3.6: Solving Linear Equations by Graphing

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3.6 Solving Linear Equations by Graphing

• Objective
• Solve a system of linear equations in two
  variables
• by graphing.

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System of Linear Equation

• 2 linear equations create a system of linear
  equations.
• Can be in slope intercept or standard form.
• Slope Intercept form y mx b
• Standard form Ax By C
• Where A, B, and C are integers
• NO FRACTIONS, NO DECIMALS
• A, B and C can be NEGATIVE

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Solving a Linear Equation

• What does this mean?
• Locating where the lines cross
• Where lines share a coordinate (x,y)
• How many solutions?
• Parallel lines no solutions
• Same equations infinite solutions
• Any other lines one solution
• Perpendicular lines

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Solving Systems by Graphing
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Intersecting Lines

• The lines cross at one point one solution
• perpendicular or not perpendicular
• How to tell if two lines are perpendicular?
• Put both equations in slope intercept form
• Multiply the slopes
• If answer -1, they are perpendicular

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Parallel Lines

• Same slope
• Lines dont cross no solution
• Solve equation answer is two different numbers
• EXAMPLE y3x4
• y3x-5
• If slope is the same and y-intercept isnt, the
  lines are parallel.

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The Same Line

• If the equations are identical
• Lines are located on top of one another
• End result The same line or coincides
• -infinite solutions

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Linear System in Two Variables

• Three possible solutions to a linear system in
  two variables
• One solution coordinates of a point
• No solutions inconsistent case
• Infinitely many solutions dependent case

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Graphing
Four steps to solve a linear system using a
graph.
Solve both equations for y, so that each equation
looks like y mx b.
Step 1 Put both equations in slope - intercept
form.
Use the slope and y - intercept for each equation
in step 1. Be sure to use a ruler and graph
paper!
Step 2 Graph both equations on the same
coordinate plane.
Step 3 Estimate where the graphs intersect.
This is the solution! LABEL the solution!
Step 4 Check to make sure your solution
makes both equations true.
Substitute the x and y values into both equations
to verify the point is a solution to both
equations.
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Example 1 (Problem Set 3.6 7)
Solve the systems of linear equation by
graphing. 3x 2y 6 x y 1
-2y -3x 6 y 3 x 3 2 (0, -3) m
3/2
(4, 3)
Answer One solution (4, 3)
-y -x 1 y x 1 (0,-1) m 1
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Example 2 (Problem Set 3.6 25)
Solve the systems of linear equation by
graphing. x y 4 2x 2y -6
y -x 4 (0, 4) m -1
Answer No solution
2y -2x - 6 y -x 3 (0,-3) m -1
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Example 3 (Problem Set 3.6 27)
Solve the systems of linear equation by
graphing. 4x - 2y 8 2x - y 4
-2y -4x 8 y 2x - 4 (0, -4) m 2
Answer Infinite solutions
-y -2x 4 y 2x 4 (0,-4) m 2
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Determine Without Graphing

• Once the equations are in slope-intercept form,
  compare the slopes and intercepts.
• One solution the lines will have different
  slopes.
• No solution the lines will have the same slope,
  but different intercepts.
• Infinitely many solutions the lines will have
  the same slope and the same intercept.

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Problem Set 3.6 (TB pp. 198-199)
Individual Practice Homework Even numbers Nos.
2, 4, 6,36