Overview

This topic explores solving systems of linear equations and inequalities using multiple methods. You'll learn how to solve using graphing, substitution, and elimination, and how to analyze systems with one, none, or infinitely many solutions. Inequality systems are covered with emphasis on shading and solution regions.

Key Concepts and Structures

• System of Equations: A set of two or more equations with the same variables. The solution is the point(s) where the equations intersect.
• Graphical Method: Plot each equation and identify the intersection point(s).
• Substitution Method: Solve one equation for a variable, then substitute into the other equation.
• Elimination Method: Add or subtract equations to eliminate one variable and solve for the other.
• Types of Solutions:
  • One solution (lines intersect)
  • No solution (parallel lines)
  • Infinitely many solutions (same line)
• Systems of Inequalities: Graph each inequality and shade the solution region. The final solution is the overlapping shaded area.
• Boundary Lines: Use a dashed line for < or >; solid for ≤ or ≥.

Step-by-Step Example

Problem: Solve the system:

  x + y = 5
  x - y = 1
      

Step 1: Use the elimination method.
Add both equations:

  (x + y) + (x - y) = 5 + 1 → 2x = 6

Step 2: Solve for x:
x = 3

Step 3: Substitute into the first equation:
3 + y = 5 → y = 2

Final Answer: (x, y) = (3, 2)

Quick Tip