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.