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Table of Contents
Teacher note
Teacher note:
Many of the Responder Questions have boxes over the answers. Have
the students take some time to solve the system prior to showing
them the multiple choice answers by clicking on the box, so they do
not just substitute in the answers.
Solve Systems by Graphing
Solve Systems by Substitution
Solve Systems by Elimination
Choosing your Strategy
Writing Systems to Model Situations
Solving Systems of Inequalities
Click on topic to go to that
section.
Strategy One:
Graphing
Return to
Table of
Contents
Some vocabulary...
A "system" is two or more
linear equations.
The "solution" to a system is an ordered pair that will work in each
equation. One way to find the solution is to graph the equations on the
same coordinate plane and find the point of intersection.
Consider this...
Suppose you are walking to school. Your friend is 5 blocks ahead of you.
You can walk two blocks per minute, your friend can walk one block per
minute. How many minutes will it take for you to catch up with your
friend?
First, make a table to represent the problem.
Friend's
Time
(min. distance
from your
)
start
(blocks)
5
0
Your distance
from your
start (blocks)
0
1
6
2
2
7
4
3
8
6
4
9
8
5
10
10
Next, plot the points on a graph.
20
Tim Friend's
distance
e
(mi from
your
n.)
15
Blocks
start
(blocks)
10
5
0
0
5
10
Time (min.)
15
Your
distance
from your
start(block
s)
0
5
0
1
6
2
2
7
4
3
8
6
4
9
8
5
10
10
The point where they intersect is the solution to the system.
20
(5,10) is the solution. In the
context of the problem this
means after 5 minutes, you
will meet your friend at
block 10.
Blocks
15
10
5
0
0
5
10
Time (min.)
15
Solve the system of equations graphically.
Solution
y = 2x -3
y=x-1
Solve the system of equations graphically.
Solution
2x + y = 3
x - 2y = 4
Solve the system of equations graphically.
Solution
3x + y = 11
x - 2y = 6
Solve using graphing
Write the
equation for
the green
dashed line
move
y = 4x+6
y =move
-3x-1
What is this point
of intersection?
(move the hand!)
(-1, 2)
Write the
equation for
the blue
solid line
Now take the ordered pair we just found and substitute it into the equation to
prove that it is a solution for both lines.
( -1,
y = -3x-1
)2
y = 4x+6
Solve by Graphing
y = 2x + 3
y = -4x - 3
(-1,1)
Solve by Graphing
y= -3x + 4
y= x - 4
(2,-2)
What's the problem here?
y= 2x + 4
y= 2x - 4
Therefore there
is no solution.
Parallel
lines
do not
intersect!
click to reveal
(
No ordered
pair that will
work in BOTH
equations
click to reveal
)
Solve by Graphing
First - transform the equations into y = mx + b
(slope-intercept form)
2x + y = 5
-2x
-2x
y = -2x + 5
2y = -4x + 10
2
2
y = -2x + 5
Now graph the two transformed lines.
form
What's the problem?
2x + y = 5
becomes
y = -2x + 5
The equations
transform to
the same line.
click to reveal
2y = 10 -4x
becomes
y = -2x + 5
So we have
infinitely many
solutions.
click to reveal
Solve the system by graphing.
y = -x + 4
y = 2x +1
Click for multiple choice answers.
A
(3,1)
B
(1,3)
C
(-1,3)
D
no solution
Solution
1
Click for multiple choice answers.
A
(0,-1)
B
(0,0)
C
infinitely many
D
no solution
Solution
Solve the system by graphing.
y = 0.5x - 1
y = -0.5x -1
2
Click for multiple choice answers.
A
(2,4)
B
(0.4, 2.2)
C
(2, -1)
D
no solution
Solution
Solve the system by graphing.
2x + y = 3
x - 2y = 4
3
Solve the system by graphing.
y = 3x + 3
y = 3x - 3
Click for multiple choice answers.
A
(0,0)
B
(3,3)
C
infinitely many
D
no solution
Solution
4
Solve the system by graphing.
y = 3x + 4
4y = 12x + 16
Click for multiple choice answers.
A
(3,4)
B
(-3,-4)
C
infinitely many
D
no solution
Solution
5
On the accompanying set of axes, graph and label the following lines:
y=5
x=-4
y = x+5
Calculate the area,
in square units,
of the triangle formed
by the three points
of intersection.
