3-1: Solving Systems by Graphing

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3-1 Solving Systems by Graphing
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What is a System of Linear Equations?
Definition A system of linear equations is
simply two or more linear equations using the
same variables.
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How to Use Graphs to Solve Linear Systems
Consider the following system
x y 1 x 2y 5
Using the graph to the right, we can see that any
of these ordered pairs will make the first
equation true since they lie on the line.
We can also see that any of these points will
make the second equation true.
However, there is ONE coordinate that makes both
true at the same time
The point where they intersect makes both
equations true at the same time.
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Three Possible Outcomesp. 154

• Two intersecting lines
• Two lines on top of each other
• Two parallel lines

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EXAMPLE 4
Writing and Using a Linear System (p. 155)
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EXAMPLE 4
Step 1 Write linear equations in standard form
Equation 1
Equation 2
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EXAMPLE 4
Step 2 Graph both equations

• Two intersecting lines one solution

• (20, 50) appears to be the solution

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EXAMPLE 4
Step 4 Check your solution
Point of intersection (20,50 ). Substitute 20
and 50 in place of x and y in both
equations y x 30 y 2.5x
Equation 1 checks.
50 20 30
Equation 2 checks.
50 2.5(20)
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Graphing to Solve a Linear System
Let's summarize! There are 4 steps to solving a
linear system using a graph.
Step 1 Put both equations in slope - intercept
form.
Solve both equations for y, so that each equation
looks like y mx b.
Step 2 Graph both equations on the same
coordinate plane.
Use the slope and y - intercept for each equation
in step 1. Be sure to use a ruler and graph
paper!
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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Practice Checking the Solution
Page 412, 11
4x y 25 -3x - 2y -16
We must ALWAYS verify that your coordinates
actually satisfy both equations.
To do this, we substitute the coordinate (6 , -1)
into both equations.
-3x - 2y -16 -3(6) - 2(-1) -18 2
-16 ?
4x y 25 4(6) (-1) 24 1
25 ?
Since (6 , -1) makes both equations true, then (6
, -1) is the solution to the system of linear
equations.
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EXAMPLE 1
Solve a system graphically
Graph the linear system and estimate the
solution. Then check the solution algebraically.
4x y 8
Equation 1
2x 3y 18
Equation 2
SOLUTION
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EXAMPLE 1
Solve a system graphically
Equation 2
Equation 1
2x 3y 18
4x y 8
The solution is (3, 4).
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Page 153 Example 1
GUIDED PRACTICE
Graph the linear system and estimate the
solution. Then check the solution algebraically.
3x 2y 4
Equation 1
x 3y 1
Equation 2
SOLUTION
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Page 153, Example 1
GUIDED PRACTICE
Equation 1
Equation 2
3x 2y 4
x 3y 1
The solution is (2, 1).
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Page 142, 23
Start with 3x 4y -10 Subtracting 3x from both
sides yields 4y 3x -10 Dividing everything by
6 gives us
While there are many different ways to graph
these equations, we will be using the slope -
intercept form.
Similarly, we can add 7x to both sides and then
divide everything by -1 in the second equation to
get
To put the equations in slope intercept form, we
must solve both equations for y.
Now, we must graph these two equations.
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Page 142, 23, cont.
Using the slope intercept form of these
equations, we can graph them carefully on graph
paper.
Start at the y intercept Note that my scale is
2 on this graph. then use the slope.
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Page 142, 23, cont.
Can you read the solution? It looks close to (2,
-4) Check to make sure.
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Graphing to Solve a Linear System
Let's do ONE moreSolve the following system of
equations by graphing.
2x 2y 3 x 4y -1
Step 1 Put both equations in slope - intercept
form.
Step 2 Graph both equations on the same
coordinate plane.
Step 3 Estimate where the graphs intersect.
LABEL the solution!
Step 4 Check to make sure your solution makes
both equations true.