### ch.8 active learning

```Active Learning Lecture Slides
For use with Classroom Response Systems
Chapter 8
Systems of
Equations and
Inequalities
© 2009 Pearson Education, Inc.
All rights reserved.
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 1
Solve the system of
equations by substitution.
a.
x  12, y  2;
b.
x  2, y  3;
c.
x  3, y  7;
d.
x  7, y  12;
Copyright © 2009 Pearson Education, Inc.
 x  7y  2

 3x  y  34
12, 2 
2, 3
3, 7 
7,12 
Slide 8 - 2
Solve the system of
equations by substitution.
a.
x  12, y  2;
b.
x  2, y  3;
c.
x  3, y  7;
d.
x  7, y  12;
Copyright © 2009 Pearson Education, Inc.
 x  7y  2

 3x  y  34
12, 2 
2, 3
3, 7 
7,12 
Slide 8 - 3
5x  3y  80
Solve the system of
equations by elimination. 2x  y  30
10,10 
a.
x  10, y  10;
b.
x  0, y  10;
0,10 
c.
x  10, y  0;
10, 0 
d.
x  0, y  0;
Copyright © 2009 Pearson Education, Inc.
0, 0 
Slide 8 - 4
5x  3y  80
Solve the system of
equations by elimination. 2x  y  30
10,10 
a.
x  10, y  10;
b.
x  0, y  10;
0,10 
c.
x  10, y  0;
10, 0 
d.
x  0, y  0;
Copyright © 2009 Pearson Education, Inc.
0, 0 
Slide 8 - 5
Solve the system of
equations.
a.
x  7, y  9;
 7x  3y  9

28x  12y  27
7, 9 
7
1  7 1
b. x  , y   ;  ,  
6
2  6 2
c.
x  4, y  3;
4, 3
d. inconsistent
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 6
Solve the system of
equations.
a.
x  7, y  9;
 7x  3y  9

28x  12y  27
7, 9 
7
1  7 1
b. x  , y   ;  ,  
6
2  6 2
c.
x  4, y  3;
4, 3
d. inconsistent
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 7
Solve the system of
equations.
 x  4 y  3

 4 x  16 y  12
x


a. x, y  y   4  3, where x is any real number 


3
or y   x  3, where x is any real number
4
b. x  0, y  0; 0, 0 
c.
x  3, y  0;
3, 0 
d. inconsistent
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 8
Solve the system of
equations.
 x  4 y  3

 4 x  16 y  12
x


a. x, y  y   4  3, where x is any real number 


3
or y   x  3, where x is any real number
4
b. x  0, y  0; 0, 0 
c.
x  3, y  0;
3, 0 
d. inconsistent
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 9
Solve the system of
equations.
5x  5y  z  16

2x  2y  z  8
5x  y  5z  4

a.
x  3, y  4, z  1;
3, 4,1
b.
x  3, y  1, z  4;
3,1, 4 
c.
x  4, y  1, z  3;
4,1, 3
d. inconsistent
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 10
Solve the system of
equations.
5x  5y  z  16

2x  2y  z  8
5x  y  5z  4

a.
x  3, y  4, z  1;
3, 4,1
b.
x  3, y  1, z  4;
3,1, 4 
c.
x  4, y  1, z  3;
4,1, 3
d. inconsistent
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 11
9x  6y  2z  5

Write the augmented

matrix of the system of 2x  5y  5z  17
5x  8y  5z  16
equations.

a.  9 2 5 5 


 6 5 8 17 
 2 5 5 16 
b.  9 6 2 


 2 5 5 
 5 8 5 
c.  9 6 2 5 


2
5
5
17


 5 8 5 16 
d.  5 2 6 9 


17
5
5
2


 16 5 8 5 
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 12
9x  6y  2z  5

Write the augmented

matrix of the system of 2x  5y  5z  17
5x  8y  5z  16
equations.

a.  9 2 5 5 


 6 5 8 17 
 2 5 5 16 
b.  9 6 2 


 2 5 5 
 5 8 5 
c.  9 6 2 5 


2
5
5
17


 5 8 5 16 
d.  5 2 6 9 


17
5
5
2


 16 5 8 5 
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 13
Write the system of
equations corresponding
to the augmented matrix.
