📚 Solving Linear and Quadratic Simultaneous Equations | 解线性与二次联立方程
em can have two solutions, one solution, or no real solutions at all. The central skill is to combine the two equations so that only one unknown remains, then solve the resulting equation accurately.
在IGCSE数学中，联立方程组要求我们求出未知变量的值，使每一个方程同时成立。当一个方程是线性方程，例如 y = 2x + 3，另一个是二次方程，例如 y = x² - 4x + 1，方程组可能有两组解、一组解或没有实数解。核心技能是把两个方程结合起来，使方程中只留下一个未知数，然后准确求解得到的方程。
1. What Are Simultaneous Equations? | 什么是联立方程
A simultaneous equation is an equation that must be solved together with at least one other equation. The solution is the set of values that satisfies all the equations at once. For a system involving two variables, x and y, each solution is usually written as an ordered pair (x, y). In IGCSE exam questions, you will often see two equations written together, and the task is to find all possible pairs of x and y that work for both equations.
联立方程是指至少需要与另一个方程一起求解的方程。解就是能够同时满足所有方程的一组值。对于含有两个变量 x 和 y 的方程组，每个解通常写成有序数对 (x, y)。在IGCSE考试题中，你经常会看到两个方程写在一起，任务就是找出所有能够同时满足两个方程的 x 和 y 的值。
A linear equation has variables raised only to the power of 1, so its graph is a straight line. A quadratic equation includes a variable raised to the power of 2, so its graph is a parabola. This difference in shape is exactly why a linear-quadratic system can have up to two real intersection points.
线性方程中变量的次数仅为1，因此它的图像是一条直线。二次方程中含有次数为2的变量，因此它的图像是一条抛物线。这种图像形状上的差异正是线性-二次方程组最多可以有两个实数交点解的原因。
2. Linear-Linear vs Linear-Quadratic Systems | 线性-线性与线性-二次方程组的区别
For two linear equations, you can often use the elimination method: add or subtract the equations to cancel one variable, then solve for the other. This method is efficient because both equations have degree 1. However, when one equation is quadratic, elimination by simple addition or subtraction is usually not possible, because the quadratic term will not cancel directly with a linear term.
对于两个线性方程，你通常可以使用消元法：将两个方程相加或相减，消去一个变量，然后求出另一个变量。这种方法之所以高效，是因为两个方程的次数都是1。然而，当一个方程是二次方程时，通常无法通过简单的相加或相减来消元，因为二次项不会直接与一次项相互抵消。
Instead, the standard technique for a linear-quadratic system is substitution. You take one variable from the linear equation and replace it in the quadratic equation. This step removes one variable and produces a single quadratic equation, which can be solved by factorising, completing the square, or using the quadratic formula.
相反，解线性-二次方程组的标准方法是代入法。你从线性方程中取出一个变量的表达式，并将它代入二次方程中替换同一个变量。这一步可以消去一个变量，得到一个一元二次方程，然后可以通过因式分解、配方法或求根公式来求解。
3. Solving a Linear-Linear System by Elimination | 用消元法解线性方程组
Before studying linear-quadratic systems, it is useful to revise the elimination method for two linear equations. Consider the system:
在学习线性-二次方程组之前，先复习一下两个线性方程的消元法会很有帮助。考虑下面的方程组：
2x + y = 7 and x - y = 2
ancel because y + (-y) = 0. This gives (2x + y) + (x - y) = 7 + 2, so 3x = 9, hence x = 3.
如果我们将两个等式的左边相加、右边相加，y 项就会抵消，因为 y + (-y) = 0。这样得到 (2x + y) + (x - y) = 7 + 2，所以 3x = 9，因此 x = 3。
, for example x - y = 2. This gives 3 - y = 2, so y = 1. Therefore the solution is (3, 1).
现在将 x = 3 代入任意一个原方程，例如 x - y = 2。得到 3 - y = 2，所以 y = 1。因此方程组的解为 (3, 1)。
2(3) + 1 = 6 + 1 = 7, which is correct. This habit of checking is very important in exams.
