Lesson 75: Introduction to Systems of Equations
A system of equations is a set of two or more equations with the same variables. The solution to a system is the set of values that satisfy all equations simultaneously.
Systems can be classified as:
- Linear Systems: Each equation represents a line in two dimensions.
- Nonlinear Systems: At least one equation is not linear (e.g., quadratic, circle, parabola).
Methods for Solving Systems
Common methods include:
- Graphing: Plot the equations and find intersection points.
- Substitution: Solve one equation for a variable and substitute it into the other.
- Elimination: Combine equations to eliminate one variable.
For nonlinear systems, substitution or factoring is often used to reduce the system to a single equation.
Examples
Example 1 — Linear System:
Solve:
x + y = 4
2x − y = 1
Solution (Substitution): y = 4 − x → 2x − (4 − x) = 1 → 3x − 4 = 1 → 3x = 5 → x = 5/3 → y = 4 − 5/3 = 7/3
Example 2 — Nonlinear System:
Solve:
y = x²
x + y = 6
Solution: Substitute y = x² → x + x² = 6 → x² + x − 6 = 0 → (x + 3)(x − 2) = 0 → x = −3, 2
Corresponding y-values: y = 9, 4 → Solutions: (−3, 9), (2, 4)
Practice Problems
- Solve the system: 2x + y = 5, x − y = 1
- Solve the nonlinear system: y = x² − 2, y = 3x − 1
- Determine if the system has one solution, no solution, or infinitely many solutions: y = 2x + 3, y = 2x − 5
- Solve using substitution: x + y = 7, 3x − y = 5
- Graph the system and find the solution: y = −x + 4, y = x²