Coconote
AI notes
AI voice & video notes
Export note
Try for free
Graphing Systems of Equations Explained
Sep 27, 2024
Solving System of Equations by Graphing
Overview
Solve a system of equations by graphing.
Equations:
( y = 2x - 3 )
( 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.
📄
Full transcript