Solving Simultaneous Linear Equations | 解线性联立方程组

📚 Solving Simultaneous Linear Equations | 解线性联立方程组

Simultaneous linear equations involve two or more equations that share two or more unknown variables. In the IGCSE Mathematics syllabus, you are typically asked to solve a pair of linear equations in two unknowns, \(x\) and \(y\), using algebraic or graphical methods. Mastering this topic is essential for higher-level mathematics, as it builds a foundation for solving systems of equations in algebra, coordinate geometry, and real-life problems.

解线性联立方程组是指求解多个含有两个或以上未知数的方程。在 IGCSE 数学课程中，通常要求你用代数方法或作图方法来解含有两个未知数 \(x\) 和 \(y\) 的一对线性方程。掌握这个专题对后续数学学习非常重要，因为它是代数、坐标几何以及实际问题中解方程组的基础。

1. Understanding Simultaneous Equations | 理解联立方程组

One linear equation with two unknowns, such as \(x + y = 5\), has infinitely many solutions. To find a unique solution we need another independent equation, e.g. \(x – y = 1\). Together they form a system of simultaneous equations. The solution is the ordered pair \((x, y)\) that satisfies both equations at the same time.

含有两个未知数的一个线性方程，如 \(x + y = 5\)，会有无穷多组解。为了找到唯一的解，我们需要另一个独立的方程，比如 \(x – y = 1\)。它们合在一起就构成了一个联立方程组。方程组的解是一个有序数对 \((x, y)\)，它同时满足两个方程。

The solution can be interpreted geometrically as the point of intersection of the two straight lines representing each equation. If the lines intersect at one point, there is exactly one solution; if they are parallel, there is no solution; if they coincide, there are infinitely many solutions.

从几何角度看，解就是两个方程所表示的直线的交点。如果两条直线交于一点，则有一个解；如果两线平行，则无解；如果两线重合，则有无穷多组解。

2. The Elimination Method | 消元法

The elimination method involves adding or subtracting the equations to eliminate one variable. The steps are:

Align the equations with like terms in columns.
Make the coefficients of one variable (say \(x\)) equal in magnitude.
Add or subtract the equations to eliminate that variable.
Solve the resulting single-variable equation.
Substitute back to find the other variable.

将方程按同类项对齐。
使某个未知数（如 \(x\)）的系数绝对值相等。
两式相加或相减，消去该未知数。
解出剩下的一元方程。
代回求得另一个未知数。

Example: Solve \(2x + 3y = 11\) and \(x – 2y = 3\). First, multiply the second equation by 2 to get \(2x – 4y = 6\). Now subtract the second from the first: \((2x+3y) – (2x-4y) = 11 – 6\), so \(7y = 5\), hence \(y = 5/7\). Substitute \(y = 5/7\) into \(x – 2y = 3\): \(x – 10/7 = 3\), so \(x = 31/7\). Solution: \((31/7, 5/7)\).

例：解方程组 \(2x + 3y = 11\) 和 \(x – 2y = 3\)。首先将第二个方程两边乘以 2，得到 \(2x – 4y = 6\)。然后用第一个方程减去第二个方程：\((2x+3y) – (2x-4y) = 11 – 6\)，得 \(7y = 5\)，因此 \(y = 5/7\)。将 \(y = 5/7\) 代入 \(x – 2y = 3\)：\(x – 10/7 = 3\)，得 \(x = 31/7\)。所以解为 \((31/7, 5/7)\)。

3. The Substitution Method | 代入法

In the substitution method, one equation is rearranged to express one variable in terms of the other. This expression is then substituted into the other equation, yielding a single equation in one unknown.

代入法的思路是：先将一个方程变形成用另一个变量表示该变量，然后把这个表达式代入另一个方程，从而得到一个仅含一个未知数的方程。

Example: Solve \(y = 2x – 1\) and \(3x + y = 10\). Since \(y\) is already isolated, substitute \(2x – 1\) for \(y\) in the second equation: \(3x + (2x – 1) = 10\), which simplifies to \(5x – 1 = 10\), so \(5x = 11\) and \(x = 11/5\). Then \(y = 2(11/5) – 1 = 22/5 – 5/5 = 17/5\). Solution: \((11/5, 17/5)\).

