EquationsAdvanced
Systems of Equations: Three Ways to Solve
When two equations share the same variables, their solution is where the lines meet. Compare graphing, substitution, and elimination.
Mathew JoaneJuly 8, 20269 min read
A system of equations is two or more equations considered together. The solution is the set of values that makes every equation true at the same time.
x + y = 6
2x - y = 3
Method 1: Substitution
- 1Solve the first equation for y: y = 6 - x.
- 2Substitute into the second: 2x - (6 - x) = 3.
- 3Simplify: 3x - 6 = 3, so x = 3.
- 4Back-substitute: y = 6 - 3 = 3.
Method 2: Elimination
Add the two equations so that one variable cancels. Here, adding removes y directly:
(x + y) + (2x - y) = 6 + 3
Method 3: Graphing
Graph both lines and look for the point where they intersect. For this system, both approaches confirm the solution is (3, 3).
Written by Mathew Joane
Algebra teacher on a mission to make math feel less like memorizing and more like understanding.