Understanding Simultaneous Equations | 理解联立方程

📚 Understanding Simultaneous Equations | 理解联立方程

Simultaneous equations are a pair (or more) of equations that share the same variables, and we seek the values of those variables that satisfy all equations at the same time. In IGCSE Mathematics, you most often deal with two linear equations in two unknowns, x and y.

联立方程是指两个（或更多）包含相同变量的方程，我们需要找到同时满足所有这些方程的变量值。在IGCSE数学中，最常见的是包含两个未知数 x 和 y 的两个线性方程。

For example, consider the system:

2x + y = 7 and x − y = 2

Here x = 3 and y = 1 satisfy both equations, because 2(3) + 1 = 7 and 3 − 1 = 2. This is called the solution of the simultaneous equations.

例如，考虑方程组：

2x + y = 7 和 x − y = 2

这里 x = 3，y = 1 同时满足两个方程，因为 2(3) + 1 = 7 且 3 − 1 = 2。这就是联立方程的解。

1. What Are Simultaneous Equations? | 什么是联立方程？

A linear equation in two variables can be written in the form ax + by = c, where a, b and c are constants. When you have two such equations, you are looking for an ordered pair (x, y) that makes both equations true simultaneously.

含两个变量的线性方程可以写成 ax + by = c 的形式，其中 a、b、c 是常数。当你拥有两个这样的方程时，你是在寻找一个有序数对 (x, y)，使得两个方程同时成立。

Key idea: one equation alone has infinitely many solutions (a whole line of points). Two equations together usually narrow this down to a single unique point, which is the intersection of the two lines.

关键思想：单独一个方程有无数个解（一条直线上的所有点）。两个方程组合在一起通常可以将范围缩小到一个唯一的点，即两条直线的交点。

Unique solution: the two lines intersect at exactly one point.
唯一解：两条直线恰好相交于一点。
No solution: the two lines are parallel and never meet.
无解：两条直线平行，永不相交。
Infinite solutions: the two lines are actually the same line.
无穷多解：两条直线实际上是同一条直线。

In this article you will learn three algebraic or graphical methods to solve these systems, plus how to handle real-life word problems. Mastering these skills is essential for the IGCSE exam.

在本文中，你将学习三种代数或图像解法，以及如何处理真实生活中的应用题。掌握这些技能对IGCSE考试至关重要。

2. The Substitution Method | 代入法

The substitution method works by solving one equation for one variable, then substituting this expression into the other equation. This reduces the system to a single equation with one unknown.

代入法的思路是：先从一个方程中解出一个变量，然后将这个表达式代入另一个方程。这样就把方程组化简为只含一个未知数的单一方程。

Follow these steps:

请按以下步骤操作：

Step 1: Choose one equation and rearrange it to make either x or y the subject.
第一步：选择一个方程，通过移项使 x 或 y 成为主题（即单独放在一边）。
Step 2: Substitute the resulting expression into the other equation.
第二步：将得到的表达式代入另一个方程。
Step 3: Solve the resulting linear equation for the remaining variable.
第三步：解这个只含一个变量的线性方程。
Step 4: Substitute this value back into one of the original equations to find the other variable.
第四步：将这个值代回原方程之一，求出另一个变量。

Worked example:

y = 2x + 1 and 3x + 2y = 10

Here y is already the subject of the first equation. Substitute y = 2x + 1 into the second equation:

这里第一个方程已经将 y 作为主题。将 y = 2x + 1 代入第二个方程：

3x + 2(2x + 1) = 10

Expanding gives 3x + 4x + 2 = 10, so 7x = 8, hence x = 8/7. Now substitute back to find y:

展开得 3x + 4x + 2 = 10，所以 7x = 8，因此 x = 8/7。再代回求出 y：

y = 2(8/7) + 1 = 16/7 + 7/7 = 23/7

So the solution is x = 8/7, y = 23/7. You can check both values in the original equations.

