Pair of Linear Equations in Two Variables

Two linear equations in x and y: solve them together, and tell from the coefficients whether there is one solution, none, or infinitely many.

Key ideas

• A pair a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 is two straight lines; a solution is a point on both lines.
• One solution (the lines cross) when a₁/a₂ ≠ b₁/b₂.
• No solution (parallel lines) when a₁/a₂ = b₁/b₂ ≠ c₁/c₂.
• Infinitely many solutions (the same line) when a₁/a₂ = b₁/b₂ = c₁/c₂.
• Substitution: make x or y the subject of one equation and put it into the other.
• Elimination: multiply so that x or y has the same coefficient in both, then add or subtract the equations.
• In word problems, first write what x and y stand for, then make two equations from the two facts given.

Remember

Watch out

• Sign slips when subtracting one equation from the other: subtract every term, including the constants.
• Finding x and forgetting to go back for y.
• Not checking the answer in both equations.

Worked example