The Main Idea

• System Definition and Purpose:
  • A system contains two or more linear equations with two or more variables
  • All equations must be satisfied simultaneously for a solution
  • Solution requires finding specific values for all variables
• Types of Solutions:
  • Independent systems: Exactly one solution (lines intersect at one point)
  • Inconsistent systems: No solution (parallel lines)
  • Dependent systems: Infinite solutions (same line)
• Solution Requirements:
  • Need at least as many equations as variables for a unique solution
  • This alone doesn’t guarantee a unique solution
  • Solution must satisfy ALL equations in the system
• Solution Verification:
  • Substitute proposed solution into ALL equations
  • Must get true statements for each equation
  • Graphically, solution point lies on all lines

The Main Idea

• Graphical Approach Steps:
  • Graph both equations on the same coordinate plane
  • Find the point of intersection (if it exists)
  • Verify the solution algebraically
  • Identify the system type based on how lines intersect
• System Types from Graphs:
  • Intersecting lines → One solution (independent)
  • Parallel lines → No solution (inconsistent)
  • Same line → Infinite solutions (dependent)
• Graphing Techniques:
  • Solve for [latex]y[/latex] to get slope-intercept form
  • Use slope and [latex]y[/latex]-intercept method
  • Use [latex]x[/latex] and [latex]y[/latex] intercepts method
  • Plot strategic points
• Solution Verification:
  • Always check intersection point in both equations
  • Both equations must be satisfied
  • Graphical solutions are approximate unless points are clearly identifiable

The Main Idea

• Method Overview:
  • Solve one equation for one variable
  • Substitute that expression into other equation
  • Solve resulting equation for remaining variable
  • Back-substitute to find first variable
• Advantages:
  • More precise than graphing
  • Works well with fractional/decimal solutions
  • Often simpler than other algebraic methods
• Solution Analysis:
  • May reveal no solution (contradictory equations)
  • May reveal infinite solutions (identical equations)
  • Check final answer in both original equations

The Main Idea

• Method Fundamentals:
  • Also called elimination method
  • Add equations to eliminate one variable
  • Variables with opposite coefficients sum to zero
  • Goal is to isolate one variable
• Key Steps:
  • Align like terms
  • Multiply equations if needed for elimination
  • Add equations to eliminate one variable
  • Solve for remaining variable
  • Back-substitute to find other variable
• Strategic Planning:
  • Look for variables with opposite coefficients
  • Find LCM of coefficients if multiplication needed
  • Choose simpler variable to eliminate first
  • Clear fractions before adding if needed
• When to Use:
  • Variables have coefficients that are opposites
  • Coefficients have a small LCM
  • Substitution would be too complex

The Main Idea

• Understanding Inconsistent Systems:
  • Parallel lines that never intersect
  • Same slope but different [latex]y[/latex]-intercepts
  • No solution exists
  • Result in contradictory equations
• Recognition Methods:
  • Converting to slope-intercept form reveals parallel lines
  • Solving leads to contradictory statement (e.g., [latex]5 = 8[/latex])
  • Algebraic solution gives impossible result
  • Graphing shows parallel lines
• Verification Techniques:
  • Write equations in slope-intercept form
  • Compare slopes and [latex]y[/latex]-intercepts
  • Attempt algebraic solution
  • Graph equations to confirm parallel lines

The Main Idea

• Understanding Dependent Systems:
  • Two equations represent the same line
  • Infinite number of solutions
  • All points on one line satisfy both equations
  • Solving yields identity (e.g., [latex]0 = 0[/latex])
• Key Characteristics:
  • Same slope and same [latex]y[/latex]-intercept
  • One equation is multiple of the other
  • Equations reduce to identity
  • Lines are coincident (overlap completely)
• Solution Format:
  • Written in parametric form: [latex](x, mx + b)[/latex]
  • [latex]x[/latex] can be any real number
  • [latex]y[/latex] is expressed in terms of [latex]x[/latex]
  • Represents all points on the line
• Identification Methods:
  • Convert to slope-intercept form
  • Compare coefficients after simplification
  • Look for scalar multiples
  • Check if equations are equivalent