所以解为 x = 8/7，y = 23/7。你可以将这两个值代入原方程验证。

Substitution is especially useful when one equation already has a variable isolated, or when dealing with non-linear equations, but for linear pairs the elimination method is often faster.

代入法尤其适用于其中一个方程已经将某个变量单独表示的情形，或者处理非线性方程时。但对于线性方程组，消元法通常更快。

3. The Elimination Method | 消元法

The elimination method (also called the addition method) involves adding or subtracting the equations to eliminate one variable. You may need to multiply one or both equations by a constant first so that the coefficients of one variable match.

消元法（也称加减法）通过将两个方程相加或相减来消去一个变量。你可能需要先将一个或两个方程乘以某个常数，使某个变量的系数相等。

Standard procedure:

标准步骤：

Step 1: Arrange both equations in the form ax + by = c.
第一步：将两个方程整理为 ax + by = c 的形式。
Step 2: If necessary, multiply one or both equations so that the coefficients of x (or y) have the same absolute value.
第二步：如有必要，将一个或两个方程乘以适当的数，使 x（或 y）的系数绝对值相等。
Step 3: Add or subtract the equations to eliminate one variable.
第三步：将两个方程相加或相减，以消去一个变量。
Step 4: Solve the remaining one-variable equation.
第四步：解剩下的一个变量方程。
Step 5: Substitute back to find the other variable.
第五步：代回求出另一个变量。

Worked example:

3x + 2y = 12 and x − 2y = 4

The coefficients of y are +2 and −2, which are opposites. Add the two equations directly:

y 的系数分别为 +2 和 −2，互为相反数。直接将两个方程相加：

3x + 2y + x − 2y = 12 + 4

This simplifies to 4x = 16, so x = 4. Substitute x = 4 into x − 2y = 4:

化简得 4x = 16，所以 x = 4。将 x = 4 代入 x − 2y = 4：

4 − 2y = 4, so −2y = 0, hence y = 0

The solution is (4, 0). Always check your answer in both original equations.

解为 (4, 0)。请务必在两个原方程中检验你的答案。

When the coefficients are not opposites, multiply first. For example, to eliminate x from 2x + 3y = 8 and 3x − y = 1, multiply the first by 3 and the second by 2 to get 6x in both, then subtract.

当系数不是相反数时，先进行乘法。例如，要从 2x + 3y = 8 和 3x − y = 1 中消去 x，可将第一个方程乘以 3，第二个方程乘以 2，使两个方程都含有 6x，然后相减。

4. The Graphical Method | 图像法

The graphical method requires plotting both equations on the same set of axes. The coordinates of the point where the two lines cross give the solution.

图像法需要在同一坐标系中绘制两个方程的图像。两条直线交点的坐标就是方程组的解。

To draw a straight line from a linear equation, you can:

要根据线性方程画直线，你可以：

Find the x-intercept by setting y = 0 and solving for x.
令 y = 0 求 x，得到 x 轴截距。
Find the y-intercept by setting x = 0 and solving for y.
令 x = 0 求 y，得到 y 轴截距。
Plot these two points and draw the line through them.
标出这两个点，并过它们画一条直线。

Worked example:

y = x + 1 and y = −2x + 4

For y = x + 1, when x = 0, y = 1; when y = 0, x = −1. Plot (0, 1) and (−1, 0). For y = −2x + 4, when x = 0, y = 4; when y = 0, x = 2. Plot (0, 4) and (2, 0).

对于 y = x + 1，当 x = 0 时 y = 1；当 y = 0 时 x = −1。标出 (0, 1) 和 (−1, 0)。对于 y = −2x + 4，当 x = 0 时 y = 4；当 y = 0 时 x = 2。标出 (0, 4) 和 (2, 0)。

Draw both lines on the same axes. They intersect at approximately (1, 2). Therefore x = 1, y = 2 is the solution.

