Notes on Systems of Linear Equations - Two Variables

Learning Objectives

  • Solve systems of equations by graphing.
  • Solve systems of equations by substitution.
  • Solve systems of equations by addition.
  • Identify inconsistent systems of equations containing two variables.
  • Express the solution of a system of dependent equations containing two variables.

Introduction to Systems of Equations

  • A system of linear equations consists of two or more linear equations with multiple variables to be considered simultaneously.
  • Goal: Find numerical values for each variable that satisfy all equations.

Types of Solutions

  1. Independent System: Exactly one solution (point of intersection).
  2. Dependent System: Infinite solutions (lines coincide).
  3. Inconsistent System: No solution (parallel lines).

Consistency of a Linear System

  • Consistent System: At least one solution exists.
  • Dependent System: Has infinitely many solutions, represented by coincident lines.
  • Inconsistent System: No solutions exist, represented by parallel lines.

Determining Solutions

  • To verify if an ordered pair is a solution to a system:
    1. Substitute the ordered pair into each equation.
    2. If the equations hold true for both, it is a solution.
Example
  • Verify if (5, 1) is a solution:
    • Substitute into equations:
    • For equation 1: 5 + 3(1) = 8 (True)
    • For equation 2: 2(5) - 9(1) = 1 (True)
    • Result: (5, 1) is a solution.

Solving by Graphing

  • Graph both equations on the same axes to find the intersection point.
  • The intersection represents the solution to the system.
  • Example: If the lines intersect at (x,y), verify the solution.

Solving by Substitution

  1. Solve one equation for one variable in terms of the other.
  2. Substitute this value into the second equation.
  3. Solve for the remaining variable.
  4. Substitute back to find the first variable.
  5. Write the solution as an ordered pair.
Example: Substitution Method
  • Given equations: y = 3x - 5 and 2x + y = 1
  • Substitute: 2x + (3x - 5) = 1
  • Solve for x and then y.

Solving by Addition (Elimination)

  1. Align equations for the addition method.
  2. Adjust coefficients to eliminate one variable when added together.
  3. Solve the resulting equation for the remaining variable.
Example
  • Given equations: 3x + 5y = 11 and 2x - y = 3.
  • Multiply to align and eliminate a variable, then solve.

Identifying Inconsistent Systems

  • Parallel lines indicate no intersections.
  • Practice determining an inconsistent system by substitution or graphical methods.

Dependent Systems

  • When both equations represent the same line, resulting in an infinite number of solutions.
  • Check by rewriting into the slope-intercept form to identify identical slopes.

Break-even Analysis Example

  • Set cost and revenue functions equal:
    • Cost function
    • Revenue function
  • The break-even point is where they intersect, representing where the costs equal revenue.

Summary of Methods for Solving Systems of Equations

  1. Graphing: Visual method for intersection.
  2. Substitution: Effective for precise values.
  3. Addition/Elimination: Useful for aligning coefficients.
Practical Application
  • Use these methods to analyze real-world problems, like profitability analyses in business scenarios.