Systems of Equations
A system of equations is two or more equations with the same variables. The solution is the point (x, y) that satisfies BOTH equations simultaneously — where the lines intersect. Systems can be solved by graphing, substitution, or elimination.
Key Concepts
Solving by Substitution
Solve one equation for one variable, substitute into the other. y = 2x + 1 and 3x + y = 16. Substitute: 3x + (2x+1) = 16 → 5x+1=16 → x=3. Then y=2(3)+1=7. Solution: (3, 7).
Solving by Elimination
Add or subtract equations to eliminate one variable. 2x + y = 10 and x − y = 2. Add them: 3x = 12 → x = 4. Substitute: 4 − y = 2 → y = 2. Solution: (4, 2). Multiply one equation to make coefficients match before eliminating.
Types of Solutions
One solution: lines intersect at one point (different slopes). No solution: lines are parallel (same slope, different intercept). Infinite solutions: same line (same slope and intercept). A system of linear equations always has one of these three outcomes.
🆕 System of Equations Solver
Enter two lines. See where they intersect!
✅ Check Your Understanding
1. The solution to a system of two equations is...
2. Two lines with the same slope but different y-intercepts have...
3. In elimination, the goal is to...