📚 Solving Simultaneous Equations | 解联立方程组
Simultaneous equations are a set of two or more equations that share the same unknown variables. Solving them means finding the values of the variables that satisfy all equations at the same time. In IGCSE Edexcel Mathematics, linear simultaneous equations appear in both Paper 1 and Paper 2, often combined with word problems or graphs.
联立方程组是一组包含相同未知数的两个或更多方程。解联立方程组就是找出同时满足所有方程的未知数的值。在 IGCSE Edexcel 数学考试中，线性联立方程经常出现在 Paper 1 和 Paper 2 中，常与应用题或图像结合考查。
1. What Are Simultaneous Equations? | 什么是联立方程组
A linear equation in two variables, such as 3x + 2y = 12, represents a straight line on a graph. When you have two such equations, the solution is the point where the two lines intersect. That point gives the single pair (x, y) that works for both equations.
一个含有两个变量的线性方程，例如 3x + 2y = 12，在图像上表示一条直线。当你拥有两个这样的方程时，解就是两条直线的交点。这个交点给出同时满足两个方程的唯一一对值 (x, y)。
x + y = 5 and 2x − y = 4 → solution: x = 3, y = 2
If the lines are parallel, there is no solution. If the two equations represent the same line, there are infinitely many solutions. You should know these three cases.
如果两条直线平行，则无解。如果两个方程表示同一条直线，则有无穷多解。你需要了解这三种情况。
2. The Graphical Method | 图像法
To solve simultaneous equations graphically, you draw both lines on the same set of axes and read the coordinates of their intersection. This method is simple but requires accurate graph paper and careful plotting.
用图像法解联立方程时，你需要在同一坐标轴中画出两条直线，然后读出交点的坐标。这种方法简单，但需要精确的坐标纸和仔细的描点。
Rearrange each equation into the form y = mx + c to make plotting easier.
Plot at least three points for each line to avoid mistakes.
The intersection point (x, y) is your solution.
将每个方程重写成 y = mx + c 的形式，便于作图。
每条直线至少描三个点，避免出错。
交点 (x, y) 就是你的解。
For example, y = x + 1 and y = −2x + 4 intersect at (1, 2). If the intersection is not an integer point, estimating from a graph may be unreliable; in an exam you are usually asked to check your answer algebraically.
例如，y = x + 1 和 y = −2x + 4 交于点 (1, 2)。如果交点不是整数点，从图像中估算可能不可靠；考试中通常要求你用代数方法检验答案。
3. Substitution Method | 代入消元法
Substitution is often used when one equation is already given in the form y = ... or can be rearranged easily. You replace y in the other equation with the expression in terms of x.
当一个方程已经表示为 y = ... 或很容易变形时，常使用代入法。你把另一个方程中的 y 替换成含有 x 的表达式。
Example: y = 2x + 1 and 3x + 2y = 11
Substitute y = 2x + 1 into the second equation:
将 y = 2x + 1 代入第二个方程：
3x + 2(2x + 1) = 11 → 3x + 4x + 2 = 11 → 7x = 9 → x = 9⁄7
Then find y = 2(9⁄7) + 1 = 18⁄7 + 7⁄7 = 25⁄7. Always substitute back into the original equation to check.
然后求 y = 2(9⁄7) + 1 = 18⁄7 + 7⁄7 = 25⁄7。始终代回原方程进行检验。
4. Elimination Method | 加减消元法
Elimination is usually the fastest method for linear equations. You multiply one or both equations by a constant so that the coefficient of one variable is the same (or opposite) in both equations. Then you add or subtract to eliminate that variable.
加减消元法通常是对线性方程最快的方法。你将一个或两个方程乘以一个常数，使某一个变量的系数在两个方程中相同（或相反）。然后通过相加或相减消去该变量。
3x + 2y = 7 and 2x − 2y = 8
Here the y coefficients are +2 and −2, so add the equations directly:
这里 y 的系数是 +2 和 −2，所以直接将两式相加：
(3x + 2y) + (2x − 2y) = 7 + 8 → 5x = 15 → x = 3
Then substitute x = 3 into either equation: 3(3) + 2y = 7 gives 2y = −2, so y = −1.
