Pair of Linear Equations in Two Variables – Solving Methods (Class 10 CBSE)
The chapter Pair of Linear Equations in Two Variables is one of the most important topics in Class 10 CBSE Mathematics. It carries significant weight in board exams and builds the foundation for higher algebra concepts.
In this detailed guide, you will learn all solving methods including the graphical method, substitution method, and elimination method with step-by-step solved problems and exam-focused tips.
1. What is a Pair of Linear Equations?
A linear equation in two variables is written in the form:
ax + by + c = 0
Where:
- a, b, c are real numbers
- a and b are not both zero
- x and y are variables
A pair of linear equations consists of two such equations:
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
2. Types of Solutions (Consistency Conditions)
- a₁/a₂ ≠ b₁/b₂ → Unique solution
- a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → No solution (parallel lines)
- a₁/a₂ = b₁/b₂ = c₁/c₂ → Infinitely many solutions
3. Method 1 – Graphical Method
In the graphical method, we plot both equations on the coordinate plane. The point of intersection gives the solution.
Example 1: Solve graphically:
2x + y = 5
x − y = 1
Solution:
Convert into table form and plot points:
For 2x + y = 5: If x = 0 → y = 5 If x = 1 → y = 3
For x − y = 1: If x = 1 → y = 0 If x = 2 → y = 1
After plotting, lines intersect at (2,1).
Solution = (2,1)
4. Method 2 – Substitution Method
In this method, one equation is solved for one variable and substituted into the other equation.
Example 2: Solve using substitution method:
x + y = 10
x − y = 2
Step 1: From first equation: x = 10 − y
Step 2: Substitute into second equation:
(10 − y) − y = 2 10 − 2y = 2 2y = 8 y = 4
Step 3: Substitute back:
x = 10 − 4 = 6
Solution = (6,4)
5. Method 3 – Elimination Method
In elimination method, coefficients of one variable are made equal and then subtracted or added to eliminate that variable.
Watch: Elimination Method Explained (Step-by-Step)
Prefer video explanation? Watch the complete elimination method solution above.
Example 3: Solve using elimination method:
3x + 2y = 11
2x + y = 6
Step 1: Multiply second equation by 2:
4x + 2y = 12
Step 2: Subtract first equation:
(4x + 2y) − (3x + 2y) = 12 − 11 x = 1
Step 3: Substitute in second equation:
2(1) + y = 6 y = 4
Solution = (1,4)
—6. Word Problems (Application-Based Questions)
Example 4: The sum of two numbers is 27 and their difference is 3. Find the numbers.
Let numbers be x and y.
x + y = 27 x − y = 3
Adding both equations: 2x = 30 x = 15
y = 27 − 15 = 12
Numbers are 15 and 12
Example 5: The cost of 2 pens and 3 pencils is ₹23. The cost of 4 pens and 6 pencils is ₹46. Find the cost of one pen and one pencil.
2x + 3y = 23 4x + 6y = 46
Notice that second equation is 2 × first equation.
Hence, infinitely many solutions.
System is dependent.
[BONUS VIDEO] Watch: Cross Multiplication Method Explained (Step-by-Step)
Prefer video explanation? Watch the complete Cross Multiplication method explanation above.
7. Important Board Exam Tips
✔ Always check consistency condition
✔ Be careful with sign changes
✔ Write full steps for method-based questions
8. Common Mistakes to Avoid
- Forgetting to multiply entire equation
- Incorrect subtraction during elimination
- Plotting wrong coordinates in graphical method
- Arithmetic calculation errors
9. Revision Strategy for Board Exams
Practice at least 10 problems daily:
- 3 substitution method
- 3 elimination method
- 2 graphical method
- 2 word problems
Consistent practice improves speed and accuracy.