📚 Student's Edition Grade 1 Lesson 16-4: Solving Simultaneous Linear Equations | 学生版 一年级 第16-4课：解二元一次方程组
In this IGCSE Mathematics lesson, we explore simultaneous linear equations - one of the most practical tools in algebra. You will learn how to find the values of two unknown variables that satisfy two different equations at the same time. This skill is essential for solving real-world problems in economics, physics, and everyday decision-making.
在这节 IGCSE 数学课中，我们将学习二元一次方程组——代数中最实用的工具之一。你将学会如何求出同时满足两个不同方程的两个未知数的值。这项技能对于解决经济学、物理学和日常决策中的实际问题至关重要。
1. What Are Simultaneous Linear Equations? | 什么是二元一次方程组？
A simultaneous linear equation is a pair of linear equations that share the same two variables, usually written as x and y. The solution is the pair of values (x, y) that makes both equations true at the same time.
二元一次方程组是由两个含有相同两个变量（通常写作 x 和 y）的线性方程组成。方程组的解就是使两个方程同时成立的一组数 (x, y)。
= 3 and y = 4 because substituting these values into both equations gives true statements.
例如，考虑方程组 x + y = 7 和 2x - y = 2。其解为 x = 3 且 y = 4，因为将这些值代入两个方程后都能得到正确的等式。
Geometrically, each equation represents a straight line on the coordinate plane. The solution of the system is the point where the two lines intersect. If the lines are not parallel, they will cross at exactly one point, giving one unique solution.
从几何角度看，每个方程都代表坐标平面上的一条直线。方程组的解就是两条直线相交的点。如果两条直线不平行，它们会恰好相交于一点，从而给出唯一解。
2. The Substitution Method | 代入消元法
The substitution method involves rearranging one equation to make one variable the subject, and then substituting that expression into the other equation. This reduces the system to a single equation in one unknown.
代入法是指先将其中一个方程变形，使某个变量成为主项，然后将该表达式代入另一个方程。这样可以把方程组化简为只含一个未知数的单个方程。
Use this method when one equation already has a variable on its own or when a variable has a coefficient of 1 or -1. This makes substitution straightforward and reduces the chance of arithmetic errors.
当一个方程中已经有一个变量单独在等号一边，或者某个变量的系数为 1 或 -1 时，使用代入法会比较方便。这样代入过程更直接，也能减少计算错误。
Worked example: plify to 7x + 2 = 16, so 7x = 14 and x = 2. Then y = 2(2) + 1 = 5. The solution is (2, 5).
例题：解方程组 y = 2x + 1 和 3x + 2y = 16。将 y = 2x + 1 代入第二个方程，得到 3x + 2(2x + 1) = 16。化简得 7x + 2 = 16，所以 7x = 14，x = 2。然后 y = 2(2) + 1 = 5。方程组的解为 (2, 5)。
3. The Elimination Method | 加减消元法
This is often the fastest method when both equations are in the standard form ax + by = c.
加减消元法通过将两个方程相加或相减来消去其中一个变量。当两个方程都写成标准形式 ax + by = c 时，这种方法通常是最快的。
Before adding or subtracting, you may need to multiply one or both equations by a constant so that the coefficients of one variable are the same or opposite. Then adding or subtracting cancels that variable.
在相加或相减之前，可能需要将其中一个或两个方程乘以一个常数，使某个变量的系数相等或互为相反数。然后通过相加或相减就可以消去该变量。
Worked example: the first equation: 3(2) + 2y = 12, so 6 + 2y = 12, 2y = 6, y = 3. The solution is (2, 3).