例：解方程组 \(y = 2x – 1\) 和 \(3x + y = 10\)。由于 \(y\) 已经用 \(x\) 表示，把 \(2x – 1\) 代入第二个方程中的 \(y\)：\(3x + (2x – 1) = 10\)，化简得 \(5x – 1 = 10\)，即 \(5x = 11\)，所以 \(x = 11/5\)。于是 \(y = 2(11/5) – 1 = 22/5 – 5/5 = 17/5\)。解为 \((11/5, 17/5)\)。

Choose the method that seems easier for each problem. If one variable has coefficient 1, substitution is usually quicker; if coefficients are ready to match, elimination may be more direct.

每道题可以选择更简便的方法。如果某个未知数的系数是 1，代入法通常较快；如果系数容易配成相等，则消元法更直接。

4. The Graphical Method | 图像法

To solve equations graphically, draw both straight lines on the same set of axes. The coordinates of the point of intersection give the solution. This method is useful for estimation, but it may not give exact values unless the solution has integer or simple fraction coordinates.

用图像法解方程组时，需要在同一坐标平面内画出两条直线，交点的坐标就是方程组的解。这种方法适合估计解，但如果解不是整数或简单分数，可能无法得到精确值。

Steps:

Rearrange each equation in the form \(y = mx + c\) or find two points from each equation.
Plot the lines accurately on graph paper.
Read off the coordinates of the intersection point.
Check the values by substituting them into the original equations.

将每个方程化成 \(y = mx + c\) 的形式，或从每个方程中找出两个点。
在坐标纸上准确作图。
读出交点的坐标。
将解代回原方程进行检验。

For example, \(y = x + 1\) and \(y = -x + 3\) intersect at (1, 2), so the solution is \(x = 1, y = 2\).

例如，直线 \(y = x + 1\) 和 \(y = -x + 3\) 相交于点 (1, 2)，因此解为 \(x = 1, y = 2\)。

5. Word Problems | 应用题

Many real-world situations can be modelled with two linear equations. The key is to define variables clearly, translate the given sentences into equations, and then solve the system.

许多现实问题可以用两个线性方程来建模。关键是要明确定义变量，把题目的文字转化成方程，然后求解方程组。

Example: The sum of two numbers is 12, and their difference is 4. Let the numbers be \(a\) and \(b\). Then \(a + b = 12\) and \(a – b = 4\). Adding the two equations gives \(2a = 16\), so \(a = 8\). Then \(b = 12 – 8 = 4\). The numbers are 8 and 4.

例：两个数的和是 12，差是 4。设这两个数为 \(a\) 和 \(b\)，则 \(a + b = 12\)，\(a – b = 4\)。两式相加得 \(2a = 16\)，所以 \(a = 8\)。于是 \(b = 12 – 8 = 4\)。这两个数是 8 和 4。

Another common type involves pricing: three apples and two oranges cost 6 dollars, while one apple and three oranges cost 5 dollars. Let \(p\) be the price of one apple and \(q\) the price of one orange. Equations are \(3p + 2q = 6\) and \(p + 3q = 5\). Solve by elimination or substitution to find \(p = 1\) and \(q = 1.5\).

另一类常见问题是价格问题：三个苹果和两个橙子共 6 元，一个苹果和三个橙子共 5 元。设一个苹果的价格为 \(p\)，一个橙子的价格为 \(q\)，则方程为 \(3p + 2q = 6\) 和 \(p + 3q = 5\)。用消元法或代入法可解得 \(p = 1\)，\(q = 1.5\)。

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

If the two equations represent parallel lines, they never meet, so the system has no solution. Algebraically, this happens when the coefficients of \(x\) and \(y\) are proportional but the constant terms are not, e.g. \(x + y = 2\) and \(2x + 2y = 5\).

如果两个方程表示两条平行直线，它们永不相交，所以方程组无解。在代数上，当 \(x\) 和 \(y\) 的系数成比例，而常数项不成比例时，就会出现这种情况，例如 \(x + y = 2\) 和 \(2x + 2y = 5\)。

If the equations are actually the same line (all coefficients and constants are proportional), they intersect everywhere, giving infinitely many solutions. For example, \(x + y = 2\) and \(2x + 2y = 4\) represent the same line.

如果两个方程实际上表示同一条直线（所有系数和常数项成同一比例），那么它们处处相交，方程组有无穷多组解。例如，\(x + y = 2\) 和 \(2x + 2y = 4\) 表示同一条直线。

When using elimination, a false statement such as \(0 = 3\) indicates no solution, while a true statement such as \(0 = 0\) indicates infinitely many solutions.

