Download MJ2A - Ch 8.9 System of Equations

MJ2A
Ch 8.9 – System of
Equations
Bellwork

Solve for y, given the value of x
Solutions
1.
2.
3.
4.
y=x+1
y=x–4
x + y = -1
x+y=0
x=2
x = -1
x=1
x=4
3
-1
-2
-4
Assignment Review

Text p. 407 – 408 # 15 – 25
Before We Begin…



Please take out your notebook and get
ready to work…
In the last lesson we worked with writing
linear equations using the slopeintercept form.
In today’s lesson we will look at pairs of
linear equations and their common
solutions…
Objective 8.9

Students will solve systems of linear
equations by the graphing & substitution
methods
Vocabulary


Systems of Equations are sets of linear
equations.
Solution – is the common ordered pair
that make the system of equations true.
System of Equations


There are several methods to solving
systems of equations.
In today’s lesson we will look at
• Solving by graphing
• Solving by substitution
Solving by Graphing
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
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When solving systems of equations by graphing
you will be presented with or asked to graph 2
linear equations in the same coordinate plane.
When analyzing the graphs there are three
possible scenarios as follows:
 One Solution
 No Solution
 Infinite Solutions
Let’s see what they look like…
One Solution


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When analyzing a graph with systems of
equations, the lines will intersect.
The intersection point represents the solution of
both equations
You can check the solution by substituting the
ordered pair into both equations and you should
get a true statement for both equations
Let’s look at an example…
One Solution
y

This is the graph of the
equations
y = -x
y=x+2

Notice that the lines
intersect at the point (-1, 1)

The ordered pair
represents the solution
to the system of
equations
x
(-1, 1)
Checking the Solution

You can check to make sure that the ordered pair
(-1, 1) is the solution by substituting the x and yvalues into the original equations as follows:
Equation #1
Equation #2
y = -x
y=x+2
1 = - (-1)
1 = -1 + 2
1=1 
1=1 
Observation


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Observe that the slope of each equation is
different.
y = -x
m = -1
y=x+2 m=1
Without doing any work, we can analyze the
slopes of the equations and predict how many
solutions there will be to a system of equations
If the slopes are different, then there will be
only one solution to the system of equations
No Solution



In a graph with 2 parallel lines there will
be no solution to the system of equations
In this instance, the lines do not intersect.
Therefore, no ordered pair will be the
solution to both equations…
Let’s see what that looks like…
No Solution

This is a graph of the equations
y = 2x + 4
y = 2x - 1
Notice that the lines are
parallel and do not intersect
Because the lines do not
intersect there is no
solution to this system of
equations
x
y
Observation


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Observe the slope an y-intercepts of each equation.
y = 2x + 4 m = 2, b = 4
y = 2x - 1
m = 2, b = -1
Without doing any work, we can analyze the slopes
and y-intercepts of the equations and predict how
many solutions there will be to the system of
equations
If the slopes are same, with different y-intercepts
then there will be no solution to the system of
equations
Infinite Solutions
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
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In a system of equations where the
graph of each line is the same you will
have an infinite number of solutions.
Because the graphs of each line overlap,
all points on each line will make the
system of equations true, therefore you
will have an infinite number of solutions
Let’s look at an example…
Infinite Solutions

This is a graph of the equations
2y = x + 6
y=½x+3
Both equations have
the same graph
Any ordered pair on
the lines will make
both equations true
x
y
Observation


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Observe that the slope an y-intercepts of each
equation.
2y = x + 6
m=½,b=3
y=½x+3
m = ½, b = 3
If you transform the first equation into the slopeintercept form you will see that both equations have
the same slope and y-intercept
If the slopes and y-intercepts are the same then
there will be an infinite number of solutions to the
system of equations
Systems of Equations by
Substitution
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You can also solve a system of equations by
the substitution method.
First, you need to know that you cannot
solve an equation with two variables.
In the substitution method you substitute the
value of one variable into the other
equation…
Let’s see what that looks like…
Example
Solve
y=x–5
y=3
In this instance the second equations gives you the value of y.
To solve, substitute the value of y into the first equation and solve algebraically as
follows:
y=x–5
3=x–5
+5
+5
8=x
The solution to the system of
equation is x = 8 and y = 3
The ordered pair that will make this
system of equation true is (8, 3)
Your Turn

In the notes section of your notebook
write and solve the system of equations
using the substitution method.
Solutions
1.
2.
y = 3x – 4
x=0
x+y=8
y=6
(0, -4)
(2, 6)
Summary
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
In the notes section of your notebook
summarize the key concepts covered in
today’s lesson
Today we discussed:
• Solving systems of equations by graphing –
•
what are the solution scenarios?
Solving systems of equations by substituting –
how does that work?
Assignment
Text p. 417 # 12 – 23
Reminder

• This assignment is due tomorrow
• I do not accept late assignments
• You must show how you got your answers or
no credit (no work = no credit)