📚 Simultaneous Linear Equations | 联立一次方程
In IGCSE Mathematics, simultaneous linear equations appear in both the core and extended tiers, usually in algebra and problem-solving questions. A pair of simultaneous equations simply means two equations that must both be true for the same values of the unknowns. This topic builds directly on your skills in rearranging formulae, solving linear equations, and interpreting straight-line graphs.
在 IGCSE 数学中，联立一次方程在核心和拓展层级都会出现，通常出现在代数和应用题中。一对联立方程指的是两个方程必须在未知数取相同值时同时成立。这一主题直接建立在你对公式变形、解一次方程以及解释直线图像等技能的基础上。
1. What Are Simultaneous Linear Equations? | 什么是联立一次方程？
constants and both a and b are not zero at the same time. When two such equations are given together, we call them simultaneous linear equations. A valid solution must satisfy both equations at once.
含有两个未知数（通常为 x 和 y）的一次方程可以写成一般形式 ax + by = c，其中 a、b 和 c 是常数，且 a 和 b 不能同时为零。当两个这样的一次方程被同时给出时，我们称它们为联立一次方程。一个有效解必须同时满足两个方程。
The standard pair looks like this:
标准形式如下：
ax + by = c
dx + ey = f
ed pair, such as (x, y) = (3, −1). This notation tells us that x equals 3 and y equals −1.
这里 a、b、c、d、e 和 f 都是数字。解通常写成有序数对，例如 (x, y) = (3, −1)。这种记号表示 x 等于 3，y 等于 −1。
2. The Graphical Interpretation | 图像解释
Every linear equation in two unknowns represents a straight line on the coordinate plane. Therefore, a pair of simultaneous linear equations represents two straight lines. The solution of the system is exactly the point where the two lines intersect.
每一个含有两个未知数的一次方程都代表坐标平面上的一条直线。因此，一对联立一次方程代表两条直线。方程组的解恰好就是两条直线相交的点。
There are three possible graphical cases. If the lines intersect at one point, the system has exactly one solution. If the lines are parallel, they never meet, so there is no solution. If the two equations describe the same line, the lines overlap completely, and there are infinitely many solutions.
图像上共有三种可能情况。如果两条直线相交于一点，则方程组有唯一解。如果两条直线平行，它们不会相交，因此无解。如果两个方程描述的是同一条直线，两条直线完全重合，则有无穷多个解。
This graphical idea is helpful because it explains why some pairs of equations behave differently. In the exam, you may be asked to find the solution algebraically, but knowing the graphical meaning helps you check whether an answer is sensible.
这一图像概念很有帮助，因为它解释了为什么有些方程对表现不同。在考试中，你可能需要用代数方法求根，但了解图像意义有助于你判断答案是否合理。
3. Solving by Elimination | 消元法
The elimination method is often the fastest way to solve simultaneous linear equations. The key idea is to make the coefficient of one unknown the same size in both equations, then add or subtract the equations so that unknown is eliminated.
消元法通常是解联立一次方程最快的方法。关键思路是使某个未知数的系数在两个方程中大小相同，然后将方程相加或相减，使该未知数被消去。
Consider the pair:
考虑以下方程组：
2x + y = 7
x − y = 2
stituting x = 3 into x − y = 2 gives 3 − y = 2, so y = 1. The solution is (x, y) = (3, 1).
y 的系数分别是 +1 和 −1，因此将两个方程相加可以立即消去 y。我们得到 3x = 9，因此 x = 3。将 x = 3 代入 x − y = 2，得到 3 − y = 2，所以 y = 1。解为 (x, y) = (3, 1)。
et 4x + 2y = 14. Now the y terms are both 2y, so subtracting gives −x = −2, meaning x = 2.
当两个未知数的系数都不匹配时，你需要先对一个或两个方程乘以适当的数。例如，在方程组 3x + 2y = 12 和 2x + y = 7 中，将第二个方程乘以 2，得到 4x + 2y = 14。此时两个方程中的 y 项都是 2y，相减得到 −x = −2，即 x = 2。
- Align the two equations so like terms are above each other.
- Multiply equations if needed to make one pair of coefficients equal or opposite.
- Add or subtract to eliminate one unknown.
- Solve for the remaining unknown, then substitute back to find the other.