在同一坐标系中画出两条直线。它们相交于点 (1, 2) 附近。因此解为 x = 1，y = 2。

Graphical solutions are often approximate, especially when the intersection is not at integer coordinates. For exact answers, use substitution or elimination. However, the graphical method helps you visualise the meaning of a solution and is useful for checking your results.

图像法的解往往是近似值，尤其是当交点不在整数坐标上时。要想得到精确答案，请使用代入法或消元法。不过，图像法有助于直观理解解的含义，也便于检查结果。

5. Solving Word Problems | 解应用题

Many real-life problems can be solved by translating them into simultaneous equations. The key is to define your variables clearly and then set up two independent equations based on the given information.

许多实际问题可以通过转化为联立方程来解决。关键在于清晰地定义变量，然后根据题目给出的信息建立两个独立的方程。

Worked example:

The sum of two numbers is 15. Three times the larger number plus twice the smaller number is 40. Find the numbers.

已知两个数之和为 15。较大数的三倍加上较小数的两倍等于 40。求这两个数。

Let the larger number be a and the smaller number be b. Then:

设较大的数为 a，较小的数为 b。则有：

a + b = 15 and 3a + 2b = 40

Use elimination. Multiply the first equation by 2:

使用消元法。将第一个方程乘以 2：

2a + 2b = 30 and 3a + 2b = 40

Subtract the first from the second: a = 10. Then b = 15 − 10 = 5. The two numbers are 10 and 5.

用第二个方程减去第一个方程：a = 10。然后 b = 15 − 10 = 5。所以这两个数是 10 和 5。

When setting up word problems, watch for words like “sum” (addition), “difference” (subtraction), “twice” (multiply by 2), and “is equal to” (=). Always state what each variable represents before writing equations.

在建立应用题方程时，要注意这类词：”sum”（和，加法）、”difference”（差，减法）、”twice”（两倍，乘以 2）、”is equal to”（等于）。在写方程之前，务必说明每个变量代表什么。

6. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无穷多解

Not every pair of simultaneous linear equations has a unique solution. There are two special cases you must recognise.

并非每对联立线性方程都有唯一解。有两种特殊情况你必须识别。

Case 1: No solution (parallel lines)

情况一：无解（平行线）

If the two equations have the same gradient (slope) but different y-intercepts, the lines are parallel and never intersect. Algebraically, this happens when the coefficients of x and y are proportional, but the constant term is not proportional in the same way.

如果两个方程的斜率相同但 y 截距不同，那么两条直线平行，永不相交。从代数上看，这发生在 x 和 y 的系数成比例，但常数项不成同样比例时。

Example:

2x + 3y = 6 and 4x + 6y = 10

Divide the second equation by 2: 2x + 3y = 5. This contradicts the first equation (6 ≠ 5), so there is no solution.

将第二个方程除以 2：2x + 3y = 5。这与第一个方程矛盾（6 ≠ 5），所以无解。

Case 2: Infinite solutions (same line)

情况二：无穷多解（同一条直线）

If one equation is a multiple of the other, then both equations represent the same line. Every point on that line is a solution.

如果一个方程是另一个方程的倍数，那么两个方程代表同一条直线。该直线上的每一个点都是解。

Example:

x + 2y = 4 and 2x + 4y = 8

The second equation is exactly 2 times the first. Therefore there are infinitely many solutions.

第二个方程恰好是第一个方程的 2 倍。因此有无穷多组解。

In the IGCSE exam, you may be asked to identify these cases from a graph or from the equations themselves. Remember that when solving by elimination, “no solution” appears as a false statement like 0 = 2, while “infinite solutions” appears as an identity like 0 = 0.

在IGCSE考试中，你可能会被要求从图像或方程本身识别这些情况。记住，用消元法求解时，”无解”表现为 0 = 2 这样的矛盾式，而”无穷多解”表现为 0 = 0 这样的恒等式。

7. Common Mistakes and Tips | 常见错误与技巧

Here are the most frequent errors students make when solving simultaneous equations, along with advice to avoid them.

