Graphing shows meaning
The intersection point satisfies both equations. Parallel lines have no solution; the same line has infinitely many.
ExampleIf two graphs meet at (3, 2), then x = 3 and y = 2 solve both equations.
Chapter 05 of 10
Systems of Equations
How do you find the values that make two equations true at once?
A system asks for the common solution of two or more equations. For two linear equations, that solution is the intersection of their lines. Graphing, substitution, and elimination reveal the same result in different ways.
The big ideas
Understand these before you worry about speed.
Substitution replaces an equal quantity
Isolate one variable, then replace it in the other equation. This is efficient when a variable is already isolated.
ExampleIf y = 2x and x + y = 9, substitute to get x + 2x = 9.
Elimination cancels a variable
Add or subtract equations whose coefficients are opposites. Multiply an equation first when needed.
Examplex + y = 7 and x − y = 1 add to 2x = 8.
Worked example
Follow the reason for each step—not just the symbols.
Problem
Solve y = 2x + 1 and 3x + y = 16.
- Substitute 2x + 1 for y: 3x + 2x + 1 = 16.
- Solve: 5x = 15, so x = 3.
- Find y: y = 2(3) + 1 = 7.
- Check: 3(3) + 7 = 16.
Answer(3, 7)
Try it yourself
Work on paper first. Open each answer only when you are ready to check.
1
Solve x + y = 9 and x − y = 3.
Check answer
(6, 3)2
Solve y = x − 2 and 2x + y = 10.
Check answer
(4, 2)3
Classify y = 3x + 1 and y = 3x − 4.
Check answer
No solution4
Classify 2x + 4y = 8 and x + 2y = 4.
Check answer
Infinitely many solutionsPractice and tools
Use the resource that matches what you need next.
Worksheet
Systems by Graphing
See solutions as intersection points.
Open resource WorksheetSystems by Substitution
Practice isolation and substitution.
Open resource WorksheetSystems by Elimination
Build fluency with scaling and cancellation.
Open resource CalculatorMatrix RREF Calculator
Preview how systems extend into row reduction.
Open resource