The Main Idea

Common Features

1. Solution Types
   • No solutions (no intersection)
   • One solution (tangent point)
   • Two solutions (line crosses through curve)
2. Solution Method
   • Always start with linear equation
   • Solve linear equation for one variable
   • Substitute into nonlinear equation
   • Solve resulting equation
   • Check all solutions in both equations

Line-Parabola Systems

1. Form
   • Line: [latex]ax + by = c[/latex]
   • Parabola: [latex]y = ax^2 + bx + c[/latex]
2. Solution Process
   • Results in quadratic equation
   • Two solutions: Line crosses parabola
   • One solution: Line tangent to parabola
   • No solutions: Line misses parabola

Line-Circle Systems

1. Form
   • Line: [latex]ax + by = c[/latex]
   • Circle: [latex](x - h)^2 + (y - k)^2 = r^2[/latex]
2. Solution Process
   • Results in quadratic equation
   • Two solutions: Line crosses circle
   • One solution: Line tangent to circle
   • No solutions: Line outside circle

The Main Idea

• Method Selection
  • Elimination preferred when both equations are nonlinear
  • Especially useful when equations have similar terms
  • More efficient than substitution for two-by-two systems
  • Works well with conic sections (circles, ellipses, etc.)
• Solution Types for Conic Intersections Circle-Ellipse Intersections can have:
  • No solutions (no intersection points)
  • One solution (curves are tangent)
  • Two solutions (curves intersect twice)
  • Three solutions (curves intersect three times)
  • Four solutions (curves intersect four times)
• Strategy
  • Look for like terms between equations
  • Multiply equations to match coefficients
  • Add/subtract to eliminate one variable
  • Solve resulting single-variable equation
  • Back-substitute to find other variable
• Key Considerations
  • Don’t need to recognize curve types to solve
  • Pay extra attention to algebra if curves unfamiliar
  • Watch for extraneous solutions
  • Check all solutions in both equations

The Main Idea

• Universal Steps for Inequality Graphing
  • Convert inequality to equation by replacing inequality symbol
  • Graph boundary line/curve
  • Use solid line for [latex]\leq[/latex] or [latex]\geq[/latex]
  • Use dashed line for [latex]\lt[/latex] or [latex]\gt[/latex]
  • Test point to determine shading region
  • Shade solution region
• Boundary Line Types
  • Linear: [latex]ax + by \leq c[/latex]
  • Parabolic: [latex]y \leq ax^2 + bx + c[/latex]
  • Circle: [latex]x^2 + y^2 \leq r^2[/latex]
  • Other nonlinear curves
• Region Testing Strategy
  • Choose point NOT on boundary
  • Pick simple point (like [latex](0,0)[/latex])
  • Test in original inequality
  • If true, shade that region
  • If false, shade opposite region
• Visual Indicators
  • Solid line: includes boundary points
  • Dashed line: excludes boundary points
  • Shaded region: all solution points
  • Multiple regions possible with systems

The Main Idea

• System Components
  • At least one nonlinear inequality
  • Two or more inequalities
  • Solution is intersection of regions
  • Feasible region where all inequalities are satisfied
• Solution Process
  • Find intersection points algebraically
  • Graph boundary curves
  • Test regions for each inequality
  • Identify overlapping solution region
• Types of Boundaries
  • Parabolas: [latex]y \leq ax^2 + bx + c[/latex]
  • Circles: [latex]x^2 + y^2 \leq r^2[/latex]
  • Other conics (ellipses, hyperbolas)
  • Mix of linear and nonlinear
• Region Analysis
  • May have multiple solution regions
  • Regions can be bounded/unbounded
  • Some regions may be disconnected
  • Check entire boundary for intersections