把解代入另一个方程检验：2(3) + 1 = 6 + 1 = 7，结果正确。这种检验习惯在考试中非常重要。
4. Substitution Method for Linear-Quadratic Systems | 线性-二次方程组的代入法
linear equation is already given as y = mx + c, which makes substitution straightforward.
当一个方程是线性方程、另一个是二次方程时，代入法几乎总是最佳方法。首先，选择线性方程，并把其中一个变量写成另一个变量的表达式。在许多IGCSE题目中，线性方程通常已经写成了 y = mx + c 的形式，这使得代入变得简单直接。
Second, take the expression for y from the linear equation and replace every y in the quadratic equation with that expression. Third, expand any brackets and collect like terms. You should now have a quadratic equation in one variable, usually x. Fourth, solve this quadratic equation using factorising, completing the square, or the quadratic formula. Finally, substitute each x-value back into the linear equation to find the matching y-value.
其次，从线性方程中取出 y 的表达式，并用它替换二次方程中的每一个 y。第三，展开所有括号并合并同类项。此时你应该得到一个关于一个变量（通常是 x）的二次方程。第四，使用因式分解、配方法或求根公式解这个二次方程。最后，将每个 x 值代回线性方程，求出对应的 y 值。
A strong substitution routine will help you avoid sign errors and lost solutions. Always write each step on a new line and label your results clearly.
一个清晰的代入流程可以帮助你避免符号错误和漏解。做题时每一步都要另起一行书写，并把结果标注清楚。
5. Worked Example: Linear-Quadratic System | 例题：线性与二次方程组
Solve the following simultaneous equations:
解下列联立方程组：
y = x + 1 and y = x² - 3x + 4
Since both expressions on the right-hand sides are equal to y, we can set them equal to each other. This gives:
由于两个等式的右边都等于 y，我们可以将它们设为相等。这样得到：
x + 1 = x² - 3x + 4
Move all terms to one side so that the quadratic equals zero. Subtract x and subtract 1 from both sides:
将所有项移到一边，使二次方程等于零。两边同时减去 x 和减去 1：
0 = x² - 3x + 4 - x - 1 = x² - 4x + 3
Now we have a standard quadratic equation. The next step is to factorise it:
现在我们得到一个标准的一元二次方程。下一步是将它因式分解：
x² - 4x + 3 = (x - 1)(x - 3) = 0
m x - 3 = 0 we get x = 3. These are the two x-values where the line and the parabola meet.
令每个因式等于零。由 x - 1 = 0 得到 x = 1，由 x - 3 = 0 得到 x = 3。这两个 x 值就是直线与抛物线相交的位置。
6. Finding the Corresponding y-Values | 求对应的 y 值
Do not substitute the x-values into the quadratic equation unless you need to check your work; the linear equation is simpler and reduces the chance of arithmetic mistakes.
求出 x = 1 和 x = 3 只完成了一半工作。我们现在必须使用线性方程 y = x + 1 来求对应的 y 值。除非需要检验，否则不要把 x 值代入二次方程；线性方程更简单，可以减少计算错误。
= 2. When x = 3, we have y = 3 + 1 = 4. Therefore the two solutions are (1, 2) and (3, 4).
当 x = 1 时，y = 1 + 1 = 2。当 x = 3 时，y = 3 + 1 = 4。因此两组解为 (1, 2) 和 (3, 4)。
Always write the solutions as coordinate pairs. In a graph question, these pairs are the coordinates of the intersection points.
请务必将解写成坐标数对。在图像题中，这些数对就是交点的坐标。
7. Using the Discriminant to Predict the Number of Solutions | 用判别式判断解的个数
tion, you can write the result in the form ax² + bx + c = 0. The discriminant is given by:
当你把线性方程代入二次方程后，可以将结果写成 ax² + bx + c = 0 的形式。判别式由下式给出：
Δ = b² - 4ac
s one repeated real root, so the line is a tangent to the parabola and there is exactly one solution. If Δ < 0, the quadratic has no real roots, so the line and the parabola do not intersect.
