2x – 3，代入法非常有用。你将该变量代入另一个方程。
Solve x + y = 9 and y = 2x – 3. Substituting gives x + (2x – 3) = 9, so 3x = 12 and x = 4. Then y = 5.
解方程组 x + y = 9 和 y = 2x – 3。代入后得到 x + (2x – 3) = 9，所以 3x = 12，x = 4。然后 y = 5。
Substitution also works when one variable is the subject of a formula, even if the other equation is non-linear. This makes it very powerful for mixed systems.
当一个变量是公式的主项时，代入法同样适用，即使另一个方程是非线性的。这使得它在处理混合方程组时非常有效。
4. Graphical Interpretation | 图像解释
Each equation represents a line or curve on the Cartesian plane. The solution of the system is the point or points where the graphs intersect.
每个方程代表笛卡尔平面上的一条直线或曲线。方程组的解就是图像相交的一个或多个点。
For two linear equations, there can be exactly one intersection, no intersection (parallel lines), or infinitely many intersections (same line).
对于两个线性方程，可能恰好有一个交点、没有交点（平行线）或有无穷多个交点（同一条直线）。
IGCSE questions may ask you to estimate solutions by drawing graphs. The accuracy depends on the scale and neatness of your plot.
IGCSE 题目可能会要求你通过作图来估计解。准确性取决于你作图的尺度和整洁程度。
Graphical methods are especially useful for checking algebraic answers. A quick sketch can help you see whether the number of solutions you found is sensible.
图像法对于检查代数答案尤其有用。快速画图可以帮助你判断所求出的解的数量是否合理。
5. Linear and Non-Linear Systems | 线性与非线性方程组
A system can include one linear and one quadratic equation, such as y = x² – 4x + 3 and y = 2x – 6. These systems can have two real solutions, one real solution, or no real solutions.
方程组可以包含一个线性方程和一个二次方程，例如 y = x² – 4x + 3 和 y = 2x – 6。这类方程组可能有两个实数解、一个实数解或没有实数解。
The substitution method is usually best here. Set the two expressions for y equal to each other and solve the resulting quadratic equation.
此时通常最好使用代入法。令两个关于 y 的表达式相等，然后求解得到的二次方程。
Once you find the x-values, substitute each one back into the simpler linear equation to find the corresponding y-values. Never use only the quadratic expression for back-substitution unless necessary.
求出 x 值后，将每个 x 值代回较简单的线性方程中以求出对应的 y 值。除非必要，不要只用二次表达式进行回代。
6. Worked Example: Quadratic-Linear System | 示例：二次-线性方程组
Solve y = x² – 2x – 3 and y = x + 1. Equating gives x² – 2x – 3 = x + 1, so x² – 3x – 4 = 0.
解方程组 y = x² – 2x – 3 和 y = x + 1。令两式相等得到 x² – 2x – 3 = x + 1，所以 x² – 3x – 4 = 0。
Factorising gives (x – 4)(x + 1) = 0, so x = 4 or x = -1. Substituting into y = x + 1 gives the solutions (4, 5) and (-1, 0).
因式分解得到 (x – 4)(x + 1) = 0，所以 x = 4 或 x = -1。代入 y = x + 1 得到解 (4, 5) 和 (-1, 0)。
Always check both pairs in the original equations. In this case both pairs satisfy y = x² – 2x – 3 and y = x + 1.
始终要在原方程组中检验这两个数对。本例中两个数对都满足 y = x² – 2x – 3 和 y = x + 1。
This type of question is very common in the Extended paper because it tests factorising, substitution, and accurate sign handling in a single question.
这类题目在扩展试卷中非常常见，因为它在一道题中同时考查了因式分解、代入法和准确的符号处理。
7. Using the Discriminant | 使用判别式
When solving a quadratic-linear system, the number of real solutions is determined by the discriminant of the resulting quadratic equation.
在解二次-线性方程组时，实解的个数由所得二次方程的判别式决定。
Δ = b² – 4ac
If Δ > 0, the line cuts the curve in two distinct points. If Δ = 0, the line touches the curve at one point (tangent). If Δ < 0, there is no intersection.
如果 Δ > 0，直线与曲线相交于两个不同的点。如果 Δ = 0，直线与曲线相切于一个点。如果 Δ < 0，则没有交点。
This is a common IGCSE extension topic: given a line and a curve, find the range of k for which the line does not intersect the curve.
这是 IGCSE 扩展部分常见的题型：给定一条直线和一条曲线，求使直线与曲线不相交的 k 的取值范围。
To do this, substitute the line equation into the curve equation, write the result in the form ax² + bx + c = 0, and then set Δ < 0 or Δ > 0 according to the condition.
要做到这一点，将直线方程代入曲线方程，将结果写成 ax² + bx + c = 0 的形式，然后根据条件令 Δ < 0 或 Δ > 0。
8. Word
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