Graphing Systems of Equations Explained

Sep 27, 2024

Solving System of Equations by Graphing

Overview

  • Solve a system of equations by graphing.
  • Equations:
    1. ( y = 2x - 3 )
    2. ( y = -\frac{2}{3}x + 5 )
  • Goal: Find the point of intersection, which is the solution.

Graphing Steps

Preparing the Graph

  • Mark the graph:
    • Y-axis up to 5
    • X-axis up to 5
  • Bigger graphs are preferable for clarity.

Graphing the First Equation (( y = 2x - 3 ))

  • Slope-intercept form: ( y = mx + b )
  • Slope ( m ): 2
  • Y-intercept ( b ): -3
  • Plotting Points:
    • Start at (0, -3)
    • Use slope: rise 2, run 1
    • Points:
      • ( (0, -3) )
      • ( (1, -1) )
      • ( (2, 1) )
      • ( (3, 3) )
      • ( (4, 5) )

Graphing the Second Equation (( y = -\frac{2}{3}x + 5 ))

  • Y-intercept: 5
  • Slope: (- \frac{2}{3} )
  • Plotting Points:
    • Start at (0, 5)
    • Use slope: rise -2, run 3
    • Points:
      • ( (0, 5) )
      • ( (3, 3) )

Finding the Intersection

  • Point of Intersection: (3, 3)
  • This is the solution to the system.

Verifying the Solution by Substitution

  • Substitute ( y = 2x - 3 ) into ( y = -\frac{2}{3}x + 5 )
  • Simplify:
    • ( 2x - 3 = -\frac{2}{3}x + 5 )
    • Multiply all terms by 3 to eliminate fraction:
      • ( 6x - 9 = -2x + 15 )
    • Solve for ( x ):
      • Add 9 and 2x to both sides: ( 8x = 24 )
      • Divide by 8: ( x = 3 )

Substitute Back to Find ( y )

  • ( y = 2(3) - 3 )
  • ( y = 6 - 3 = 3 )
  • Thus, ( y = 3 )

Conclusion

  • The solution of the system is ( (3, 3) ). This confirms the graphing result.