判别式可以在你解方程之前就告诉你方程组有多少个实数解。如果 Δ > 0，二次方程有两个不同的实数根，因此直线与抛物线相交于两点。如果 Δ = 0，二次方程有一个重复的实数根，因此直线是抛物线的切线，恰好有一个解。如果 Δ < 0，二次方程没有实数根，因此直线与抛物线不相交。
This is especially useful for exam questions that ask you to determine whether a line intersects, touches, or misses a given curve without necessarily finding the exact coordinates.
这在考试中尤其有用，当题目要求你判断一条直线是与给定曲线相交、相切还是不相交，而不一定要求出具体坐标时，判别式可以快速给出结论。
8. Graphical Interpretation | 图形解释
² - 3x + 4 represents a parabola. The simultaneous solutions are exactly the coordinates of the points where the line and the parabola meet.
在坐标平面上，线性方程如 y = x + 1 表示一条斜率为1、y截距为1的直线。二次方程如 y = x² - 3x + 4 表示一条抛物线。联立方程组的解就是直线与抛物线相交点的坐标。
If the discriminant is positive, the line crosses the parabola at two distinct points. If the discriminant is zero, the line just touches the curve at one point, and the straight line is a tangent. If the discriminant is negative, the line misses the parabola completely and there is no real point of intersection.
如果判别式为正，直线与抛物线相交于两个不同的点。如果判别式为零，直线在一点处刚好接触曲线，这条直线就是切线。如果判别式为负，直线完全不与抛物线相交，没有实数交点。
Sketching a quick diagram can help you check whether your algebraic answers are sensible. For example, if you find two solutions but your sketch shows the line clearly below the parabola, you may have made a sign error.
快速画出示意图有助于检查代数答案是否合理。例如，如果你求出两组解，但草图显示直线明显在抛物线下方，那么你可能出现了符号错误。
9. Common Mistakes and How to Avoid Them | 常见错误及如何避免
ial correctly: (2x - 1)² = 4x² - 4x + 1, not 2x² - 1. Always expand brackets step by step.
代入后展开括号时最容易出错。例如，如果你用 (2x - 1) 替换 y，而方程中含有 y²，你必须正确平方整个二项式：(2x - 1)² = 4x² - 4x + 1，而不是 2x² - 1。请务必逐步展开括号。
Another common mistake is stopping after finding the x-values. A complete solution to a simultaneous equation must include both coordinates. Some students also mix up the pairing of x and y values when there are two solutions. To avoid this, always substitute each x-value into the original linear equation immediately and record the full coordinate pair.
另一个常见错误是求出 x 值后就不再继续。联立方程的完整解必须包含两个坐标。有些学生在有两组解时还会弄混 x 和 y 的对应关系。为避免这种情况，应始终将每个 x 值立即代入原线性方程，并记录完整的坐标数对。
Sign errors when rearranging terms from one side of an equation to the other are also frequent. For instance, when moving x from the right side to the left side, its sign must change. Write each rearrangement clearly and check by substituting the final answers back into both original equations.
移项时的符号错误也很常见。例如，把 x 从等式右边移到左边时，它的符号必须改变。请清楚书写每一步变形，并将最终答案代回两个原方程进行检验。
10. Exam-Style Practice Question | 考试模拟练习
Try the following IGCSE-style question before reading the solution below:
在阅读下面解答之前，请先尝试以下IGCSE风格的题目：
Solve y = x² - 2x + 2 and y = 2x - 1
First, set the two expressions for y equal to each other:
首先，令两个关于 y 的表达式相等：
x² - 2x + 2 = 2x - 1
Move all terms to one side. Subtract 2x from both sides and add 1 to both sides:
将所有项移到一边。两边同时减去
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图