Systems of Equations Study Notes

SYSTEMS OF EQUATIONS

Definition

  • Systems of Equations: A collection of two or more equations with the same set of variables.

Types of Solutions

  • Same Variables: Each equation in the system involves the same set of variables, typically denoted as (x, y).

Graphical Representation

  • Graphically: The solution to a system of equations is represented by the intersection point (x, y) of the lines corresponding to the equations.
  • Algebraically: The solution (x, y) is the coordinate that satisfies both equations in the system, making them true.

Classification of Lines and Their Solutions

  1. Intersecting Lines
    • Description: These are lines that cross each other at one point.
    • Solution: There is one solution.
  2. Parallel Lines
    • Description: These lines never cross and have the same slope.
    • Solution: There is no solution.
  3. Same Line
    • Description: Both equations represent the same line.
    • Solution: There are infinitely many solutions.

Summary of Types of Solutions

  • One Solution: Occurs when lines intersect at exactly one point.
  • No Solution: Occurs when lines are parallel; hence they never meet.
  • Infinite Solutions: Occurs when the equations represent the same line.

SOLVING SYSTEMS BY GRAPHING

Steps to Solve by Graphing

  1. Rewrite: Rewrite the equations in slope-intercept form (y = mx + b).
  2. Graph: Graph each equation on the same set of axes.
  3. Identify the Solution: Locate the intersection point of the graphs, if it exists.

Example Problems

Problem 1

  1. Equations:
    • y = x - 8
    • y = -2x + 1
  2. Graph the lines based on the equations.
  3. Solution: The intersection point (solution) is at (3, -5).

Problem 2

  1. Equations:
    • y = x + 9
    • y = -x + 6
  2. Graph the lines according to the equations.
  3. Solution: The intersection point (solution) is at (2, 8) indicating that (2, 8) is the solution to this system.

References

  • Gina Wilson (All Things Algebra, LLC), 2012-2016