 4 7 7 2 


9
0
5
4


 9 3 0 2 
a.  4x  7y  7z  2

5z  4
9x
9x  3y
 2

b.  4x  7y  7z  2

5z  4
9x
9x  3y
2

c.  4x  7y  7z  2

5z  4
9x
9x
 3z  2

d.  4x  7y  7z  2

5z  4
9x
9x
 3z  2

Copyright © 2009 Pearson Education, Inc.
Slide 8 - 14
Write the system of
equations corresponding
to the augmented matrix.
 4 7 7 2 


9
0
5
4


 9 3 0 2 
a.  4x  7y  7z  2

5z  4
9x
9x  3y
 2

b.  4x  7y  7z  2

5z  4
9x
9x  3y
2

c.  4x  7y  7z  2

5z  4
9x
9x
 3z  2

d.  4x  7y  7z  2

5z  4
9x
9x
 3z  2

Copyright © 2009 Pearson Education, Inc.
Slide 8 - 15
(a) R2  4r1  r2
Perform
(b) R3  2r1  r3
the row
operations. (c) R3  6r2  r3
 1 3 5 2 


 4 5 4 5 
 2 5
4 6 
a.  1 3 5 2 


 0 7 16 13 
 0 18 30 23 
b.  1 3 5 2 


 0 8 9 3 
 0 37 40 8 
c.  1 3 5 2 


0
7
16
13


 0 53 110 88 
d.  1 3 5 2 


0
17
24
7


 0 113 158 52 
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 16
(a) R2  4r1  r2
Perform
(b) R3  2r1  r3
the row
operations. (c) R3  6r2  r3
 1 3 5 2 


 4 5 4 5 
 2 5
4 6 
a.  1 3 5 2 


 0 7 16 13 
 0 18 30 23 
b.  1 3 5 2 


 0 8 9 3 
 0 37 40 8 
c.  1 3 5 2 


0
7
16
13


 0 53 110 88 
d.  1 3 5 2 


0
17
24
7


 0 113 158 52 
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 17
Solve the system of
equations using
Cramer’s Rule.
6x  2y  8

 4 x  y  7
a.
x  3, y  5;
3, 5 
b.
x  5, y  3;
5, 3
c.
x  5, y  3;
d.
x  3, y  5;
Copyright © 2009 Pearson Education, Inc.
5, 3
3, 5 
Slide 8 - 18
Solve the system of
equations using
Cramer’s Rule.
6x  2y  8

 4 x  y  7
a.
x  3, y  5;
3, 5 
b.
x  5, y  3;
5, 3
c.
x  5, y  3;
d.
x  3, y  5;
Copyright © 2009 Pearson Education, Inc.
5, 3
3, 5 
Slide 8 - 19
Find the value of the
determinant.
5 1
1
1
2
5 4
3 5
a. 62
b. –70
c. –62
d. –192
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 20
Find the value of the
determinant.
5 1
1
1
2
5 4
3 5
a. 62
b. –70
c. –62
d. –192
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 21
Find the value of x y z
the determinant u v w  52
second
1 2 3
determinant.
1 2 3
u v
x y
w ?
z
a. 52
b. –52
c. 0
d. cannot be determined
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 22
Find the value of x y z
the determinant u v w  52
second
1 2 3
determinant.
1 2 3
u v
x y
w ?
z
a. 52
b. –52
c. 0
d. cannot be determined
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 23
 7 4 8 
 2 6 1 




Let A  6 5 1 and B  7 4 3 .




6 3
 0
 3 9 5 
Find A – B.
a.  9 2 9 
b.  9 2 9 
 1 9 4 
1 9 2 




 3 15 2 
 3 15 4 
c.  5 10 7 
d.  5 10 7 
 13 1 8 
 13 1

2




 3 3 2 
 3 3 8 
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 24
 7 4 8 
 2 6 1 




Let A  6 5 1 and B  7 4 3 .




6 3
 0
 3 9 5 
Find A – B.
a.  9 2 9 
b.  9 2 9 
 1 9 4 
1 9 2 




 3 15 2 
 3 15 4 
c.  5 10 7 
d.  5 10 7 
 13 1 8 
 13 1

2




 3 3 2 
 3 3 8 
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 25
 3 4 
Let A  
.