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Solution
6
Strategy Two:
Substitution
Return to
Table of
Contents
Solve the system of equations graphically.
NOTE
y = x + 6.1
y = -2x - 1.4
Substitution Explanation
Graphing can be inefficient or approximate.
Another way to solve a system is to use substitution.
Substitution allows you to create a one variable equation.
Solve the system using substitution.
Why was it difficult to solve this system by graphing?
y = x + 6.1
y = -2x - 1.4
y = -2x - 1.4 -start with one equation
x + 6.1 = -2x - 1.4 -substitute x + 6.1 for y in equation
+2x -6.1 +2x - 6.1
3x = -7.5 -solve for x
x = -2.5
Substitute -2.5 for x in either equation and solve for y.
y = x + 6.1
y = (-2.5) + 6.1
y = 3.6
Since x = -2.5 and y = 3.6, the solution is (-2.5, 3.6)
CHECK: See if (-2.5, 3.6) satisfies the other equation.
y = -2x - 1.4
3.6 = -2(-2.5) - 1.4
3.6 = 5 - 1.4
3.6 = 3.6
?
?
NOTE
Solve the system using substitution.
y=
-2x +14
(
)
Solution
-3 y + 3x = 21
Solve the system using substitution.
x=
-5y - 39
(
)
Solution
x = -y - 3
Examine each system of equations.
Which variable would you choose to substitute?
Why?
y = 4x - 9.6
y = -2x + 9
y = -3x
7x - y = 42
y = 4x + 1
x = 4y + 1
7
Examine the system of equations.
Which variable would you substitute?
Solution
2x + y = 5
2y = 10 - 4x
A
x
B
y
Examine the system of equations.
Which variable would you substitute?
2y - 8 = x
y + 2x = 4
A
x
B
y
Solution
8
Examine the system of equations.
Which variable would you substitute?
x - y = 20
2x + 3y = 0
A
x
B
y
Solution
9
Sometimes you need to rewrite one of the equations so that you can use the substitution method
For example:
The system: Is equivalent to:
3x -y = 5 y = 3x -5
2x + 5y = -8 2x + 5y = -8
Using substitution you now have:
2x + 5(3x-5) = -8 -solve for x
2x + 15x - 25 = -8 -distribute the 5
17x - 25 = -8 -combine x's
17x = 17 -at 25 to both sides
x = 1 - divide by 17
Substitute x = 1 into one of the equations.
2(1) + 5y = -8
2 + 5y = -8
5y = -10
y = -2
The ordered pair (1,-2) satisfies both equations in the original system.
3x -y = 5
2x + 5y = -8
3(1) - (-2) = 5 2(1) + 5(-2) = -8
3 + 2 = 5 2 - 10 = -8
-8 = -8
Your class of 22 is going on a trip. There are four drivers and two types of
vehicles, vans and cars. The vans seat six people, and the cars seat four people,
including drivers. How many vans and cars does the class need for the trip?
Let v = the number of vans
and c = the number of cars
Set up the system:
Drivers: v + c = 4
People: 6v + 4c = 22
Solve the system by substitution.
v + c = 4 -solve the first equation for v.
v = -c + 4 -substitute -c + 4 for v in the
6(-c + 4) + 4c = 22
second equation
-6c + 24 + 4c = 22 -solve for c
-2c + 24 = 22
-2c = -2
c=1
v+c=4
v + 1 = 4 -substitute for c in the 1st equation
v = 3 -solve for v
Since c = 1 and v = 3, they should use 1 car and 3 vans.
Check the solution in the equations:
v + c = 4 6v + 4c = 22
3 + 1 = 4 6(3) + 4(1) = 22
4=4
18 + 4 = 22
22 = 22
Now solve this system using substitution. What happens?
x+y=6
5x + 5y = 10
x + y = 6 -solve the first equation for x
x=6-y
5(6 - y) + 5y = 10 -substitute 6 - y for x in 2nd equation
30 - 5y + 5y = 10 -solve for y
30 = 10 -FALSE!
Since 30 = 10 is a false statement,
the system has no solution.
Now solve this system using substitution. What happens?
x + 4y = -3
2x + 8y = -6
x + 4y = -3 - solve the first equation for x
x = -3 - 4y
2(-3 - 4y) + 8y = -6 - sub. -3 - 4y for x in 2nd equation
-6 - 8y + 8y = -6 - solve for y
-6 = -6 - TRUE!
