📚 Solving Systems of Linear Equations: Substitution and Elimination | 线性联立方程组的解法：代入与消元
Systems of linear equations appear throughout mathematics, from simple word problems to advanced engineering models. This lesson focuses on two foundational methods for solving them: substitution and elimination.
线性联立方程组贯穿数学学习的始终，从简单的应用题到复杂的工程模型都会遇到。本课重点讲解两种基础解法：代入法与消元法。
1. What Is a System of Linear Equations? | 什么是线性联立方程组？
A system of linear equations consists of two or more linear equations involving the same variables. The solution is the set of values that satisfies all equations simultaneously.
线性联立方程组由包含相同变量的两个或多个线性方程组成。解是同时满足所有方程的变量取值。
For example:
2x + 3y = 7
x − y = 1
, the solution is x = 2, y = 1 because substituting these values satisfies both equations.
例如，上述方程组中 x = 2, y = 1 就是解，因为代入后两个方程均成立。
2. When to Use Substitution | 何时使用代入法
o isolate. For instance, in the equation x − y = 1, rearranging to x = y + 1 is immediate.
当一个变量的系数为 1 时，代入法最为简便。例如在方程 x − y = 1 中，直接变形为 x = y + 1 即可。
This method is also preferred when one equation is already solved for a variable — for example, y = 3x + 2.
当某一方程已经解出某个变量时，也应优先使用代入法，例如 y = 3x + 2 的形式。
3. Substitution Method Step by Step | 代入法逐步详解
Step 1: Solve one equation for one variable. Step 2: Substitute the resulting expression into the other equation. Step 3: Solve for the remaining variable. Step 4: Back-substitute to find the other variable. Step 5: Check your answer in both original equations.
第一步：在其中一个方程中解出一个变量。 第二步：将所得表达式代入另一个方程。 第三步：求出剩下的变量。 第四步：回代求出另一个变量。 第五步：将答案代回原方程组检验。
Example: Solve 2x + 3y = 7 and x − y = 1
From x − y = 1, we get x = y + 1. Substitute into 2x + 3y = 7:
2(y + 1) + 3y = 7
2y + 2 + 3y = 7
5y = 5
y = 1
Then x = 1 + 1 = 2. The solution is (2, 1).
由 x − y = 1 得 x = y + 1。代入 2x + 3y = 7，解得 y = 1，再得 x = 2。解为 (2, 1)。
4. When to Use Elimination | 何时使用消元法
Elimination (also called the addition method) is ideal when both equations contain the variables in a form that can be added or subtracted to cancel one variable. It works especially well when coefficients of matching variables are already opposites or equal.
消元法（又称加减法）适用于两个方程中的变量系数可以通过相加或相减而抵消的情况。当两方程中同一变量系数互为相反数或相等时，消元法尤为高效。
and 3x − 2y = 4, adding the equations immediately eliminates both x and y to give 6x = 12.
例如，在方程组 3x + 2y = 8 与 3x − 2y = 4 中，将两式相加可直接消去 y，得到 6x = 12。
5. Elimination Method Step by Step | 消元法逐步详解
Step 1: Arrange both equations in the form Ax + By = C. Step 2: If necessary, multiply one (or both) equations so that the coefficients of one variable become opposite numbers. Step 3: Add or subtract the equations to eliminate that variable. Step 4: Solve for the remaining variable. Step 5: Back-substitute to find the other variable.
第一步：将两个方程整理为 Ax + By = C 的形式。 第二步：如有必要，将某个方程（或两个方程）乘以适当的数，使某一变量的系数互为相反数。 第三步：将两式相加或相减，消去该变量。 第四步：解出剩下的变量。 第五步：回代求出另一个变量。
Example: Solve 3x + 2y = 12 and 2x − 2y = 8
Adding the two equations:
(3x + 2x) + (2y − 2y) = 12 + 8
5x = 20
x = 4
Substitute x = 4 into 3x + 2y = 12:
12 + 2y = 12
2y = 0
y = 0
Solution: x = 4, y = 0.
将两式相加得到 5x = 20，解得 x = 4。回代得 y = 0。解为 x = 4, y = 0。
6. Multiplying Equations for Elimination | 乘数策略：使系数匹配
When coefficients do not match, multiply one or both equations by a suitable constant. The goal is to make the coefficients of one variable identical in magnitude but opposite in sign.