 0 2
Find 3A.
a.
 9 12 
0 2


b.
 9 12 
0 6


c.
 9 4 
 0 2


d.
0 7 
3 5


Copyright © 2009 Pearson Education, Inc.
Slide 8 - 26
 3 4 
Let A  
.

 0 2
Find 3A.
a.
 9 12 
0 2


b.
 9 12 
0 6


c.
 9 4 
 0 2


d.
0 7 
3 5


Copyright © 2009 Pearson Education, Inc.
Slide 8 - 27
Compute the product.
 3 0
 1 3 1 

1
1
2 0 5  



 0 5 
a.
 0 2 
 6 25 


b.
c.
 3 3 0 
 0 0 25 


d. Not defined
Copyright © 2009 Pearson Education, Inc.
 2 0 
 25 6 


Slide 8 - 28
Compute the product.
 3 0
 1 3 1 

1
1
2 0 5  



 0 5 
a.
 0 2 
 6 25 


b.
c.
 3 3 0 
 0 0 25 


d. Not defined
Copyright © 2009 Pearson Education, Inc.
 2 0 
 25 6 


Slide 8 - 29
The matrix is
nonsingular. Find the
inverse of the matrix.
 3 3 1 
 2 2 1


 4 5 2 
a.  1 1 0 
 2 3 0 


 0 2 1
b.  1 3 1 
 0 2 1 


 2 3 0 
c.  1 1 1 
 2 3 0 


 0 2 1 
d.  1 1 1 
 0 2 1 


 2 3 0 
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 30
The matrix is
nonsingular. Find the
inverse of the matrix.
 3 3 1 
 2 2 1


 4 5 2 
a.  1 1 0 
 2 3 0 


 0 2 1
b.  1 3 1 
 0 2 1 


 2 3 0 
c.  1 1 1 
 2 3 0 


 0 2 1 
d.  1 1 1 
 0 2 1 


 2 3 0 
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 31
x  13
x  3x  5 
Write the partial fraction
decomposition.
a.
4
5

x3 x5
5
4
c.

x3 x5
Copyright © 2009 Pearson Education, Inc.
b.
4
5

x3 x5
d.
5
4

x3 x5
Slide 8 - 32
x  13
x  3x  5 
Write the partial fraction
decomposition.
a.
4
5

x3 x5
5
4
c.

x3 x5
Copyright © 2009 Pearson Education, Inc.
b.
4
5

x3 x5
d.
5
4

x3 x5
Slide 8 - 33
Write the partial fraction
decomposition.
3x 2  9x  8
x  2 x  1
2
2
1
2
2
1
2


a.


2 b.
2
x  2 x  1 x  1
x  2 x  1 x  1
2
1
2
2
1
2
c.
d.




2
2
x  2 x  1 x  1
x  2 x  1 x  1
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 34
Write the partial fraction
decomposition.
3x 2  9x  8
x  2 x  1
2
2
1
2
2
1
2


a.


2 b.
2
x  2 x  1 x  1
x  2 x  1 x  1
2
1
2
2
1
2
c.
d.




2
2
x  2 x  1 x  1
x  2 x  1 x  1
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 35
12x  3
2
x  1 x  x  1
Write the partial fraction
decomposition.
a.
5
5
2


x 1 x 1 x 1
c.
5
5x  2
 2
x 1 x  x 1
Copyright © 2009 Pearson Education, Inc.


b.
5
5x  2
 2
x 1 x  x 1
d.
5
2x  5
 2
x 1 x  x 1
Slide 8 - 36
12x  3
2
x  1 x  x  1
Write the partial fraction
decomposition.
a.
5
5
2


x 1 x 1 x 1
c.
5
5x  2
 2
x 1 x  x 1
Copyright © 2009 Pearson Education, Inc.


b.
5
5x  2
 2
x 1 x  x 1
d.
5
2x  5
 2
x 1 x  x 1
Slide 8 - 37
Write the partial fraction
decomposition.
x
2
5

2

3x

3
15x

15
b. 2

2
2
x 5
x 5
3x

3
15x

15
c. 2

2
2
x 5
x 5
3x

3
15x

15
d.

