### ch.8 active learning

```Active Learning Lecture Slides
For use with Classroom Response Systems
Chapter 8
Systems of
Equations and
Inequalities
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;
 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;
 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;
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;
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
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
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
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
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
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
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 
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 
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

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

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 
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 
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;
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;
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
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
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
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
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 
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 
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


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


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
 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
 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 
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 
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
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
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
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
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


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


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







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







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 
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 
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
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
Slide 8 - 43
Graph 3x  4y  12.
a.
b.
c.
d.
Slide 8 - 44
Graph 3x  4y  12.
a.
b.
c.
d.
Slide 8 - 45
x  2y  6
Graph the solution set of 
.
 3x  2y  18
a.
b.
c.
d.
Slide 8 - 46
x  2y  6
Graph the solution set of 
.
 3x  2y  18
a.
b.
c.
d.
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
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
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
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