例题：解方程组 3x + 2y = 12 和 4x - 2y = 2。注意到 y 的系数分别是 2 和 -2，互为相反数。将两个方程相加：(3x + 2y) + (4x - 2y) = 12 + 2，得到 7x = 14，所以 x = 2。将 x = 2 代入第一个方程：3(2) + 2y = 12，6 + 2y = 12，2y = 6，y = 3。方程组的解为 (2, 3)。
4. Graphical Method | 图像法
The graphical method involves drawing the two straight lines on the same coordinate grid and reading off the coordinates of their intersection point. This method gives a visual understanding of why simultaneous equations have a unique solution.
图像法是在同一个坐标网格上画出两条直线，然后读出它们交点的坐标。这种方法能让我们直观地理解为什么二元一次方程组会有唯一解。
e through them. For example, to graph x + y = 5, you can use the points (0, 5) and (5, 0).
要画出一个线性方程的图像，需要找到满足方程的两个点，然后通过它们画出直线。例如，要画出 x + y = 5 的图像，可以使用点 (0, 5) 和 (5, 0)。
The main limitation of the graphical method is accuracy. If the solution is not a whole number or the lines are nearly parallel, reading the intersection point from a hand-drawn graph can be difficult. Algebraic methods are more reliable for exact answers.
图像法的主要局限在于精度。如果解不是整数，或者两条直线几乎平行，从手绘图上读取交点会比较困难。对于精确答案，代数方法更为可靠。
5. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无穷多解
Not every pair of linear equations has exactly one solution. If the two lines are parallel, they never intersect, so the system has no solution. In algebra, this happens when the coefficients of x and y are proportional, but the constant terms are not.
并不是每一对线性方程都恰好有一个解。如果两条直线平行，它们永远不会相交，因此方程组无解。在代数中，当 x 和 y 的系数成比例，但常数项不成比例时，就会出现这种情况。
second equation by 2 gives 2x + 3y = 5, which contradicts the first equation 2x + 3y = 6.
例如，方程组 2x + 3y = 6 和 4x + 6y = 10 无解，因为第二个方程除了常数项外是第一个方程的两倍。将第二个方程除以 2 得到 2x + 3y = 5，这与第一个方程 2x + 3y = 6 矛盾。
initely many solutions. For example, 2x + 3y = 6 and 4x + 6y = 12 represent the same line.
如果两个方程互为倍数关系，它们表示同一条直线。在这种情况下，直线上的每一个点都是方程组的解，因此方程组有无穷多解。例如，2x + 3y = 6 和 4x + 6y = 12 表示同一条直线。
6. Forming Simultaneous Equations from Word Problems | 从应用题建立方程组
Many IGCSE exam questions require you to translate a real-world situation into a pair of simultaneous equations. Look for two unknown quantities and two separate pieces of information that connect them.
许多 IGCSE 考试题目要求你将实际问题转化为二元一次方程组。你需要找出两个未知量，以及连接这两个未知量的两条独立信息。
Example: A cinema sells adult tickets for £6 and child tickets for £4. On one day, 80 tickets were sold, and the total revenue was £420. Let a be the number of adult tickets and c be the number of child tickets. Form two equations.
例题：一家电影院成人票每张 £6，儿童票每张 £4。某天共售出 80 张票，总收入为 £420。设 a 表示成人票数量，c 表示儿童票数量。建立两个方程。
s 6a + 4c = 420. These two equations can then be solved using substitution or elimination.
售票总数给出方程 a + c = 80。总收入给出方程 6a + 4c = 420。然后就可以使用代入法或加减消元法来解这个方程组。
7. Choosing the Best Method | 选择最佳解法
There is no single 'best' method for every system, but you can use a quick checklist. If one variable is already isolated or has a coefficient of 1, substitution is usually easiest. If both equations are in standard form and no variable has coefficient 1, elimination is often faster.
对于每个方程组来说，并不存在唯一的“最佳”方法，但你可以使用一个快速检查清单。如果某个变量已经被分离出来，或者某个变量的系数为 1，通常代入法最简单。如果两个方程都是标准形式，且没有变量的系数为 1，那么加减消元法通常更快。
If the equation involves fractions or decimals, it is often helpful to multiply the entire equation by a common denominator or by a power of 10 to clear them before applying any method.