当系数不匹配时，需将方程乘以适当的常数。目标是使某一变量的系数绝对值相同但符号相反。
Consider the system:
2x + 3y = 8
x + 2y = 5
Multiply the second equation by −2:
−2x − 4y = −10
Now add to the first equation:
(2x − 2x) + (3y − 4y) = 8 − 10
−y = −2
y = 2
Then from x + 2(2) = 5, we get x = 1.
考虑方程组：将第二个方程乘以 −2，再与第一式相加即可消去 x，解得 y = 2，进而得到 x = 1。
7. Choosing the Best Method | 如何选择最佳方法
Generally, use substitution when a variable has coefficient 1 or when an equation is already solved for a variable. Use elimination when coefficients are already matched or can be easily matched with small multipliers.
一般来说，当变量系数为 1 或方程已解出某变量时，用代入法；当系数已匹配或只需乘以较小的数即可匹配时，用消元法。
= 15, substitution is faster. With 5x + 2y = 10 and 5x − 2y = 6, elimination is immediate.
例如，面对 y = 2x + 3 与 4x + y = 15，代入法更快；面对 5x + 2y = 10 与 5x − 2y = 6，消元法立即可用。
8. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无穷多解
Some systems have no solution. This occurs when the lines are parallel — the equations represent parallel lines with different intercepts. For example:
有些方程组无解。这种情况出现在两直线平行时——方程代表斜率相同但截距不同的平行线。例如：
x + y = 3
x + y = 5
These equations cannot both be true. The system is inconsistent.
这两个方程不可能同时成立。该方程组为矛盾方程组。
Other systems have infinitely many solutions. This happens when both equations represent the same line. For example:
另一些方程组有无数个解。这发生在两个方程表示同一条直线时。例如：
2x + 2y = 6
x + y = 3
These are equivalent equations. Every point on the line is a solution.
这两个方程等价。直线上任意一点都是解。
9. Solving Word Problems with Systems | 用方程组解应用题
Word problems require translating verbal statements into equations. Identify two unknown quantities, define variables, then write two equations based on the given relationships.
应用题需要将文字叙述转化为方程。先找出两个未知量，设定变量，再根据题目给出的关系列出两个方程。
Example: The sum of two numbers is 12. Twice the larger minus the smaller is 15. Find the numbers.
示例：两数之和为 12。大数的两倍减去小数为 15。求这两个数。
Let x be the larger number and y the smaller. Then:
x + y = 12
2x − y = 15
Using elimination, add the equations:
3x = 27
x = 9
Then y = 12 − 9 = 3. The numbers are 9 and 3.
设大数为 x，小数为 y。列出方程组后相加消去 y，解得 x = 9，从而 y = 3。两数为 9 和 3。
10. Checking Your Solution | 验证解的正确性
Always verify your solution by substituting the values into both original equations, not just one. This ensures no arithmetic or algebraic error has occurred.
务必把解代回原方程组中的两个方程进行检验，而不仅是一个方程。这能确保没有算术或代数错误。
For x = 2, y = 1 in the earlier example:
2(2) + 3(1) = 4 + 3 = 7 ✓
2 − 1 = 1 ✓
Both equations hold, confirming the solution.
对于前例中 x = 2, y = 1，两个方程均成立，验证解正确。
11. Common Mistakes to Avoid | 常见易错点
Failing to distribute correctly when substituting — always multiply into every term inside brackets.
代入时忘记正确去括号——需将系数乘到括号内每一项。
Adding instead of subtracting when eliminating variables with matching signs.
消元时系数符号相同却用了加法而非减法。
Forgetting to multiply every term when scaling an equation.
对方程进行倍乘时遗漏某一项。
Substituting solutions back incorrectly — check each step carefully.
回代时出错——务必仔细检查每一步。
12. Summary and Practice Tips | 总结与练习建议
Both substitution and elimination are reliable methods for solving linear systems. Substitution works best when a variable is easy to isolate; elimination works best when coefficients align neatly. Master both, and you will be able to choose the most efficient path for any system.
代入法和消元法都是解线性方程组的可靠方法。变量容易分离时用代入法；系数整齐匹配时用消元法。熟练掌握两种方法后，面对任何方程组你都能选择最有效的路径。
Practice tip: Solve every system twice — once with each method — to build deep confidence and verify your answers.
练习建议：每道题分别用两种方法各解一次，既能加深理解，也能相互验证答案。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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