∸ Systems of Equations

What Is a System?

A system of equations is two or more equations with the same variables. The solution is the (x, y) pair that satisfies ALL equations simultaneously. Graphically, it's the intersection point.

Substitution Method

Solve one equation for one variable, then substitute into the other. Example: y = 2x − 1 and y = −x + 5. Set equal: 2x − 1 = −x + 5 → 3x = 6 → x = 2, y = 3.

Elimination Method

Add or subtract equations to eliminate one variable. Multiply equations by constants if needed. Once one variable is gone, solve for the remaining one.

Special Cases

• No solution: parallel lines (same slope, different intercepts). The system is inconsistent.
• Infinite solutions: same line (identical equations). Dependent system.