Linear Systems
Copyright 2014 Scott Storla
A 2 x 2 linear system is a set containing two
linear functions.
A linear system can be presented symbolically,
as a graph, as a data table or in English.
x
4
4
y  x 3
y  2x  1
Copyright 2014 Scott Storla
y1
10
10
y2
1
7
An ordered pair “solves” a system if the
ordered pair makes both equations true.
Copyright 2014 Scott Storla
Checking the solution of a
system of linear equations
Copyright 2014 Scott Storla
An ordered pair solves a system if the ordered
pair makes both equations true.
y  2x  3
Does  2, 7  solve the system
?
y  3x  2
y  3x  2
y  2x  3
7  2  2   3
7  3  2   2
7  7
7  8
 2, 7 
does not solve the system.
Copyright 2014 Scott Storla
Does 1, 1 solve the system
y  2x  3
?
y  3x  2
y  4x  2
Does  3,14  solve the system
?
y  4 x  2
Copyright 2014 Scott Storla
“Solving” a system graphically
Copyright 2014 Scott Storla
A student in a math class has decided to hire a tutor. One
tutor, Matt, asks $100 for the semester and then charges
$10 per hour. A second tutor, Keisha, charges a flat rate of
$15 per hour. Which tutor is the “better buy”?
For Matt y  10x 100
Matt
For Keisha
y  15x  0
Keisha
Copyright 2014 Scott Storla
A student in a math class has decided to hire a tutor. One
tutor, Matt, asks $100 for the semester and then charges
$10 per hour. A second tutor, Keisha, charges a flat rate of
$15 per hour. Which tutor is the “better buy”?
Matt is the better buy.
Keisha is the better buy.
Copyright 2014 Scott Storla
Graph each system to estimate the quadrant where the lines
intersect and the point of intersection.
x
5
2
y1 y 2
9 3
6 0
Copyright 2014 Scott Storla
Graph each system to estimate the quadrant where the
lines intersect and the point of intersection.
x
4
4
y1
10
10
y2
1
7
Copyright 2014 Scott Storla
Graph each system to estimate the quadrant where the
lines intersect and the point of intersection.
x
3
6
y1 y 2
1 4
2 0
Copyright 2014 Scott Storla
Graphing using the intercept method
Copyright 2014 Scott Storla
Graph using the intercept method.
y  2x  4
x
y
0 4
2 0
Copyright 2014 Scott Storla
Graph using the intercept method.
y  x  1
x
y
0
1
1
0
Copyright 2014 Scott Storla
Graph using the intercept method.
1
y  x2
2
x
y
0 2
4 0
Copyright 2014 Scott Storla
Graph using the intercept method.
y  x
x
y
0
0
3 3
0
0
Copyright 2014 Scott Storla
Graph each system to estimate the quadrant where
the lines intersect and the point of intersection.
y  x 3
y  2x  1
y  2x  7
y  x4
y  x 7
y  x  5
y  x  2
y  2x
Copyright 2014 Scott Storla
Applying Linear Systems
Copyright 2014 Scott Storla
A student in a math class has decided to hire a tutor. One
tutor, Matt, asks $100 for the semester and then charges
$10 per hour. A second tutor, Keisha, charges a flat rate of
$15 per hour. Which tutor is the “better buy”?
Copyright 2014 Scott Storla
A student in a math class has decided to hire a tutor. One
tutor, Matt, asks $100 for the semester and then charges $10
per hour. A second tutor, Keisha, charges a flat rate of $15 per
hour. Which tutor is the “better buy”?
For Matt
y  10x 100
Matt
For Keisha
y  15x  0
Keisha
Property – The Transitive Property of Equality
Two expressions, equal to a third expression, are equal to
each other.
Copyright 2014 Scott Storla
A student in a math class has decided to hire a tutor. One
tutor, Matt, asks $100 for the semester and then charges $10
per hour. A second tutor, Keisha, charges a flat rate of $15 per
hour. Which tutor is the “better buy”?