将两个方程上下对齐，使同类项对应；如有必要，对方程乘以倍数，使某一对系数相等或互为相反数；相加或相减以消去一个未知数；求出剩余未知数后，代回原方程求另一个未知数。
4. Solving by Substitution | 代入法
The substitution method is useful when one equation already has one unknown isolated, or can be rearranged easily. You replace that unknown in the other equation with an expression, leaving a single equation in one unknown.
当一个方程中已经有一个未知数被单独表示，或者很容易变形出这种形式时，代入法非常有用。你用该表达式替换另一个方程中的这个未知数，从而得到一个只含一个未知数的方程。
uation: 3x + 2x + 1 = 16. This simplifies to 5x + 1 = 16, so x = 3. Then y = 2(3) + 1 = 7.
例如，取 y = 2x + 1 和 3x + y = 16。由于 y 已经用 x 表示，将第二个方程中的 y 替换为 2x + 1：3x + 2x + 1 = 16。化简得 5x + 1 = 16，所以 x = 3。然后 y = 2(3) + 1 = 7。
This method is especially helpful when coefficients are fractions or decimals, because substitution often avoids the need to multiply entire equations by a large common multiple.
当系数为分数或小数时，代入法尤其有用，因为代入通常可以避免将整个方程乘以一个较大的公倍数。
In general, follow these steps: rearrange one equation to make x or y the subject, substitute this expression into the other equation, solve the resulting single equation, and finally substitute back to find the remaining unknown.
一般来说，遵循以下步骤：将其中一个方程变形，使 x 或 y 成为主体；将该表达式代入另一个方程；解所得的一元方程；最后将结果代回求另一个未知数。
5. Checking Your Answers | 检验答案
Many IGCSE questions explicitly ask you to check your solution, and even when they do not, checking is an excellent habit. You should substitute your values for x and y into both original equations separately.
许多 IGCSE 题目明确要求你检验答案，即使没有要求，检验也是一个很好的习惯。你应当把 x 和 y 的值分别代入两个原方程。
3x + y = 16 gives 3(3) + 7 = 16, which is also true. Therefore the solution is confirmed.
对于上一节得到的解 x = 3 和 y = 7，第一个原方程 y = 2x + 1 给出 7 = 2(3) + 1，成立。第二个原方程 3x + y = 16 给出 3(3) + 7 = 16，也成立。因此解得到确认。
If either substitution does not work, you have made an arithmetic or rearrangement error. It is safer to go back through the steps than to ignore the mismatch.
如果有任何一个代入不成立，就说明你在计算或变形中出现了错误。回到步骤中检查比忽视这种不一致要安全得多。
6. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无穷多解
Not every pair of simultaneous equations has one unique solution. Some pairs are inconsistent, meaning they have no solution. This happens when the two lines are parallel but not identical.
并不是每一对联立方程都有唯一解。有些方程对是不相容的，意味着它们没有解。当两条直线平行但不重合时，就会出现这种情况。
ible. The equations contradict each other, so there is no pair (x, y) that satisfies both.
考虑 x + y = 2 和 x + y = 5。如果我们将第一个方程从第二个方程中减去，得到 0 = 3，这是不可能的。两个方程相互矛盾，因此不存在同时满足两者的数对 (x, y)。
represent the same line. Every point on that line is a solution, so there are infinitely many solutions.
另一些方程对是相关的，即一个方程只是另一个方程的倍数。例如，x + y = 2 和 2x + 2y = 4 表示同一条直线。这条直线上的每一个点都是解，因此有无穷多个解。
In an IGCSE answer, you should state clearly whether the system has 'one solution', 'no solution' or 'infinitely many solutions' when the question asks you to classify it.
在 IGCSE 答案中，当题目要求你判断类型时，你应当清楚说明方程组是“有唯一解”“无解”还是“有无穷多个解”。
7. Word Problems and Modelling | 应用题与建模
Simultaneous equations are often hidden in real-life word problems. You need to define two variables clearly, translate the written conditions into two linear equations, and then solve the system using either elimination or substitution.
