Special Cases in Elimination with Multiplication

Property When you scale equations using multiplication and add them together, it is possible for both variables to cancel out simultaneously. Contradiction: If the result is a false statement (e.g., $0 = c$, where $c \neq 0$), the lines are parallel, resulting in No Solution . Identity: If the result is a true statement (e.g., $0 = 0$), the equations represent the exact same line, resulting in Infinitely Many Solutions .

Examples No Solution: Solve $2x + 3y = 6$ and $4x + 6y = 15$. Multiply the top equation by $ 2$ to eliminate $x$: $ 2(2x + 3y = 6) \rightarrow 4x 6y = 12$ Add this to the second equation: $( 4x + 4x) + ( 6y + 6y) = 12 + 15 \rightarrow 0 = 3$. This is a false statement, so there is no solution. Infinitely Many Solutions: Solve $3x y = 4$ and $6x 2y = 8$. Multiply the top equation by $ 2$: $ 2(3x y = 4) \rightarrow 6x + 2y = 8$ Add this to the second equation: $( 6x + 6x) + (2y 2y) = 8 + 8 \rightarrow 0 = 0$. This is a true statement, so there are infinitely many solutions.

Explanation When both variables vanish, your algebra is uncovering a secret about the two equations. If you multiply an equation and it suddenly looks identical to the other one (giving $0=0$), it means they were the exact same line the entire time—just disguised by a scale factor! If multiplying makes the left sides match but the right sides completely different (giving $0=3$), it means they share a slope but have different intercepts, proving they are parallel lines that will never cross.