然后将 x = 3 代入任意一个方程：3(3) + 2y = 7 得到 2y = −2，所以 y = −1。
5. Elimination When Coefficients Need Scaling | 需要调整系数的消元法
If the coefficients are not the same, you must multiply first. For example, solve:
如果系数不相同，你必须先进行乘法调整。例如，解：
2x + 3y = 8 and 5x − 4y = 20
To eliminate x, multiply the first equation by 5 and the second by 2:
为了消去 x，将第一个方程乘以 5，第二个方程乘以 2：
10x + 15y = 40 and 10x − 8y = 40
Subtract: 23y = 0, so y = 0. Substitute back: 2x + 0 = 8, so x = 4.
相减：23y = 0，所以 y = 0。代回：2x + 0 = 8，所以 x = 4。
Be careful with signs. If the signs of the chosen coefficients are the same, subtract; if they are different, add.
注意符号。如果所选系数同号，则相减；如果异号，则相加。
6. Equations Involving Fractions | 含分数的方程
When a simultaneous equation contains fractions, first clear the denominators by multiplying both sides of each equation by a common multiple. This gives you whole-number coefficients to work with.
当联立方程中含有分数时，首先通过在等式两边同时乘以公倍数来去分母。这样你就得到了可以操作的整数系数。
x⁄2 + y⁄3 = 4 and x⁄4 − y⁄2 = 0
Multiply the first equation by 6 and the second by 4:
第一个方程乘以 6，第二个方程乘以 4：
3x + 2y = 24 and x − 2y = 0
Add the two equations: 4x = 24, so x = 6. Then x − 2y = 0 gives y = 3. Always check that the fractions work in the original equations.
两式相加：4x = 24，所以 x = 6。然后 x − 2y = 0 得到 y = 3。始终检查原方程中的分数是否成立。
7. Solving Non-Linear Simultaneous Equations | 解非线性联立方程组
IGCSE Edexcel sometimes asks you to solve one linear and one quadratic equation together. For example, y = 2x + 1 and y = x2 + 3x − 5. The graphical solution is the intersection of a line and a curve.
IGCSE Edexcel 有时会要求你同时解一个线性方程和一个二次方程。例如，y = 2x + 1 和 y = x2 + 3x − 5。图像解就是一条直线与一条曲线的交点。
Use substitution: replace y in the quadratic with the linear expression.
使用代入法：将二次方程中的 y 替换为线性表达式。
2x + 1 = x² + 3x − 5 → x² + x − 6 = 0 → (x + 3)(x − 2) = 0
So x = −3 or x = 2. Substitute these into the linear equation to find y = −5 and y = 5. The solutions are (−3, −5) and (2, 5).
所以 x = −3 或 x = 2。将这些值代入线性方程求出 y = −5 和 y = 5。解为 (−3, −5) 和 (2, 5)。
You may need to use the quadratic formula if the quadratic does not factorise easily:
如果二次式不易因式分解，你可能需要使用二次公式：
x = (−b ± √b² − 4ac) ⁄ 2a
8. Word Problems Using Simultaneous Equations | 用联立方程组解应用题
Exam questions often present a real-world scenario. You need to define variables, translate the information into two equations, then solve them.
考试题目通常给出实际生活场景。你需要定义变量，将信息转化为两个方程，然后求解。
Example: The total cost of 3 apples and 2 bananas is $7. The cost of 5 apples and 1 banana is $9. Find the cost of one apple and one banana.
例子：3 个苹果和 2 根香蕉的总价是 7 美元。5 个苹果和 1 根香蕉的总价是 9 美元。求 1 个苹果和 1 根香蕉的单价。
Let a be the price of an apple and b the price of a banana.
设 a 为苹果单价，b 为香蕉单价。
3a + 2b = 7 and 5a + b = 9
Multiply the second equation by 2: 10a + 2b = 18. Subtract the first: 7a = 11, so a = 11⁄7. Then b = 9 − 5(11⁄7) = 63⁄7 − 55⁄7 = 8⁄7. One apple costs $1.57 and one banana costs $1.14 approximately.
第二个方程乘以 2：10a + 2b = 18。减去第一个：7a = 11，所以 a = 11⁄7。然后 b = 9 − 5(11⁄7) = 63⁄7 − 55⁄7 = 8⁄7。1 个苹果约 1.57 美元，1 根香蕉约 1.14 美元。
Always state your final answer in the context of the question, with appropriate units.
始终在最终答案中结合题目语境，注明适当单位。
9. Common Mistakes and Key Tips | 常见错误与关键技巧
Many students lose marks on simultaneous equations due to small sign errors or careless arithmetic. Here are the most important checks.