2
2
2
x 5 x 5
a.
3x  3 15x  15

2
2
2
x 5
x 5
3x 3  3x 2


Copyright © 2009 Pearson Education, Inc.





Slide 8 - 38
Write the partial fraction
decomposition.
x
2
5

2

3x

3
15x

15
b. 2

2
2
x 5
x 5
3x

3
15x

15
c. 2

2
2
x 5
x 5
3x

3
15x

15
d.

2
2
2
x 5 x 5
a.
3x  3 15x  15

2
2
2
x 5
x 5
3x 3  3x 2


Copyright © 2009 Pearson Education, Inc.





Slide 8 - 39
Solve the system of
equations using substitution.
a.
c.
x  5, y  6
b.
 xy  30

 x  y  11
x  5, y  6
x  6, y  5
x  6, y  5
(5, 6), 6, 5 
(5, 6), 6, 5 
x  5, y  6
d.
x  5, y  6
x  6, y  5
x  6, y  5
(5, 6), 6, 5 
(5, 6), 6, 5 
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 40
Solve the system of
equations using substitution.
a.
c.
x  5, y  6
b.
 xy  30

 x  y  11
x  5, y  6
x  6, y  5
x  6, y  5
(5, 6), 6, 5 
(5, 6), 6, 5 
x  5, y  6
d.
x  5, y  6
x  6, y  5
x  6, y  5
(5, 6), 6, 5 
(5, 6), 6, 5 
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 41
The sum of the squares of two numbers is 65.
The sum of the two numbers is 3. Find the two
numbers.
a. –4 and 7 or –7 and 4
b. –4 and 7
c. –7 and 4
d. 4 and 7 or –7 and –4
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 42
The sum of the squares of two numbers is 65.
The sum of the two numbers is 3. Find the two
numbers.
a. –4 and 7 or –7 and 4
b. –4 and 7
c. –7 and 4
d. 4 and 7 or –7 and –4
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 43
Graph 3x  4y  12.
a.
b.
c.
d.
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 44
Graph 3x  4y  12.
a.
b.
c.
d.
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 45
x  2y  6
Graph the solution set of 
.
 3x  2y  18
a.
b.
c.
d.
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 46
x  2y  6
Graph the solution set of 
.
 3x  2y  18
a.
b.
c.
d.
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 47
Mrs. Jones needs 7 hours to knit a hat and 3 for
an afghan. She has no more than 46 hours,
enough material for no more than 10 items and
needs at least two afghans. Write a system of
inequalities that describes these constraints.
a. 7x  3y  46
x  y  10
b. 7x  3y  46
x  y  10
y2
c.
7x  3y  46
x  y  10
x2
Copyright © 2009 Pearson Education, Inc.
x2
d.
7x  3y  46
x  y  10
y2
Slide 8 - 48
Mrs. Jones needs 7 hours to knit a hat and 3 for
an afghan. She has no more than 46 hours,
enough material for no more than 10 items and
needs at least two afghans. Write a system of
inequalities that describes these constraints.
a. 7x  3y  46
x  y  10
b. 7x  3y  46
x  y  10
y2
c.
7x  3y  46
x  y  10
x2
Copyright © 2009 Pearson Education, Inc.
x2
d.
7x  3y  46
x  y  10
y2
Slide 8 - 49
A candy company has 145 pounds of cashews
and 190 pounds of peanuts. The deluxe mix
contains half cashews and half peanuts and sells
for \$8 per pound. The economy mix has one
third cashews and two thirds peanuts and sells
for \$5.70 per pound. How many pounds of each
mix should be prepared for maximum revenue?
a. 100 deluxe,
45 economy
b. 300 deluxe,
90 economy
c. 200 deluxe,
135 economy
d. 145 deluxe,
0 economy
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 50
A candy company has 145 pounds of cashews
and 190 pounds of peanuts. The deluxe mix
contains half cashews and half peanuts and sells
for \$8 per pound. The economy mix has one
third cashews and two thirds peanuts and sells
for \$5.70 per pound. How many pounds of each
mix should be prepared for maximum revenue?
a. 100 deluxe,
45 economy
b. 300 deluxe,
90 economy
c. 200 deluxe,
135 economy
d. 145 deluxe,
0 economy
Copyright © 2009 Pearson Education, Inc.
Slide 8 - 51
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