Often neither variable is ready to cancel: in $2x + 3y = 12$ and $5x - 2y = 11$, no coefficients match. You can make them match. This extends elimination with one extra step.
Why it works
Multiplying both sides of an equation by the same nonzero number gives an equivalent equation with exactly the same solutions; it is the same line, written differently. So you may scale either equation, or both, until one variable has opposite coefficients. The least common multiple of the two coefficients tells you the smallest numbers to use.
Key idea
Pick a variable. Multiply one or both equations (every term, on both sides) so its coefficients become opposites. Add, solve, substitute back, check. If both variables cancel, $0 = 0$ means infinitely many solutions and $0 = 5$ means none.
Worked examples
Worked example: Scale both equations
Solve the system $$\begin{aligned} 2x + 3y &= 12 \\ 5x - 2y &= 11 \end{aligned}$$
- Eliminate $y$: the coefficients $3$ and $-2$ can both become $6$ and $-6$.
- First equation times $2$: $4x + 6y = 24$. Second equation times $3$: $15x - 6y = 33$.
- Add: $19x = 57$, so $x = 3$.
- Substitute: $2(3) + 3y = 12$, so $3y = 6$ and $y = 2$. Check: $5(3) - 2(2) = 11$.
Answer: $(3, 2)$
Worked example: Scale one equation
Solve the system $$\begin{aligned} x + 4y &= 2 \\ 3x - 5y &= 23 \end{aligned}$$
- Eliminate $x$: multiply the first equation by $-3$: $-3x - 12y = -6$.
- Add to the second: $-17y = 17$, so $y = -1$.
- Substitute: $x + 4(-1) = 2$, so $x = 6$.
Answer: $(6, -1)$
Worked example: When everything cancels
Solve the system $$\begin{aligned} 2x - y &= 3 \\ -4x + 2y &= 5 \end{aligned}$$
- Multiply the first equation by $2$: $4x - 2y = 6$.
- Add to the second: $0 = 11$, which is false.
- No pair $(x, y)$ makes both true: the lines are parallel.
Answer: No solution.
Common mistake
Multiplying only the left side of an equation. “Times $3$” must reach the constant too: $3(5x - 2y) = 3(11)$ gives $15x - 6y = 33$, not $15x - 6y = 11$. A wrong right side moves the line and gives a wrong answer, which a check in the original equations will catch.