Pair of Linear Equations in Two Variables

Five pencils and seven pens together cost ₹250, while seven pencils and five pens cost ₹230. Form a pair of linear equations for this situation.

Final Answer: 5x + 7y = 250 and 7x + 5y = 230, where x = cost of a pencil and y = cost of a pen.

Common Mistake: Mixing up which quantity goes with which price and swapping the coefficients.

Exam Tip: Define the variables in words first — forming the equations then becomes mechanical.

Solve 2x + y = 8 and x − y = 1 by the elimination method, and state what kind of solution it is.

Formula: Unique solution when a₁/a₂ ≠ b₁/b₂

Final Answer: x = 3, y = 2 — a unique solution (the equations are consistent).

Common Mistake: Adding the equations when the coefficients are the same sign — that does not eliminate the variable.

Exam Tip: Always substitute your answer back into the other equation to check before writing the final line.

By comparing ratios of coefficients, decide whether the pair 3x + 2y = 5 and 6x + 4y = 9 has a solution.

Formula: a₁/a₂ = b₁/b₂ ≠ c₁/c₂ ⇒ no solution (parallel, inconsistent)

Final Answer: No solution — the lines are parallel, so the pair is inconsistent.

Common Mistake: Forgetting to move the constant to the left side, which gives the wrong sign for c.

Exam Tip: Write all three ratios in lowest terms before comparing them.