MATH 1314 - Test 4 Review - Dave Rice
Graph the equation and identify the x- and y-intercepts.
1) 3x + 5y = 15
x-intercept (x,y): __________
5
y-intercept (x,y): __________
y
4
3
2
1
-5
-4
-3
-2
-1
-1
1
2
3
4
5 x
-2
-3
-4
-5
Solve the problem.
2) Joey borrows $2700 from his grandfather and pays the money back in monthly payments of $100.
a. Write a linear function that represents the remaining money owed L(x) after x months.
b. Evaluate L(8) and interpret the meaning in the context of this problem.
3) A bakery makes and sells pastries. The fixed monthly cost to the bakery is $750. The cost for labor, taxes, and
ingredients for the pastries amounts to $0.50 per pastry. The pastries sell for $1.70 each.
a. Write a linear cost function representing the cost for producing and selling x pastries.
b. Write a linear revenu function representing the revenue for producing and selling x pastries.
c. Write a linear profit function representing the profit for producing and selling x pastries.
Write the augmented matrix of the given system of equations.
9x + 2y = 84
4)
3x + 2y = 36
5)
6)
4x - y = 0
-4x + y - 8 = 0
7x +4y +5z = 58
9x +3y -2z = 7
2x +8y +8z = 90
MATH 1314 - Test 4 Review
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Dave Rice
7) 2x + 7z = 5
-2y + 4z = -8
7x + 6y + 5z = 49
8) x - 6y - 7z = 6
-9y - 4z = -2
4z = -6
8x
-3 = -6z
9) -5x
-2z = -5 -6y
-2x -4y +8z = 2
Write a system of linear equations represented by the augmented matrix.
-4 9 -6
10)
4 6 5
11)
12)
13)
6 7 -3
-2 2 0
3 0 -7
2
7
3
6 6 6 -2
8 0 9 4
3 9 0 2
1
0 0
9
0
1 0
4
0
0 1
1
3
For the given augmented matrix, determine the number of solutions corresponding to the system of equations.
1 0 -9
14)
-4
0 1
6
7
0 0
0
-1
Determine the solution set for the system represented by the augmented matrix.
15)
1 0 -5 -2
0 1 -8 -2
0 0 0
0
MATH 1314 - Test 4 Review
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Dave Rice
1 0 -4 4
16)
0 1 -9 1
0 0 0 1
Solve the system using matrices. Also, show the first and final matrices.
17) x + 4y = -7
4x + 3y = 11
First Matrix:
Final Matrix:
Solution: __________________________________________________
18) x + y + z = -11
x - y + 5z = -9
5x + y + z = -27
First Matrix:
Final Matrix:
Solution: __________________________________________________
19) -4x - 6y - z = -25
x - 4y + 3z = 9
-7x + y + z = -7
First Matrix:
Final Matrix:
Solution: __________________________________________________
20) -3x + 4y - 4z = -8
-6x + 4y -6z = -12
-6x + 6y - 7z = 15
First Matrix:
Final Matrix:
Solution: __________________________________________________
MATH 1314 - Test 4 Review
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Dave Rice
21) y + 3z = 3
x - y - 5z = 1
2x + 2y + 2z = 14
22) A summer jazz festival was held and 1500 concert tickets were sold. Tickets cost $25 for covered pavillion seats
and $15 for lawn seats. The total receipts from ticket sales were $28,500. How many tickets of each type were sold?
a) Write the system of equations.
b) Solve the system using matrices. Show the first and final matrices.
c) Give the the solution in terms of # of each type sold.
23) A basketball fieldhouse seats 15,000. Courtside seats sell for $8, goal for $6, and balcony for $4. The total for a
sell-out is $76,000. If half the courtside, all the goal, and half the balcony seats are sold, then the ticket sales total
$44,000. How many of each type of seat are there?
a) Write the system of equations.
b) Solve the system using matrices. Show the first and final matrices.
c) Give the the solution in terms of # of each type sold.
Solve the problem.
24) A concession
stand at a city park sells hamburgers, hot dogs, and drinks. Three patrons buy the
following food and drink combinations for the following prices.
Patron
1
2
3
Hamburgers
0
3
1
Hot Dogs
1
1
1
Drinks Revenue
2
$5
11
$22
5
$11
Let x, y, and z represent the cost for hamburgers, hot dogs, and drinks, respectively. Set up an
augmented matrix for the system and solve for x, y, and z. Explain what the stand manager knows
about the reported earnings. Show the first and final matrices.
The nth term of a sequence is given. Find the indicated term.
25) an = 4 n + 7; a6
The nth term of a sequence is given. Write the first four terms of the sequence.
(-1)n+1
26) an =
n2 + 8
MATH 1314 - Test 4 Review
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Dave Rice
Find the sum.
6
27) ∑ (n2 - 1)
n=4
28)
3
∑
(k - 4)(k + 1)
k=1
MATH 1314 - Test 4 Review
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Dave Rice
Answer Key
Testname: MATH 1314 TEST 4 REVIEW & ANSWERS
1) x-intercept:
y-intercept:
Set y = 0
Set x = 0
⇒ 3x + 5(0) = 15
⇒ 3(0) + 5y = 15
⇒ 3x = 15
⇒ 5y = 15
⇒ x=5
⇒ x=3
x-intercept: (5, 0); y-intercept: (0, 3)
5
y
4
3
2
1
-5
-4
-3
-2
-1
-1
1
2
3
4
5 x
-2
-3
-4
-5
2) L(x) = -100x + 2700; L(8) = 1900, This represents the amount Joey still owes his grandfather after 8 months.
