Solve the Following Pair of Linear Equations by Elimination Method and Substitution Method
Question:
-
x + y = 5
2x − 3y = 4 -
3x + 4y = 10
2x − 2y = 2 -
3x − 5y = 4
9x = 2y + 7 -
x/2 + 2y/3 = −1
x − y/3 = 3
Solution (i)
Given Equations
Equation (1): x + y = 5
Equation (2): 2x − 3y = 4
Step 1: Elimination Method
Multiply Equation (1) by 3.
3x + 3y = 15
Add Equation (2).
(3x + 3y) + (2x − 3y) = 15 + 4
5x = 19
x = 19/5
Step 2: Substitution Method
Substitute x into Equation (1).
19/5 + y = 5
y = 25/5 − 19/5
y = 6/5
Final Answer
x = 19/5, y = 6/5
Solution (ii)
Given Equations
Equation (1): 3x + 4y = 10
Equation (2): 2x − 2y = 2
Step 1: Elimination Method
Multiply Equation (2) by 2.
4x − 4y = 4
Add Equation (1).
(4x − 4y) + (3x + 4y) = 4 + 10
7x = 14
x = 2
Step 2: Substitution Method
Substitute x = 2 into Equation (2).
2(2) − 2y = 2
4 − 2y = 2
2y = 2
y = 1
Final Answer
x = 2, y = 1
Solution (iii)
Given Equations
Equation (1): 3x − 5y = 4
Equation (2): 9x − 2y = 7
Step 1: Elimination Method
Multiply Equation (1) by 3.
9x − 15y = 12
Subtract Equation (2).
(9x − 15y) − (9x − 2y) = 12 − 7
−13y = 5
y = −5/13
Step 2: Substitution Method
Substitute y into Equation (1).
3x − 5(−5/13) = 4
3x + 25/13 = 4
3x = 27/13
x = 9/13
Final Answer
x = 9/13, y = −5/13
Solution (iv)
Given Equations
Equation (1): x/2 + 2y/3 = −1
Equation (2): x − y/3 = 3
Step 1: Remove Fractions
LCM of 2 and 3 = 6
Multiply Equation (1) by 6.
3x + 4y = −6
Multiply Equation (2) by 3.
3x − y = 9
Step 2: Elimination Method
Subtract the equations.
(3x − y) − (3x + 4y) = 9 − (−6)
−5y = 15
y = −3
Step 3: Substitution Method
Substitute y into Equation (2).
3x − (−3) = 9
3x + 3 = 9
3x = 6
x = 2
Final Answer
x = 2, y = −3
Key Concepts
- Elimination Method removes one variable by making coefficients equal.
- Substitution Method replaces one variable using another equation.
- Always verify the obtained values in both equations.
- Clear fractions before solving whenever possible.
- Write equation numbers to avoid mistakes during elimination.
Exam Tips
- Choose the variable that is easier to eliminate.
- Multiply equations carefully to avoid sign errors.
- Keep positive and negative signs under control.
- After finding one variable, substitute into the simplest equation.
- Practice both elimination and substitution methods as both are important for CBSE Board exams.