Math 112 Practice Problems for Test #4
Remember to look over your homework as well as the problems on this sheet.
13) x - y + z = 4
x + y + z = -4
x + y - z = 2
Determine the order of the matrix.
1) 9 5 -3
-4 6 1
2)
18
-5 2
1
3) 6
-6
7
14) 4x + 2y + z = 13
4x - 3y - z = 28
5x + y + 4z = 4
-7
0
9
-1
15) -3x - 3y + 5z = -16
-12x - 11y + 21z = 2
-6x - 5y + 11z = -9
2 0 4
4) -2 5 -5
0 8 -7
16) 3x + 3y + 3z = 3
9x + 10y + 11z = 12
6x + 7y + 8z = 9
Decompose into partial fractions.
3x - 21
17)
(x + 5)(x - 4)
Write the augmented matrix for the system.
5) 8x + 8y = 48
5x - 2y = 16
6) 6x + 2y + 8z = 34
2x + 3y + 4z = 29
4x + 8y + 4z = 68
18)
7) 7x - 2y = 57
9y = 27
19)
Solve the system of equations using Gaussian elimination
or Gauss-Jordan elimination.
8) 4x + 8y = 48
-2x + 6y = 36
5x - 19
2
x - 7x + 10
- 31x - 46
2x2 + 10x + 8
20)
-2x2 + 3x + 8
(x + 1)2 (3x + 4)
21)
26x2 - 119x + 139
(x - 3)(2x - 4)2
10) 5x + 9y = -31
10x = -33 - 18y
22)
3x - 16
(x - 4)2
11) 4x + 8y = 8
6x + 12y = 12
23)
-3x2 - 2x - 33
(x - 3)(x2 + 2)
12) x + y + z = 1
x - y + 4z = 17
5x + y + z = -3
24)
4x2 - 37x + 18
(x2 + 3x - 6)(x - 4)
25)
28x2 + 45x - 11
(5x2 + 1)(6 - x)
9) 7x + 49 = -7y
2x - 5y = 14
1
33) x ≥ -1
Solve the system of equations. List all solutions as
ordered pairs.
26) x2 + y2 = 85
y
10
x + y = -13
5
27) y = x2 - 14x + 49
x + y = 27
-10
-5
28) xy - x2 = -20
x - 2y = 3
5
10
x
5
10
x
-5
-10
29) x2 + y2 = 90
x2 - y2 = -72
34) y ≤ 2
y
30) x2 + y2 = 20
y2 = 2x + 12
10
5
Graph the linear inequality.
31) -2x - 3y ≤ -6
-10
y
-5
10
-5
5
-10
-10
-5
5
10
x
Graph the system of inequalities, and find the coordinates
of the vertices.
35) 2x + y ≤ 4
x - 1 ≥ 0
-5
-10
4
y
32) x + y < -6
y
10
4x
-4
5
-10
-5
5
10
x
-4
-5
-10
2
36) 2x + y ≤ 4
y - 1 ≤ 0
4
39) 3y + x ≥ 0,
y + 2x ≤ 10,
y ≥ 0
y
y
10
8
6
4
2
4x
-4
-10 -8 -6 -4 -2
-2
2
4
6
8
x
2
4
6
8
x
-4
-6
-4
-8
-10
37) 3x - 2y ≥ -6
x - 1 < 0
5
40) 3y + x ≥ -6,
y + 2x ≤ 8,
y ≤ 0,
x ≥ 0
y
y
10
8
6
5x
-5
4
2
-10 -8 -6 -4 -2
-2
-5
-4
-6
-8
38) 3y - x ≤ 9,
y + 2x ≤ 10,
y ≥ 0
-10
10
Graph the inequality.
41) (x - 5)2 + (y - 4)2 > 4
y
8
6
y
4
10
2
-10 -8 -6 -4 -2
-2
2
4
6
8
x
5
-4
-6
-10
-8
-10
-5
5
-5
-10
3
10
x
42) y > x2 + 6
45) x2 + y2 ≤ 49
7x + 5y ≤ 35
y
10
y
10
5
5
-10
-5
10 x
5
-10
-5
-5
5
10
x
-5
-10
-10
Graph the solution set of the system of inequalities or
indicate that the system has no solution.
43) x2 + y2 ≤ 64
Find the maximum or minimum value of the given
objective function of a linear programming problem. The
figure illustrates the graph of the feasible points.
46) Objective Function: z = 5x + 8y
Find maximum and minimum.
x2 + y2 ≥ 49
y
6
4
2
-8
-6
-4
y
-2
2
4
6
8x
-2
-4
(0, 9)
(9, 9)
-6
44) y > x2
3x + 6y ≤ 18
(9, 3)
(0, 3)
y
(3, 0)
10
5
-10
-5
5
10
x
Find the maximum or minimum value of the given
objective function of a linear programming problem. The
figure illustrates the graph of feasible points.
x
-5
-10
47) Objective Function: z = -x - 8y
Find maximum.
