Find the intersection of two lines using substitution and elimination.
A system of two linear equations has three possible outcomes: one solution (lines intersect at one point), no solution (parallel lines), or infinitely many solutions (same line). We solve systems using substitution or elimination.
Substitution Method
Elimination Method
Use substitution when one variable is already isolated; use elimination when coefficients can be easily matched.
Multiply the second equation by 3: 12x − 3y = 6. Add to first: 14x = 18 → x = 9/7. Back-substitute: 2(9/7) + 3y = 12 → y = (12 − 18/7)/3 = 22/7.
Remember This!
Always verify your solution by substituting both values into both original equations. If either equation is not satisfied, recheck your work.
What is the solution to the system: x + y = 7 and x − y = 3?