联立方程常常隐藏在实际生活的应用题中。你需要清楚地定义两个变量，将文字条件转化为两个一次方程，然后用消元法或代入法求解方程组。
For example, suppose 3 apples and 2 bananas cost $4.50, while 2 apples and 3 bananas cost $4.00. Let a be the cost of one apple and b be the cost of one banana. The equations are:
例如，假设 3 个苹果和 2 根香蕉共花费 4.50 美元，而 2 个苹果和 3 根香蕉共花费 4.00 美元。设 a 为一个苹果的价格，b 为一根香蕉的价格。方程如下：
3a + 2b = 4.50
2a + 3b = 4.00
ubtracting gives 5a = 5.50, so a = 1.10. Substituting back gives b = 0.60. One apple costs $1.10 and one banana costs $0.60.
将第一个方程乘以 3，第二个方程乘以 2，得到 9a + 6b = 13.50 和 4a + 6b = 8.00。相减得 5a = 5.50，所以 a = 1.10。代回得 b = 0.60。一个苹果 1.10 美元，一根香蕉 0.60 美元。
Always state what each variable represents before writing the equations, and remember to give your final answer in the context of the problem, including units where appropriate.
在写方程之前，一定要说明每个变量代表什么，并且记住在问题情境中给出最终答案，包括适当的单位。
8. Common Mistakes to Avoid | 常见错误
limination. If you multiply 2x + y = 7 by 3, you must write 6x + 3y = 21, not 6x + y = 21.
一个常见错误是在使用消元法时忘记乘以方程中的每一项。如果将 2x + y = 7 乘以 3，你必须写成 6x + 3y = 21，而不是 6x + y = 21。
Another frequent error is losing a negative sign during subtraction. When subtracting one entire equation from another, be careful to subtract all terms, including constants. Writing brackets around the equation being subtracted can help.
另一个常见错误是在相减时丢失负号。将整个方程相减时，要小心减去所有项，包括常数项。在被减的方程外加括号会有所帮助。
Some students solve for one unknown but forget to substitute it back to find the other. A complete solution must give values for both x and y. Finally, always check your solution in both original equations, because an answer that works in one equation may fail in the other.
有些学生求出一个未知数后忘记代回求另一个未知数。完整解必须同时给出 x 和 y 的值。最后，一定要在两个原方程中检验答案，因为只在一个方程中成立的答案可能在另一个方程中不成立。
9. Exam-Style Practice Question | 考试风格练习
Try this IGCSE-style question before looking at the guidance below:
在查看下面的提示之前，先尝试这道 IGCSE 风格的题目：
4x + 3y = 24
2x − y = 2
, so x = 3. Substitute x = 3 into 2x − y = 2: 6 − y = 2, so y = 4. The solution is (3, 4).
一种高效的方法是消元法。将第二个方程乘以 3，使 y 的系数互为相反数：6x − 3y = 6。将其与第一个方程相加以消去 y：10x = 30，所以 x = 3。将 x = 3 代入 2x − y = 2：6 − y = 2，所以 y = 4。解为 (3, 4)。
− 2) = 24, leading to the same result. Being flexible with methods is useful in the exam.
注意这道题也可以用代入法求解。将第二个方程改写为 y = 2x − 2，代入第一个方程得到 4x + 3(2x − 2) = 24，最终得到相同结果。在考试中灵活选择方法很有帮助。
10. Summary and Key Formulae | 总结与关键公式
To solve a pair of simultaneous linear equations, you need to find the unique pair of values that satisfies both equations. The two main algebraic methods are elimination and substitution, and both rely on simplifying the problem to one equation with one unknown.
要解一对联立一次方程，你需要找到同时满足两个方程的唯一数值对。两种主要代数方法是消元法和代入法，二者都依赖将问题化简为一个只含一个未知数的方程。
. The slope is −a/b and the y-intercept is c/b. Comparing slopes helps you determine whether two lines intersect, are parallel, or coincide.
对于一次方程 ax + by = c，只要 b ≠ 0，就可以将其变形为斜截式 y = −(a/b)x + c/b。斜率为 −a/b，y 轴截距为 c/b。比较斜率有助于判断两条直线是相交、平行还是重合。
slope = −a/b, y-intercept = c/b
Remember the three possible outcomes: one solution when lines intersect, no solution when lines are parallel, and infinitely many solutions when lines are identical. In every case, checking your solution by substitution remains one of the most reliable ways to avoid losing marks.
记住三种可能结果：两线相交时有一个解；两线平行时无解；两线重合时有无穷多个解。无论如何，代入检验仍然是避免失分最可靠的方法之一。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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