4
An objective function and a system of linear inequalities
representing constraints are given. Graph the system of
inequalities representing the constraints. Find the value
of the objective function at each corner of the graphed
region. Use these values to determine the maximum value
of the objective function and the values of x and y for
which the maximum occurs.
48) Objective Function
z = 12x + 5y
Constraints
0 ≤ x ≤ 10
0 ≤ y ≤ 5
3x + 2y ≥ 6
49) Objective Function
Constraints
50) Objective Function
Constraints
z = 3x + 5y
x ≥ 0
y ≥ 0
2x + y ≤ 15
x - 3y ≥ -3
z = 6x - 5y
x ≥ 0
0 ≤ y ≤ 2
x - y ≤ 3
x + 2y ≤ 6
Solve the problem.
51) Your computer supply store sells two types of
laser printers. The first type, A, has a cost of
$86 and you make a $45 profit on each one.
The second type, B, has a cost of $130 and you
make a $35 profit on each one. You expect to
sell at least 100 laser printers this month and
you need to make at least $3850 profit on
them. How many of what type of printer
should you order if you want to minimize
your cost?
5
Answer Key
Testname: 112REVTEST4FALL09
1)
2)
3)
4)
2 × 3
2 × 2
4 × 2
3 × 3
5) 8 8 48
5 -2 16
6 2 8 34
6) 2 3 4 29
4 8 4 68
7) 7 -2 57
0 9 27
8) (0, 6)
9) (-3, -4)
10) No solution
11) (2 - 2y, y)
12) (-1, -2, 4)
13) (3, -4, -3)
14) (5, -1, -5)
15) No solution
16) (z - 2, -2z + 3, z)
1
4
- 17)
x + 5 x - 4
18)
3
2
+ x - 2 x - 5
19) - 20)
4
3
2
+ - 3x + 4 (x + 1)2 x + 1
21) - 22)
23)
24)
25)
5
13
- 2x + 2 x + 4
5
(2x - 4)2
+ 5
4
+ 2x - 4 x - 3
3
4
- x - 4 (x - 4)2
3x + 7
x2 + 2
- 6
x - 3
7x
x2 + 3x - 6
- 3
x - 4
7x - 3
7
- 2
x - 6
5x + 1
26) (-6, -7), (-7, -6)
27) (2, 25), (11, 16)
11
28) (5, 1) and -8, - 2
29) (3, 9), (-3, 9), (3, -9), (-3, -9)
30) (2, 4), (-4, 2), (2, -4), (-4, -2)
6
Answer Key
Testname: 112REVTEST4FALL09
31)
y
10
5
-10
-5
5
10
x
5
10
x
5
10
x
-5
-10
32)
y
10
5
-10
-5
-5
-10
33)
y
10
5
-10
-5
-5
-10
7
Answer Key
Testname: 112REVTEST4FALL09
34)
y
10
5
-10
-5
5
10
x
-5
-10
35)
4
y
(1, 2)
4x
-4
-4
36)
4
y
(3/2, 1)
4x
-4
-4
37)
5
y
(1, 9/2)
5x
-5
-5
8
Answer Key
Testname: 112REVTEST4FALL09
38)
y
10
8
6
4
(3, 4)
2
(-9, 0)
(5, 0)
-10 -8 -6 -4 -2
-2
2
4
2
4
6
8
x
-4
-6
-8
-10
39)
y
10
8
6
4
2
(0, 0)
(5, 0)
-10 -8 -6 -4 -2
-2
6
8
x
-4
-6
-8
-10
40)
y
10
8
6
4
2
(0, 0) (4, 0)
-10 -8 -6 -4 -2
-2
(0, -2)
-4
2
4
6
x
8
(6, -4)
-6
-8
-10
y
10
5
-10
-5
5
10
x
-5
-10
41)
9
Answer Key
Testname: 112REVTEST4FALL09
42)
y
10
5
-10
-5
10 x
5
-5
-10
43)
y
6
4
2
-8
-6
-4
-2
2
4
6
8x
-2
-4
-6
44)
y
10
5
-10
-5
5
10
x
5
10
x
-5
-10
45)
y
10
5
-10
-5
-5
-10
10
Answer Key
Testname: 112REVTEST4FALL09
46) maximum value: 117; minimum value: 15
47) maximum: -20
48) Maximum: 145; at (10, 5)
49) Maximum 33; at (6, 3)
50) Maximum: 19; at (4, 1)
51) order 100 type A printers
11