How to Solve Systems of Equations (3 Methods)

A system of equations is two or more equations with the same variables. The solution is where they all agree. Three methods solve them.

Quick answer

Graphing works when the solution is a clean integer pair. Substitution is fastest when one equation is already solved for a variable. Elimination is fastest when coefficients line up easily.

Method 1: Graphing

Graph each equation. The solution is the intersection point. Best for visual confirmation. Limited if the solution isn’t at clean grid lines.

Method 2: Substitution

Example: y = 2x + 1 and 3x + y = 11. Substitute the first into the second: 3x + (2x + 1) = 11 → 5x = 10 → x = 2. Plug back: y = 5. Solution: (2, 5).

Method 3: Elimination

Example: 2x + 3y = 12 and 2x − y = 4. Subtract: 4y = 8 → y = 2. Plug back: x = 3. Solution: (3, 2).

What if there’s no solution?

Parallel lines never intersect (no solution). Identical lines have infinite solutions (any point). After substituting, if you get a false statement (like 5 = 7), no solution. If you get a true statement (like 5 = 5), infinitely many.

Common mistakes

• Forgetting to plug back to find the other variable.
• Sign errors when distributing the elimination step.
• Solving for the wrong variable first.