3) a. C(x) = 0.50x + 750
b. R(x) = 1.70x
c. P(x) = R(x) - C(x)
P(x) = 1.70x -(0.50x + 750)
P(x) = 1.70x -0.50x - 750
P(x) = 1.20x - 750
4) 9 2
3 2
5)
6)
7)
84
36
4x - y = 0 ⇒
-4x + y - 8 = 0
4x - y = 0
-4x + y = 8
⇒
4 -1
-4 1
0
8
7 4 5 58
9 3 -2 7
2 8 8 90
2 0 7
5
0 -2 4 -8
7 6 5 49
MATH 1314 - Test 4 Review
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Dave Rice
Answer Key
Testname: MATH 1314 TEST 4 REVIEW & ANSWERS
1 -6 -7
6
8) 0 -9 -4 -2
0 0 4 -6
8x
-3 = -6z
9) -5x
-2z = -5 -6y
-2x -4y +8z = 2
⇒
8x
+6z = 3
-5x +6y -2z = -5
-2x -4y +8z = 2
⇒
8
-5
-2
0 6
6 -2
-4 8
3
-5
2
10) -4x + 9y = -6
4x + 6y = 5
11)
12)
6x + 7y - 3z = 2
-2x + 2y
=7
3x
- 7z = 3
6x + 6y + 6z = -2
8x
+ 9z = 4
3x + 9y
= 2
13) x = 9
y=4
1
z=
3
14) 0 or no solutions
15) {(-2 + 5z, -2 + 8z, z) | z is any real number}
16) { } or ∅
17) First Matrix:
1 4 -7
4 3 11
Final Matrix:
1 0 5
0 1 -3
Solution: x = 5, y = -3
MATH 1314 - Test 4 Review
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Dave Rice
Answer Key
Testname: MATH 1314 TEST 4 REVIEW & ANSWERS
18) First Matrix:
1 1 1
-11
1 -1 5
-9
5 1 1 -27
Final Matrix:
1
0
0
-4
0
1
0
-5
0
0
1
-2
Solution: x = -4, y = -5, z = -2
19) First Matrix:
-4 -6 -1 -25
1 -4
3
9
1
1
-7
-7
Final Matrix:
1
0
0
2
0
1
0
2
0
0
1
5
Solution: x = 2, y = 2, z = 5
20) First Matrix:
-3 4 -4
-8
-6 4 -6 -12
-6 6 -7
15
Final Matrix:
1
0
2/3 0
0
1 -1/2 0
0
0
0
1
Solution: No Solution or { } or ∅
MATH 1314 - Test 4 Review
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Dave Rice
Answer Key
Testname: MATH 1314 TEST 4 REVIEW & ANSWERS
21) First Matrix:
0
1
3
3
1 -1 -5
1
2
14
2
2
Final Matrix:
1
0 -2 4
x - 2z = 4
⇒ y + 3z = 3 ⇒
0 1 3 3
0
0
0 0
0=0
x = 2z + 4
y = -3z + 3
z
Solution: {(2z + 4, -3z + 3, z) | z is any Real #)}
22) a) x + y = 1500
25x +15y = 28,500
b) First Matrix:
1 1 1500
25 15 28500
Final Matrix:
1 0 600
0 1 900
x = 600, y = 900
c) 600 Covered Pavillion Seats, 900 Lawn Seats
MATH 1314 - Test 4 Review
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Dave Rice
Answer Key
Testname: MATH 1314 TEST 4 REVIEW & ANSWERS
23) a) x + y + z = 15,000
8x + 6y + 4z = 76,000
8(x/2) +6y + 4(z/2) = 44,000
x + y + z = 15,000
8x + 6y + 4z = 76,000
4x + 6y + 2z = 44,000
b) First Matrix:
1 1 1 15000
8 6 4
76000
4
44000
6 2
Final Matrix:
1
0
0
3000
0
1
0
2000
0
0
1
10000
x = 3000, y = 2000, z = 10,000
c) 3000 courtside seats, 2000 goal seats, 10,000 balcony seats
0
24) 3
1
1
1
2
5
1 11 22
5 11
⇒
1
0 3 0
0
1 2 0
0
0 0 1
⇒ No Solution
Because there is no solution, the connession stand manager knows that there was an error in the record keeping.
25) a6 = 4 6 + 7 = 4096 + 7 = 4103
26)
(-1)1+1 (-1)2+1 (-1)3+1 (-1)4+1 (-1)2 (-1)3 (-1)4 (-1)5
1
1 1
1
,
,
,
,
,
,
,
,=
= ,1 + 8 4 + 8 9 + 8 16 + 8 9
12 17 24
1 2 + 8 22 + 8 3 2 + 8 4 2 + 8
27) (4 2 - 1) + (5 2 - 1) + (6 2 - 1) = (16 - 1) + (25 - 1) + (36 - 1) = 15 + 24 + 35 = 74
28) (1 - 4) (1 + 1) + (2 - 4) (2 + 1) + (3 - 4) (3 + 1) =(-3) (2) + (-2) (3) + (-1) (4) = -6 - 6 - 4 = -16
MATH 1314 - Test 4 Review
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Dave Rice