Transcript Ch 7.1 Solving Linear Systems by Graphing

Algebra 1
Ch 7.1 – Solving Linear
Systems by Graphing
Before we begin…
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In previous chapters you learned how to
transform and graph linear equations using
slope-intercept…
In this chapter, we will use that information to
solve systems of linear equations in two
variables…
The solution to a system of linear equations is an
ordered pair or pairs that make both statements
true…
To be successful here it is important to lay out
your work in a logical sequential manner and not
skip any steps.
Process
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The steps below are the process to solving a
system of linear equations by graphing
Write each equation in a format that is easy to
graph.
Graph both equations on the same coordinate
plane
Determine the coordinates of the point of
intersection
Check the coordinates algebraically by
substituting them into each equation of the
original system of linear equations.
Note: In order for the ordered pair to be a solution, after
substituting, BOTH equations MUST be a true statement!
Example #1
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Solve the linear system graphically. Check
your solution algebraically.
x + y = -2
2x – 3y = -9
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The first step is to write the equations in a
format that is easy to graph…
Step 1 – Rewrite Equations
x + y = -2
-x
-x
y = -x - 2
2x – 3y = -9
-2x
-2x
- 3y = -2x – 9
-3
-3
y = 2/3x + 2
m = -1
m = 2/3
b = -2
b = +2
Now that you have both equations in slope-intercept form,
the next step is to graph the equations on the same
coordinate plane and determine the point where the 2 lines
intersect
Step 2 – Graph Equations
y = -x - 2
y
m = -1, b = -2
y = 2/3x + 2
m = 2/3, b = +2
x
Step 3 – Determine Intersection
y
After graphing, it appears
that the 2 lines intersect
at the point (-3, 1)
x
The next step is to
substitute this point into
both equations and solve
algebraically
Step 4 – Check Solution Algebraically
Substitute the Ordered pair (-3, 1) into each
equation and solve algebraically.
x+y=-2
-3 + 1 = - 2
-2 = - 2 √
TRUE
2x – 3y = -9
2(-3) – 3(1) = -9
-6 – 3 = - 9
-9 = - 9 √
TRUE
Because (-3, 1) is the solution to BOTH
equations it is the solution of this system of
linear equations
Comments
The method used to solve the example is
called the graph & check method.
 Sometimes you are given a graph with the
system of linear equations already
graphed.
 If that is the case, determine the
intersection point and substitute the
ordered pair into both equations to
determine if it is the solution
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Comments
In this example you could probably figure
out the solution in your head…However,
that is not the most efficient way to work
with these types of problems…
 Additionally, the graph and check method
is not the only way to solve a system of
linear equations…
 We will explore more methods as we go
through this chapter…
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Comments
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On the next couple of slides are some practice
problems…The answers are on the last slide…
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Do the practice and then check your answers…If
you do not get the same answer you must
question what you did…go back and problem
solve to find the error…
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If you cannot find the error bring your work to
me and I will help…
Your Turn
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Determine if the ordered pair is a solution to the
system of linear equations
1.
3x – 2y = 11
-x + 6y = 7
6x – 3y = -15
2x + y = -3
x + 3y = 15
4x + y = 6
-5x + y = 19
x – 7y = 3
-15x + 7y = 1
3x – y = 1
2.
3.
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5.
(5, 2)
(-2,1)
(3, -6)
(-4, -1)
(3 , 5)
Your Turn
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Graph & check to solve the linear system
6.
y = -x + 3
y=x+1
y = 2x – 4
y=-½x+1
5x + 4y = 16
y = -16
3x + 6y = 15
-2x + 3y = -3
1/5x + 3/5y = 12/5
-1/5x + 3/5y = 6/5
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Your Turn Solutions
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Solution
Solution
Not a solution
Solution
Not a solution
(1, 2)
7. (2,0)
8. (16,-16)
9. (3,1)
10. (3,3)
6.
Summary
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A key tool in making learning effective is being able
to summarize what you learned in a lesson in your
own words…
In this lesson we talked about solving linear
systems by graphing. Therefore, in your own
words summarize this lesson…be sure to include
key concepts that the lesson covered as well as any
points that are still not clear to you…
I will give you credit for doing this lesson…please
see the next slide…
Credit
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I will add 25 points as an assignment grade for you working
on this lesson…
To receive the full 25 points you must do the following:
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Have your name, date and period as well a lesson number
as a heading.
Do each of the your turn problems showing all work
Have a 1 paragraph summary of the lesson in your own
words
Please be advised – I will not give any credit for work
submitted:
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Without a complete heading
Without showing work for the your turn problems
Without a summary in your own words…