- there are infinitely many solutions
How can you quickly decide the number of solutions
a system has?
1 Solution
Different slopes
No Solution
Same slope; different yintercept (Parallel Lines)
Infinitely Many
Same slope; same y-intercept
(Same Line)
3x - y = -2
y = 3x + 2
Solution
10
A
1 solution
B
no solution
C
infinitely many solutions
11
3x + 3y = 8
y= x 1
A
1 solution
B
no solution
C
infinitely many solutions
Solution
3
y = 4x
2x - 0.5y = 0
A
1 solution
B
no solution
C
infinitely many solutions
Solution
12
3x + y = 5
6x + 2y = 1
Solution
13
A
1 solution
B
no solution
C
infinitely many solutions
y = 2x - 7
y = 3x + 8
Solution
14
A
1 solution
B
no solution
C
infinitely many solutions
Solve each system by substitution.
y=x-3
y = -x + 5
Click for multiple choice answers.
A
(4,9)
B
(-4,-9)
C
(4,1)
D
(1,4)
Solution
15
Solve each system by substitution.
y=x-6
y = -4
Solution
16
Click for multiple choice answers.
A
(-10,-4)
B
(-4,2)
C
(2,-4)
D
(10,4)
Solve each system by substitution.
y + 2x = -14
y = 2x + 18
Solution
17
Click for multiple choice answers.
A
(1,20)
B
(1,18)
C
(8,-2)
D
(-8,2)
Solve each system by substitution.
4x = -5y + 50
x = 2y - 7
Solution
18
Click for multiple choice answers.
A
(6,6.5)
B
(5,6)
C
(4,5)
D
(6,5)
Solve each system by substitution.
y = -3x + 23
-y + 4x = 19
Click for multiple choice answers.
A
(6,5)
B
(-7,5)
C
(42,-103)
D
(6,-5)
Solution
19
Strategy Three:
Elimination
Return to
Table of
Contents
When both linear equations of a system are in Standard Form,
Ax + By = C, you can solve the system using elimination.
You can add or subtract the equations to eliminate a variable.
How do you decide which variable to eliminate?
First, look to see if one variable has the same or opposite
coefficients. If so, eliminate that variable.
Second, look for which coefficients have a simple least common
multiple. Eliminate that variable.
If the variables have the same coefficient, you can subtract the two equations to eliminate
the variable.
If the variables have opposite coefficients, you add the two equations to eliminate the
variable.
Sometimes, you need to multiply one, or both, equations by a number in order to create a
common coefficient.
Solve by Elimination - Click on the terms to eliminate and they
will disappear, then add the two equations together.
5x + y = 44
( -4x - y = -34
)
Solve by Elimination - Click on the terms and they will
disappear then add the two equations together.
3x + y = 15
-3x -3y
( = -21
)
One method is to recognize that the y-coefficient
is the same. You can multiply the second
equation by -1. This
opposite
5x + will
y = create
17
coefficients for -2x
the +
yy
variable.
Then, add the
=
-4
)
-1 (
two equations.
5x + y = 17
2x - y = 4
-
(
5x + y = 17
-2x + y = -4
7x
= 21
Subtraction
Multiplication by
-1
Solve by Elimination - There are 2 ways to complete this
problem. See both examples.
)
Solve the system by elimination.
Pull
4x + 3y = 16
2x - 3y = 8
20
Solution
Solve each system by elimination.
x+y=6
x-y=4
Click for multiple choice answers.
A
(5,1)
B
(-5,-1)
C
(1,5)
D
no solution
Solve each system by elimination.
2x + y = -5
2x - y = -3
Solution
21
Click for multiple choice answers.
A
(-2,1)
B
(-1,-2)
C
(-2,-1)
D
infinitely many
22
Solution
Solve each system by elimination.
2x + y = -6
3x + y = -10
(4,2) choice answers.
A Click for multiple
B
(3,5)
C
(2,4)
D
(-4,2)
23
Solution
Solve each system by elimination.
4x - y = 5
x - y = -7
Click for multiple choice answers.
A
no solution
B
(4,11)
C
(-4,-11)
D
(11,-4)
24
Click for multiple choice answers.
A
(2,-7)
B
(7,2)
C
(2,7)
D
infinitely many
Solution
Solve each system by elimination.
