Use Elimination Method to Solve Linear Equations (No Solution Case) | Class 10 Maths
The Elimination Method is one of the easiest techniques to solve a pair of linear equations in two variables. In this method, we eliminate one variable by making its coefficients equal and then subtracting or adding the equations.
In this example, we will observe a special case where the equations have no solution. Such questions are very important for CBSE Class 10 Board Exams because they help students identify whether a pair of linear equations is consistent or inconsistent.
Question
Use the Elimination Method to find all possible solutions of the following pair of linear equations:
2x + 3y = 8
4x + 6y = 7
Step 1: Write the Equations
Equation (1):
2x + 3y = 8
Equation (2):
4x + 6y = 7
Step 2: Make the Coefficients Equal
Multiply Equation (1) by 2.
2(2x + 3y) = 2(8)
We get:
4x + 6y = 16
Now compare:
Equation (3): 4x + 6y = 16
Equation (2): 4x + 6y = 7
Step 3: Eliminate the Variables
Subtract Equation (2) from Equation (3):
(4x + 6y) − (4x + 6y) = 16 − 7
0 = 9
This statement is impossible because 0 can never be equal to 9.
Final Answer
Since we obtain an impossible statement,
- The pair of linear equations has No Solution.
- The equations are Inconsistent.
- The two lines are Parallel to each other.
Why Does This Happen?
When both equations have the same coefficients of x and y but different constants, they represent two different parallel lines. Parallel lines never intersect, so there is no common solution.
Conditions for Pair of Linear Equations
| Condition | Nature of Lines | Number of Solutions |
|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | Intersecting Lines | One Unique Solution |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Coincident Lines | Infinitely Many Solutions |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Parallel Lines | No Solution |
Quick Exam Trick
- If eliminating variables gives 0 = 0, the equations have infinitely many solutions.
- If eliminating variables gives 0 = any non-zero number (like 0 = 9), the equations have no solution.
- If eliminating variables gives the value of a variable, continue solving to get the unique solution.
Common Mistakes Students Make
- Forgetting to multiply every term of the equation.
- Making sign errors while subtracting equations.
- Thinking 0 = 9 means x = 9.
- Not identifying the equations as inconsistent.
- Confusing parallel lines with coincident lines.
Important Formula to Remember
For two linear equations:
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
The equations have no solution if:
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Board Exam Tips
- Always write equation numbers.
- Show the multiplication step clearly.
- Write every subtraction carefully.
- After getting 0 = non-zero, immediately conclude "No Solution".
- Mention that the pair is Inconsistent and the lines are Parallel to score full marks.
Practice Questions
- Solve using elimination method: x + y = 4 and 2x + 2y = 8.
- Find the solution of 3x + 2y = 7 and 6x + 4y = 15.
- Determine whether the equations 5x − y = 8 and 10x − 2y = 16 have one solution, no solution, or infinitely many solutions.
- Solve: 2x + y = 5 and 4x + 2y = 12.
Conclusion
The elimination method is a fast and reliable technique for solving pairs of linear equations. Whenever eliminating the variables results in an impossible statement like 0 = 9, it clearly indicates that the equations represent parallel lines and therefore have no solution. Recognizing these special cases quickly can save valuable time in the CBSE Class 10 Board Examination.