Systems of Two Linear Equations

One intersection, many methods

Two lines can intersect at one point (unique solution), be parallel (none), or be the same line (infinitely many). Algebra finds that intersection without graphing.

• Substitution: solve one equation for a variable, plug into the other.
• Elimination: add/subtract multiples so one variable cancels.
• Always substitute back to find the second variable, then check both equations.

Worked example — elimination

Solve: 2x + 3y = 12 and 4x − 3y = 6. Add: 6x = 18 → x = 3. Plug into first: 6 + 3y = 12 → y = 2. Solution (3, 2).