Systems of Equations
A system of equations is two or more equations with the same variables. The solution is the point (or points) that satisfies all equations simultaneously.
Solving by Graphing
Graph both lines. The solution is the intersection point. Parallel lines = no solution. Same line = infinite solutions.
Substitution Method
Solve one equation for one variable, substitute into the other.
y = 2x + 1 and 3x + y = 11
3x + (2x+1) = 11 → 5x = 10 → x = 2, y = 5
Elimination Method
Add or subtract equations to eliminate one variable.
2x + 3y = 12 and 2x − y = 4
Subtract: 4y = 8 → y = 2, x = 3
Interpreting Solutions
One solution → lines intersect (consistent independent).
No solution → parallel lines (inconsistent).
Infinite solutions → same line (dependent).
FAQ
Which method is best? Substitution when one variable is isolated; elimination when coefficients match or are easy to match.
Quick Quiz
Test what you just learned. Choose the best answer for each question.