Elimination method for systems of equations

The elimination method solves a system of two equations in two unknowns by combining them so that one variable disappears. Line the equations up column by column, make the coefficient of one variable match (multiplying an equation if needed), then add or subtract each column to eliminate that variable.

Take the system Multiply the first equation by $2$ so the $y$ terms match, which gives the new system Now subtract column by column. Each column carries a minus sign, and the $y$ terms cancel: So $x = 2$, and substituting back gives $2(2) + y = 8$, so $y = 4$.

Pick whichever variable is easiest to match. If a variable already has the same coefficient in both equations, subtract to remove it; if the coefficients are opposites, just add. Watch the signs: subtracting a negative term flips it to a plus, so the sign under that column becomes a plus.

The result also tells you the type of system. Take Adding the equations makes both variables cancel: The false statement $0 = 4$ means there is no solution. If instead both variables cancel and you get a true statement like $0 = 0$, the two equations describe the same line, so there are infinitely many solutions.

Solve the system Add the equations term by term (the $y$ terms are opposite and cancel): So $x = 7$.
Substitute back: $7 + y = 10$, so $y = 3$.

Solve the system Multiply the first equation by $2$ to match the $x$ terms, giving the new system Subtract column by column (each column takes a minus, and the $x$ terms cancel): So $y = 3$.
Substitute back: $4x + 3 = 11$, so $x = 2$.

Solve the system Add the equations term by term (both variables cancel): The statement $0 = 5$ is impossible, so the system has no solution.