Pair of Linear Equations in Two Variables (NCERT Ch. 3)

• Solve a pair of linear equations using graphical, substitution, and elimination methods.
• Determine whether a pair of equations has a unique solution, no solution, or infinitely many solutions.

1. A pair of linear equations a₁x+b₁y=c₁ and a₂x+b₂y=c₂ has a unique solution if a₁/a₂ ≠ b₁/b₂ (lines intersect).
2. It has no solution if a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (lines are parallel).
3. It has infinitely many solutions if a₁/a₂ = b₁/b₂ = c₁/c₂ (lines coincide).
4. Substitution method: solve one equation for one variable, substitute into the other.
5. Elimination method: multiply equations to make coefficients of one variable equal, then add/subtract to eliminate it.

Solve: x + y = 5 and x - y = 1 (elimination method)

Adding both equations: 2x = 6 -> x = 3
Substituting back: 3 + y = 5 -> y = 2
Solution: x = 3, y = 2

• Sign errors when adding/subtracting equations in the elimination method.
• Forgetting to check the ratio condition to determine the type of solution before solving.
• Substituting into the wrong equation after solving for one variable.

• Solve: 2x + y = 7 and x - y = 2.
• Does 2x + 3y = 6 and 4x + 6y = 12 have a unique solution?

Open the Practice tab for graded questions on Pair of Linear Equations in Two Variables (NCERT Ch. 3).