A solution to a system satisfies both equations at once. Use graphing for visual insight, substitution when a variable is already isolated, and elimination when coefficients can cancel efficiently.
- Pick the method that minimizes arithmetic work.
- Check the ordered pair in both original equations.
- Parallel lines mean no solution, same line means infinitely many solutions.
Know when each method is efficient
Substitution is fastest when one equation already has x or y isolated. Elimination is strong when coefficients are opposites or easy to scale into opposites.
Graphing is useful for estimation and context, but exact solutions usually come from algebraic methods unless intersections are integer lattice points.
Interpret the intersection
In applications, each equation represents a rule or constraint. The intersection point is the scenario where both rules hold. In a pricing problem, it can represent break-even quantity and cost.
- One intersection: one solution.
- No intersection: no solution.
- Overlapping lines: infinitely many solutions.
Find where two plans cost the same
Plan A costs y = 15x + 40 and Plan B costs y = 20x + 10, where x is sessions.
- Set equations equal: 15x + 40 = 20x + 10.
- Subtract 15x: 40 = 5x + 10, then subtract 10: 30 = 5x.
- Solve x = 6 and substitute into either plan: y = 130.
- Verify in both equations to confirm matching cost.
Common mistakes
- Stopping after finding x and forgetting to find y.
- Adding equations incorrectly during elimination.
- Using rounded graph estimates as exact answers.
- Not checking the final pair in both equations.
Try one
Solve by elimination: x + y = 9 and x - y = 1.
Add equations to eliminate y: 2x = 10 so x = 5, then y = 4.
Sources
- OpenStax: College AlgebraOpen textbook with full chapter coverage, worked examples, and practice problems.
- Khan Academy: Algebra and Algebra 2Official lessons and practice sets for core algebra skills.