以下是学生在解联立方程时最常犯的错误，以及避免这些错误的建议。

Mistake 1: Forgetting to multiply every term when preparing for elimination.
错误一：在进行消元准备时，忘记将每一项都乘以同一个数。
Mistake 2: Making sign errors when subtracting one equation from another, especially with negative coefficients.
错误二：用一个方程减去另一个方程时出现符号错误，特别是含负系数时。
Mistake 3: Substituting back into the wrong equation, leading to an incorrect value.
错误三：代回了错误的方程，导致得出错误的值。
Mistake 4: Not checking the solution in both original equations.
错误四：没有在两个原方程中检验解。

Useful tips:

实用技巧：

Always rearrange equations into the same form, such as ax + by = c, before applying elimination.
在应用消元法前，务必将方程整理成相同的形式，如 ax + by = c。
If one equation already has a variable isolated, substitution is usually quicker than elimination.
如果某个方程已经将变量单独表示出来，代入法通常比消元法更快。
When the coefficients are fractions, multiply the whole equation by the common denominator first.
当系数是分数时，先将整个方程乘以公分母。
After finding x and y, substitute both into the original equations. This takes 30 seconds and prevents careless errors.
求出 x 和 y 后，将两者代入原方程验证。这只需要 30 秒，可以防止粗心错误。

8. Practice Questions | 练习题

Solve these problems on your own, then check the answers below each one.

请自行解答以下问题，然后对照每题下方的答案。

Question 1:

2x + y = 8 and x − y = 1

Answer: Add the equations to get 3x = 9, so x = 3. Then y = 2. Solution: x = 3, y = 2.

答案：两式相加得 3x = 9，所以 x = 3。然后 y = 2。解为 x = 3，y = 2。

Question 2:

5x + 2y = 20 and 3x − y = 1

Answer: Multiply the second equation by 2: 6x − 2y = 2. Add to the first: 11x = 22, so x = 2. Then 3(2) − y = 1 gives y = 5. Solution: x = 2, y = 5.

答案：将第二个方程乘以 2：6x − 2y = 2。与第一个方程相加：11x = 22，所以 x = 2。由 3(2) − y = 1 得 y = 5。解为 x = 2，y = 5。

Question 3: The cost of 2 apples and 3 bananas is $3.40. The cost of 1 apple and 4 bananas is $2.90. Find the price of one apple and one banana.

问题三：2 个苹果和 3 根香蕉共 3.40 美元。1 个苹果和 4 根香蕉共 2.90 美元。求一个苹果和一根香蕉的价格。

Let a = price of one apple and b = price of one banana. Then:

2a + 3b = 3.40 and a + 4b = 2.90

Multiply the second equation by 2: 2a + 8b = 5.80. Subtract the first equation: (2a + 8b) − (2a + 3b) = 5.80 − 3.40, so 5b = 2.40, hence b = 0.48. Then a = 2.90 − 4(0.48) = 0.98. So an apple costs $0.98 and a banana costs $0.48.

设 a 为一个苹果的价格，b 为一根香蕉的价格。则有：

2a + 3b = 3.40 和 a + 4b = 2.90

将第二个方程乘以 2：2a + 8b = 5.80。用其减去第一个方程：(2a + 8b) − (2a + 3b) = 5.80 − 3.40，所以 5b = 2.40，因此 b = 0.48。则 a = 2.90 − 4(0.48) = 0.98。所以一个苹果 0.98 美元，一根香蕉 0.48 美元。

Remember these four steps for any simultaneous equation problem:

解任何联立方程问题都要记住以下四步：

Recognise which method to use (substitution, elimination, or graphical).
识别使用哪种方法（代入法、消元法或图像法）。
Carry out the algebra carefully, checking signs at every step.
仔细进行代数运算，每一步都检查符号。
Write the solution as an ordered pair (x, y).
用有序数对 (x, y) 写出解。
Verify by substituting back into both original equations.
代回两个原方程进行验证。

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Understanding Simultaneous Equations | 理解联立方程