许多学生在联立方程上丢分，原因是符号错误或计算粗心。以下是最重要的检查点。
| Mistake | Correction |
| Dropping minus signs when subtracting equations | Write each expression carefully and distribute the minus sign to every term |
| Forgetting to multiply every term when scaling an equation | Multiply both sides, not just one term |
| Not checking the answer in the original equations | Always substitute both values back into both original equations |
| 常见错误 | 改正方法 |
| 减去方程时丢掉负号 | 仔细写出每个表达式，并将减号分配给每一项 |
| 对方程乘以倍数时忘记乘以所有项 | 两边同时乘以，而不是只乘以某一项 |
| 没有在原始方程中检验答案 | 始终将两个值代回两个原始方程中检验 |
Tip: In the elimination method, if both coefficients are positive, subtract; if one is positive and one is negative, add. This rule works with aligned signs.
技巧：在消元法中，如果两个系数同为正，则相减；如果一正一负，则相加。这个规则适用于符号对齐的情况。
10. Exam Practice and Full Walkthrough | 真题练习与完整解析
Here is a typical exam-style question with a full solution.
下面是一道典型考试风格题目，附完整解答。
Question: Solve the simultaneous equations 4x + y = 9 and 2x − 3y = 1.
题目：解联立方程组 4x + y = 9 和 2x − 3y = 1。
Step 1: Multiply the first equation by 3 to make the y coefficients opposite: 12x + 3y = 27.
步骤 1：将第一个方程乘以 3，使 y 的系数相反：12x + 3y = 27。
Step 2: Add this to the second equation:
步骤 2：将它与第二个方程相加：
(12x + 3y) + (2x − 3y) = 27 + 1 → 14x = 28 → x = 2
Step 3: Substitute x = 2 into 4x + y = 9: 8 + y = 9, so y = 1.
步骤 3：将 x = 2 代入 4x + y = 9：8 + y = 9，所以 y = 1。
Step 4: Check: 4(2) + 1 = 9 and 2(2) − 3(1) = 1. Both equations are satisfied.
步骤 4：检验：4(2) + 1 = 9，2(2) − 3(1) = 1。两个方程都满足。
Answer: x = 2, y = 1.
答案：x = 2, y = 1。
11. Practice Questions | 自测练习
Try these questions on your own, then check your answers below.
请独立尝试以下问题，然后对照下方答案。
1. Solve 5x + 2y = 16 and 3x + 2y = 12.
2. Solve x + 2y = 7 and x2 + y2 = 10. (One linear, one quadratic.)
3. The sum of two numbers is 15 and their difference is 3. Find the numbers.
1. 解 5x + 2y = 16 和 3x + 2y = 12。
2. 解 x + 2y = 7 和 x2 + y2 = 10。（一个线性，一个二次。）
3. 两个数的和是 15，差是 3。求这两个数。
Answers: 1. x = 2, y = 3 | 2. (x, y) = (1, 3) or (2.6, 2.2) | 3. 9 and 6
For question 2, the exact solutions come from substitution: x = 7 − 2y, so (7 − 2y)2 + y2 = 10, giving 5y2 − 28y + 39 = 0. This factorises to (5y − 13)(y − 3) = 0. Check each pair in the original equations.
对于第 2 题，精确解来自代入法：x = 7 − 2y，所以 (7 − 2y)2 + y2 = 10，得到 5y2 − 28y + 39 = 0。因式分解为 (5y − 13)(y − 3) = 0。在原始方程中检验每一组解。
12. Summary and Final Advice | 总结与最终建议
You now have a full toolkit for solving simultaneous equations. Remember these four key steps: choose an appropriate method, show clear working, solve accurately, and always check your answers.
你现在已经掌握了解决联立方程组的完整工具包。记住四个关键步骤：选择合适的方法、展示清晰的过程、准确求解、始终检验答案。
Use the graphical method only when a graph is required or for verification.
Use substitution when one variable is already isolated.
Use elimination when coefficients are easy to align.
For non-linear systems, always expect two solutions.
仅在需要画图或验证时使用图像法。
当一个变量已经被分离时使用代入法。
当系数容易对齐时使用加减消元法。
对于非线性方程组，通常会有两个解。
With regular practice, simultaneous equations become one of the most reliable topics to score full marks in your IGCSE Edexcel Mathematics exam.
通过定期练习，联立方程组会成为你在 IGCSE Edexcel 数学考试中最容易获得满分的主题之一。
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