使用消元法时，如果出现错误的等式如 \(0 = 3\)，则说明无解；如果出现恒等式如 \(0 = 0\)，则说明有无穷多组解。

7. Solving Systems with Fractions and Decimals | 含分数与小数的方程组

Equations may contain fractions or decimals. It is often easier to clear these first by multiplying every term by a suitable number. For example, if an equation contains \(\frac{1}{2}x + \frac{1}{3}y = 2\), multiply by 6 to obtain \(3x + 2y = 12\).

方程组中可能会含有分数或小数。通常先乘以适当的数来化简比较方便。例如，方程 \(\frac{1}{2}x + \frac{1}{3}y = 2\)，两边同乘 6 可得 \(3x + 2y = 12\)。

For decimals such as \(0.2x – 0.1y = 0.3\), multiply by 10 to get \(2x – y = 3\). This makes the arithmetic simpler and less error-prone.

对于小数方程，如 \(0.2x – 0.1y = 0.3\)，两边同乘 10，得到 \(2x – y = 3\)，这样计算更简单，也更不容易出错。

Always check your final solution by substituting it into the original equations, especially when you have multiplied both sides of an equation during simplification.

解出结果后，一定要将解代回原方程进行检验，特别是当你在化简过程中将方程两边都乘了某个数时。

8. Common Mistakes to Avoid | 常见错误及避免方法

Students often make the following mistakes:

Forgetting to multiply the constant term when multiplying an equation.
Sign errors when subtracting equations: always distribute the minus sign carefully.
Substituting the found value into a rearranged equation incorrectly.
Not checking the solution in both original equations.
Confusing the coordinates when reading a graph.

在对方程两边乘以某个数时，忘记乘常数项。
相减时出现符号错误：要小心分配负号。
把求得的数值代入变形式时出错。
没有把解代回两个原方程进行检验。
从图上读取坐标时混淆横纵坐标。

To avoid these, write clearly, use brackets when subtracting expressions, and always substitute your values back into the original equations. For graphical solutions, use a sharp pencil and read the coordinates carefully.

为了避免这些错误，书写要工整，相减时给整个表达式加括号，并且始终将所求数值代回原方程验证。对于图像法，请使用削尖的铅笔，并且仔细读取坐标。

9. Practice Questions | 练习题

Try solving these systems using any method:

\(x + y = 7\), \(x – y = 3\)
\(2x + y = 10\), \(x – y = 2\)
\(3x – 2y = 4\), \(2x + y = 5\)
\(y = 3x – 1\), \(2y + 3x = 12\)
Find two numbers whose sum is 20 and product is 96. (Hint: set up one linear equation for the sum and one quadratic? No, this is a linear topic, so use \(a+b=20\) and \(a-b=4\) etc.)

解方程组：\(x + y = 7\)，\(x – y = 3\)
解方程组：\(2x + y = 10\)，\(x – y = 2\)
解方程组：\(3x – 2y = 4\)，\(2x + y = 5\)
解方程组：\(y = 3x – 1\)，\(2y + 3x = 12\)
求两个数，使得它们的和为 20，且它们的差为 4。（提示：设两个数为 \(a\) 和 \(b\)，列方程组。）

Answers:

1. \(x = 5, y = 2\)

2. \(x = 4, y = 2\)

3. \(x = 2, y = 1\)

4. \(x = 2, y = 5\)

5. \(a = 12, b = 8\)

答案：

1. \(x = 5, y = 2\)

2. \(x = 4, y = 2\)

3. \(x = 2, y = 1\)

4. \(x = 2, y = 5\)

5. \(a = 12, b = 8\)

10. Summary | 总结

Solving simultaneous linear equations is a fundamental skill in IGCSE Mathematics. The three main methods are elimination, substitution, and graphical. Each has its advantages, and you should be comfortable with all three. For algebraic methods, always check your final answer. Special cases, including no solution and infinitely many solutions, must be recognised and interpreted correctly.

解线性联立方程组是 IGCSE 数学的核心技能。三种主要方法是消元法、代入法和图像法。每种方法各有优势，你应当熟练掌握这三种方法。使用代数方法时，始终要检验最终答案。对于无解和无穷多组解等特殊情况，必须正确识别并理解其含义。

Remember the golden rule: the solution must satisfy every equation in the system. Practice regularly with a variety of problems, including word problems, and you will build strong problem-solving skills for both exams and future mathematics.

请记住黄金法则：解必须满足方程组中的每一个方程。定期练习各种类型的题目，包括应用题，你就能培养强大的解题能力，为考试以及今后的数学学习打下坚实基础。

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Solving Simultaneous Linear Equations | 解线性联立方程组