3x + 6y = 48
-5x + 6y = 32
Sometimes, it is not possible to eliminate a variable by adding or subtracting
the equations.
When this is the case, you need to multiply one or both equations by a
nonzero number in order to create a common coefficient. Then add or subtract
the equations.
Examine each system of equations.
Which variable would you choose to eliminate?
What do you need to multiply each equation by?
2x + 5y = -1
x + 2y = 0
3x + 8y = 81
5x - 6y = -39
3x + 6y = 6
2x - 3y = 4
In order to eliminate the y, you need to multiply first.
3x + 4y = -10
5x - 2y = 18
Multiply the second equation by 2 so the coefficients are opposites.
2(5x - 2y = 18)
10x - 4y = 36
Now solve by adding the equations together.
3x + 4y = -10
10x - 4y = 36
13x = 26
x=2
+
Solve for y, by substituting x = 2 into one of the equations.
3x + 4y = -10
3(2) + 4y = -10
6 + 4y = -10
4y = -16
y = -4
So (2,-4) is the solution.
Check:
3x + 4y = -10
5x - 2y = 18
3(2) + 4(-4) = -10
5(2) - 2(-4) = 18
6 + -16 = -10
10 + 8 = 18
-10 = -10 18 = 18
Now solve the same system by eliminating x. What do you multiply the two equations
by?
3x + 4y = -10
5x - 2y = 18
Multiply the first equation by 5 and the second equation by 3 so the coefficients will
be the same
5(3x + 4y = -10) 3(5x - 2y = 18)
15x + 20y = -50
15x - 6y = 54
Now solve by subtracting the equations.
15x + 20y = -50
15x - 6y = 54
26y = -104
y = -4
-
Solve for x, by substituting y = -4 into one of the equations.
3x + 4y = -10
3x + 4(-4) = -10
3x + -16 = -10
3x = 6
x=2
So (2,-4) is the solution. Check:
3x + 4y = -10
5x - 2y = 18
3(2) + 4(-4) = -10
5(2) - 2(-4) = 18
6 + -16 = -10
10 + 8 = 18
-10 = -10 18 = 18
25
A
x
B
y
9x + 6y = 15
-4x + y = 3
Solution
Which variable can you eliminate with the least amount of work?
26
A
x
B
y
3x - 7y = -2
-6x + 15y = 9
Solution
Which variable can you eliminate with the least amount of work?
27
A
x
B
y
x - 3y = -7
2x + 6y = 34
Solution
Which variable can you eliminate with the least amount of work?
28
What will you multiply the first equation by in order to solve this system using
elimination?
2x + 5y = 20
3x - 10y = 37
Now solve it....
2
(11, - )5
click for answer
29
What will you multiply the first equation by in order to solve this system using
elimination?
3x + 2y = -19
x - 12y = 19
Now solve it....
(-5,-2)
click for answer
30
What will you multiply the first equation by in order to solve this system using
elimination?
x + 3y = 4
3x + 4y = 2
Now solve it....
(-2,2)
click for answer
Choose Your Strategy
Return to
Table of
Contents
Altogether 292 tickets were sold for a basketball game. An adult ticket costs
$3. A student ticket costs $1. Ticket sales were $470.
Let a = adults
s = students
Set up the system:
number of tickets sold: a + s = 292
money collected:
3a + s = 470
First eliminate one variable.
a + s = 292 - in both equations s has the same
3a + s = 470
coefficient so you subtract the 2
-2a+ 0 = -178 equations in order to eliminate it.
a = 89 -solve for a
)
-(
Then, find the value of the eliminated variable.
a + s = 292
89 + s = 292 -substitute 89 for a in 1st equation
s = 203 -solve for s
There were 89 adult tickets and 203 student tickets sold.
(89, 203)
Check:
a + s = 292 3a + s = 470
89 + 203 = 292 3(89) + 203 = 470
292 = 292 267 + 203 = 470
470 = 470
A piece of glass with an initial temperature of 99 F is cooled at a rate of 3.5 F/min.
At the same time, a piece of copper with an initial temperature of 0 F is heated at a
rate of 2.5 F/min. Let m = the number of minutes and t = the temperature in F.
Which system models the given information?
Solution
31
A
B
C
t = 99 + 3.5m
t = 0 + 2.5m
t = 99 - 3.5m
t = 0 + 2.5m
t = 99 + 3.5m
t = 0 - 2.5m
32
Which method would you use to solve the system?