如果方程中含有分数或小数，通常可以先将整个方程乘以公分母或 10 的幂，以清除分数或小数，然后再使用任何方法求解。
Always check your solution by substituting both values back into the original equations. This final step catches most arithmetic mistakes and is quick to do in an exam.
一定要将求得的两个值代回原方程组进行检验。这个最后的步骤可以快速发现大多数计算错误，在考试中非常值得做。
8. Worked Exam-Style Question | 考试题型精讲
Question: Solve the simultaneous equations 5x - 2y = 11 and 3x + 4y = 17.
题目：解方程组 5x - 2y = 11 和 3x + 4y = 17。
Solution: 3x + 4y = 17. The y terms cancel: 10x + 3x - 4y + 4y = 22 + 17, so 13x = 39, giving x = 3.
解答：我们可以将第一个方程乘以 2，使 y 的系数互为相反数。得到 10x - 4y = 22。现在把它与第二个方程 3x + 4y = 17 相加。y 项消去：10x + 3x - 4y + 4y = 22 + 17，所以 13x = 39，得到 x = 3。
ond equation: 3(3) + 4y = 17, so 9 + 4y = 17, 4y = 8, y = 2. The solution is x = 3, y = 2.
将 x = 3 代入原第二个方程：3(3) + 4y = 17，即 9 + 4y = 17，4y = 8，y = 2。方程组的解为 x = 3，y = 2。
equation: 5(3) - 2(2) = 15 - 4 = 11, which is correct. This confirms the solution (3, 2).
代入第一个方程检验：5(3) - 2(2) = 15 - 4 = 11，结果正确。这验证了解 (3, 2)。
9. Common Mistakes to Avoid | 常见错误与避免方法
A common error is forgetting to multiply every term in an equation when using elimination. If you multiply one equation by a constant, make sure the entire equation is multiplied, including the constant term on the right-hand side.
一个常见错误是在使用加减消元法时忘记乘以方程中的每一项。如果将某个方程乘以一个常数，必须确保整个方程都被乘，包括等号右边的常数项。
Another frequent mistake is substituting a value into the wrong equation. Always substitute back into one of the original equations, not a manipulated version, to reduce the risk of carrying forward an earlier error.
另一个常见错误是将数值代入错误的方程。一定要代入原方程之一，而不是变形后的方程，这样可以降低延续之前错误的风险。
Students sometimes stop after finding one variable and forget to find the second variable. Remember that a complete solution for a system of two equations must give values for both variables, and these values must be checked.
学生们有时在求出一个变量后就停下来，忘记求另一个变量。请记住，二元一次方程组的完整解必须给出两个变量的值，并且要对这些值进行检验。
10. Key Points Summary | 重点总结
Simultaneous linear equations model situations where two conditions must be satisfied at the same time. The three main solution methods are substitution, elimination, and graphing, and each has its own strengths.
二元一次方程组用于描述两个条件必须同时满足的情境。三种主要解法分别是代入法、加减消元法和图像法，每种方法各有优势。
Substitution is best when one variable is easily isolated; elimination is best when coefficients can be made to cancel quickly; graphing is useful for visual understanding but less precise for exact answers.
当某个变量容易分离出来时，代入法最好；当系数可以快速消去时，加减消元法最好；图像法有助于直观理解，但在求精确答案时精度较低。
Always remember to check your final solution by substituting both values into both original equations. This habit not only catches mistakes but also builds confidence in your answer.
一定要记住，将两个变量的值代入两个原方程进行检验。这个习惯不仅能发现错误，还能增强你对答案的信心。
Solution of ax + by = c and dx + ey = f → (x, y) = (x₀, y₀)
方程组 ax + by = c 和 dx + ey = f 的解 → (x, y) = (x₀, y₀)
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