Solving Systems by Elimination

A system of linear equations asks for values satisfying several equations at once. Geometrically, two equations in two unknowns are two lines; a solution is where they cross.

Three outcomes are possible:

• One solution — the lines cross at a single point.
• No solution — the lines are parallel and never meet.
• Infinitely many — the two equations describe the same line.

Gaussian elimination solves systems by combining equations to remove variables. Adding a multiple of one equation to another is a legal row operation — it never changes the solution set.

Worked example. Solve $2x+y=7$ and $x-y=2$.

Add the two equations (the $y$ terms cancel): $3x = 9$, so $x=3$. Back-substitute into $x-y=2$: $3-y=2$, so $y=1$. Solution: $(3,1)$.