A
graphing
B
substitution
t = 99 - 3.5m
t = 0 + 2.5m
C
elimination
click for equations
Now solve it...
m = 16.5
t = 41.25
This means that in 16.5 minutes, the
temperatures will both be 41.25℃.
click for answer
33
What method would you choose to solve the system?
Solution
4s - 3t = 8
t = -2s -1
A
graphing
B
substitution
C
elimination
Now solve the system!
Solution
34
Click for multiple choice answers.
A
(
1
, 2-2)
1
2)
2
B
( ,
C
(2 , -2)
D
4s - 3t = 8
t = -2s -1
(-2, )
1
2
A
graphing
B
substitution
C
elimination
y = 3x - 1
y = 4x
Solution
What method would you choose to solve the system?
35
Solution
Now solve it!
36
Click for multiple choice answers.
A
(1, 4)
B
(-4, -1)
C
(-1, 4)
D
(-1, -4)
y = 3x - 1
y = 4x
What method would you choose to solve the system?
Solution
37
A
graphing
B
substitution
C
elimination
3m - 4n = 1
3m - 2n = -1
Solution
Now solve it!
38
Click for multiple choice answers.
A
(-2, -1)
B
(-1, -1)
C
(-1, 1)
D
(1, 1)
3m - 4n = 1
3m - 2n = -1
A
graphing
B
substitution
C
elimination
y = -2x
y = -0.5x + 3
Solution
What method would you choose to solve the system?
39
Now solve it!
Solution
40
Click for multiple choice answers.
A
(-6, 12)
B
(2, -4)
C
(-2, 4)
D
(1, -2)
y = -2x
y = -0.5x + 3
A
graphing
B
substitution
C
elimination
2x - y = 4
x + 3y = 16
Solution
What method would you choose to solve the system?
41
Now solve it!
Solution
42
Click for multiple choice answers.
Click for multiple choice answers.
(6, 5)
AClick for multiple
choice answers.
B
(-4, 7)
C
(-4, 4)
D
(4, 4)
2x - y = 4
x + 3y = 16
A
graphing
B
substitution
C
elimination
u = 4v
3u - 3v = 7
Solution
What method would you choose to solve the system?
43
Now solve it!
Solution
44
Click for multiple choice answers.
A
(
, 289 )
B
(
,
C
(28, 7)
D
(7,
)
7
9
)
7
9
28
9
7
4
u = 4v
3u - 3v = 7
Choose a strategy and then answer the question.
What is the value of the y-coordinate of the solution to the system of equations x − 2y = 1
and x + 4y = 7?
A
1
B
-1
C
3
D
4
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Solution
45
Modeling
Situations
Return to
Table of
Contents
A group of 148 people is spending five days at a summer camp. The cook ordered 12 pounds of food
for each adult and 9 pounds of food for each child. A total of 1,410
pounds of food was ordered.
Pull
Part A: Write an equation or a system of equations that describes the above situation and define your
variables.
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Part B: Using your work from part A, find:
(1) the total number of adults in the group
Pull
(2) the total number of children in the group
Pull
Tanisha and Rachel had lunch at the mall. Tanisha ordered three slices of pizza and two colas. Rachel
ordered two slices of pizza and three colas. Tanisha’s bill was $6.00,
and Rachel’s bill was $5.25. What was the price of one slice of pizza? What was the price of one cola?
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Pull
Sharu has $2.35 in nickels and dimes.
If he has a total of thirty-two coins, how many of each coin does he have?
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Pull
Ben had twice as many nickels as dimes.
Altogether, Ben had $4.20.
How many nickels and how many dimes did Ben have?
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Your class receives $1105 for selling 205 packages of greeting cards and
gift wrap. A pack of cards costs $4 and a pack of gift wrap costs $9
Set up a system and solve. How many packages of cards were sold?
You will answer how many
packages of gift wrap in the next
question.
Solution
46
Your class receives $1105 for selling 205 packages of greeting cards and
gift wrap. A pack of cards costs $4 and a pack of gift wrap costs $9
Set up a system and solve. How many packages of gift wrap were sold?
Solution
47
48
A
16
B
31
C
32
D
36
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Solution
The sum of two numbers is 47, and their difference is 15. What is the larger number?