Matt's Function
y  10 x  100
yMatt  yKeisha
10x 100 15x
100  5x
20  x
Keisha's Function
y  15x  0
y  10x 100
Matt
y  10(20)100
Matt
y  300
Matt
If you need 20 hours of tutoring or less go with Keisha. If
you need more than 20 hours of tutoring go with Matt.
Copyright 2014 Scott Storla
Company A charges $200 plus $50 a day to rent a backhoe,
while Company B charges $350 plus $35 a day to rent the
same type of backhoe. Which company is the better buy?
Company A
y A  200  50x
Company B
yB  350  35x
y A  yB
yB  350  35x
200  50x  350  35x
15x 150
yB  350  35(10)
x 10
If you rent for 10 days or less go with Company
A otherwise go with Company B.
Copyright 2014 Scott Storla
yB  700
A student can drive to school or take the bus. If they drive
they estimate the cost is $3.50 per day for gas and $2.50 per
day for parking. If they take the bus the cost is $30.00 a
month for a bus card which allows them unlimited rides for
$2.00 per day. If they want to save money, which is their best
choice for the month?
Drive y  6x
Bus y  2x  30
Drive  Bus
6x  2x  30
4x  30
x  7.5
y  6x
y  2x  30
y  6(7)
y  2(7)  30
y  42
y  44
y  6x
y  2x  30
y  6(8)
y  2(8)  30
y  48
y  46
If they go to school 7 days a month or less they
should drive otherwise they should take the bus.
Copyright 2014 Scott Storla
The data below shows meat consumption per person per year in
the United States. The first column, year, is the independent
item. The two columns that follow, beef and chicken, should both
be considered a separate dependent item. Estimate the year
chicken consumption will overtake beef consumption.
Copyright 2014 Scott Storla
By 1970 people had been advised to change their diets to
avoid certain diet related diseases. The data below shows
milk consumption per person per year in the United States.
Estimate the year skim milk consumption will overtake whole
milk consumption.
Copyright 2014 Scott Storla
Predict the year that high fructose corn syrup
consumption will equal sugar consumption.
Year
since
1985
Sugar
High fructose
consumption corn syrup
(pounds)
(pounds)
5
63.5
50.0
10
63.5
57.6
Copyright 2014 Scott Storla
In January 2010 CD sales were at 45,000 and
began dropping by 800 per month.
In January 2010 MP3 sales began at 5,000
and began rising by 1,200 per month.
Build a linear function for both CD sales and
MP3 sales using t for the number of months
since January 2010 and S for the sales.
Use a linear system to find the year and month
within that year that MP3 sales will overtake
CD sales. Assume sales depends on the
number of months since January 2010.
Copyright 2014 Scott Storla
Some Special Cases
Copyright 2014 Scott Storla
What if the system shares no common point.
Both Matt and Keisha charge $15 per hour and Matt
still charges $100 for the semester.
The Lines are parallel ( have the same
slope) but have different y-intercepts.
Matt's Line  Keisha's Line
100 15x  0 15x
100  0
The system is inconsistent,
there is no solution.
Copyright 2014 Scott Storla
What if the system shares all points in common.
Both Matt and Keisha charge $100 per semester plus
$15 per hour.
The lines are parallel and
have the same y-intercept.
Matt's Line  Keisha's Line
100 15x 100 15x
00
The system is consistent
and dependent.
Copyright 2014 Scott Storla
The vocabulary of solutions.
If the slopes are different the system is
consistent (it has at least one solution) and
independent (it has only one solution).
If the slopes are the same and the y-intercepts
are the same the system is consistent (it has
at least one solution) and dependent (the
system has an infinite number of solutions).
If the slopes are the same and the y-intercepts
are different the system is inconsistent (there
is no solution).
Copyright 2014 Scott Storla
Use the slope and y-intercept to decide if the
system is inconsistent, consistent and
dependent or consistent and independent. If
the system is consistent find the solution(s).
y  2x  2
y  3x  2
x
6
4
y1
10
10
y2
1
7
x
4
2
y1
6
4
y2
2
12
1
2
y  1 x 1
4
5
y  1.25 x  1.4
y  24x  150
y  18x  270
Copyright 2014 Scott Storla