Graphing: see the crossing
The graphing method turns each equation into a line and reads off the intersection. It builds the clearest intuition, but it is only as accurate as your drawing — a crossing at (2.5, 1.7) is hard to read from a sketch. Rewrite each line in slope-intercept form y = mx + b so it plots quickly.
Substitution: solve, then replace
The substitution method shines when one variable is already alone or cheap to isolate. You use isolation to express one variable, then perform a substitution into the other equation, collapsing two unknowns to one.
Solve: y = 2x - 1 (already solved for y)
3x + y = 9
Substitute y = 2x - 1 into the second equation:
3x + (2x - 1) = 9
5x - 1 = 9
5x = 10
x = 2
Back into y = 2x - 1:
y = 2(2) - 1 = 3
Solution: (2, 3). Check: 3(2) + 3 = 9 ✓Elimination: add the equations away
The elimination method adds or subtracts whole equations so that one variable cancels. Scale each equation so the coefficients of one variable are opposites, then add. This is the method that scales best to larger systems.
Solve: 2x + 3y = 12
4x - 3y = 6
The +3y and -3y are already opposites. Add the equations:
(2x + 4x) + (3y - 3y) = 12 + 6
6x = 18
x = 3
Back-substitute x = 3 into 2x + 3y = 12:
6 + 3y = 12 -> 3y = 6 -> y = 2
Solution: (3, 2). Check in eq 2: 4(3) - 3(2) = 6 ✓