49
Ramon rented a sprayer and a generator. On his first job, he used each piece of equipment
for 6 hours at a total cost of $90. On his second job, he used the sprayer for 4 hours and
the generator for 8 hours at a total cost of $100. What was the hourly cost for the sprayer?
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
50
What is true of the graphs of the two lines
3y - 8 = -5x and 3x = 2y -18?
A
no intersection
B
intersect at (2,-6)
C
intersect at (-2,6)
D
are identical
51
You have 15 coins in your pocket that are either quarters or nickels. They total
$2.75. Set up a system to solve. Which method will you use? (Solving it comes
later...)
A
graphing
B
substitution
C
elimination
You have 15 coins in your pocket that are either quarters or nickels. They total
$2.75. How many quarters do you have?
Solution
52
You have 15 coins in your pocket that are either quarters or nickels. They total
$2.75. How many nickels do you have?
Solution
53
54
Julia went to the movies and bought one jumbo popcorn and two chocolate chip cookies
for $5.00. Marvin went to the same movie and bought one jumbo popcorn and four
chocolate chip cookies for $6.00. How much does one chocolate chip cookie cost?
A
$0.50
B
$0.75
C
$1.00
D
$2.00
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Mary and Amy had a total of 20 yards of material from which to make costumes. Mary
used three times more material to make her costume than Amy used, and 2 yards of
material was not used. How many yards of material did Amy use for her costume?
Solution
55
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
The tickets for a dance recital cost $5.00 for adults and $2.00 for children. If the total
number of tickets sold was 295 and the total amount collected was $1220, how many
adult tickets were sold?
Solution
56
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Solving Systems of Inequalities
Return to
Table of
Contents
Two or more linear inequalities form a system of inequalities.
A solution to the system is an ordered pair that is a solution of each inequality
in the system.
Since inequalities have more than one solution, the solutions are best shown
in a graph.
Graphing a System of Linear Inequalities
1.
Graph the boundary lines of each inequality.
(Remember use a dashed line for < and >
and a solid line for < and >)
2. Shade the half-plane for each inequality.
3. The intersection of the half-planes is the solution.
Solve the system of inequalities.
x + 2y < 6
-x + y < 0
Pull
Solve the system of inequalities.
2x + y > -4
x - 2y < 4
Pull
Solve the system of inequalities.
4x + 2y < 8
4x + 2y > -8
Pull
Graph the following system of inequalities on the set of axes shown below
solution set S.
y > −x + 2
and label the
3
Pull
y≤ 2x+5
Pull
A company manufactures bicycles and skateboards. The company’s daily production of bicycles cannot
exceed 10, and its daily production of skateboards must be less than or equal to 12. The combined
number of bicycles and skateboards cannot be more than 16. If x is the number of bicycles and y is the
number of skateboards, graph on the accompanying set of axes the region that contains the number of
bicycles and skateboards the company can manufacture daily.
Pull
Write a system of inequalities from the graph.
57
Solve the system of linear inequalities.
A
Solution
y > -2x + 1
y<x+2
B
C
Solve the system of linear inequalities.
Solution
58
x>2
y<5
A
B
C
Solve the system of linear inequalities.
Solution
59
-2x - 2y < 4
y - 2x > 1
A
B
C
Solve the system of linear inequalities.
Solution
60
-5x + y > -2
4x + y < 1
A
B
C
Solve the system of linear inequalities.
Solution
61
3x + 2y < 12
2x - 2y < 20
A
B
C
Which point is in the solution set of the system of inequalities shown in
the accompanying graph?
Solution
62
A
(0,4)
B
(2,4)
C
D
(-4,1)
(4,-1)
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Which ordered pair is in the solution set of the system of inequalities shown in the
accompanying graph?
Solution
63
A
(0, 0)
B
(0, 1)
C
(1, 5)
D
(3, 2)
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
64
Which ordered pair is in the solution set of the following system of linear inequalities?
A
(0,3)
B
(2,0)
C
(−1,0)
D
(−1,−4)
Solution
y < 2x + 2
y ≥ −x − 1
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 1
June, 2011.
Mr. Braun has $75.00 to spend on pizzas and soda for a picnic. Pizzas cost $9.00 each
and the drinks cost $0.75 each. Five times as many drinks as pizzas are needed. What is
the maximum number of pizzas that Mr. Braun can buy?
Solution
65
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.