1. Solve the system
x2 + 4y − x = 8
x + y = 3.
Check your work by substituting your answer in the original system.
solution.
(x, y) = (1, 2) and (4, −1).
2. At a grocery store, mozzarella cheese is sold in half pound blocks for $2 each. One pound blocks
of cheddar cheese cost $3 each. If a shopper spent $13 on cheddar and mozzarella cheese to buy 4
pounds of cheese, how many blocks of cheddar and mozzarella cheese were purchased?
solution. Two blocks of mozzarella and three blocks of cheddar.
3. Solve the system
x + 2y − z = 10
2y + z = 2
2x + 5y − 5z = 28.
Check your work by substitution.
solution.
(x, y, z) = (4, 2, −2).
4. Rewrite the given matrix as a system of equations with variables x, y, z and w.


4
1 5 −2
3
 3 2
5
1 11 


 0 6 11 −1
2 
7 2
2
0 −1
5.


2 1
A = 4 3 
0 −2
0 2 −1
B=
2 5 0
Compute AB and BA.
solution.


2
9
−2
23 −4
AB =  6
−4 −10 0
8 8
BA =
.
24 17
6. Find the inverse of

1
A = 0
1
4
1
1
Verify your answer by computing A times A−1 .
1

1
1
−1
solution.
A−1

−2
= 1
−1
5
−2
3

3
−1
1
7. Simplify
3
solution.
2
1
3
1
−
−2
−1
5
4
0
.
2
9
−8
8. Compute the determinant of

3
1
0

−1
2 .
1
0
5
1
solution. 8
9. Compute
7
X
(−1)i
i=1
solution. −1
10. Simplify
11!
.
8!
solution.
11 · 10 · 9 = 990.
11. Compute
5
X
2i−1
i=1
solution. 31
12. Suppose that a sequence is defined recursively as follows: a1 = 2, a2 = −1 and ak+2 = ak + ak+1 .
Write down the first five terms of this sequence.
solution.
a1 = 2, a2 = −1, a3 = 1, a4 = 0, a5 = 1.
13. Write an expression for the nth term of the sequence:
2
2
2 2
− , ,−
,
,...
5 25 125 625
solution.
n
1
.
an = 2 −
5
2
A few more examples of inverses:
14.
15.
16.
17.


1 2 −1
A = 1 1 0 
2 0 1


1 −2 1
A−1 = −1 3 −1
−2 4 −1

3 −1
A = 1 0
1 −1

1 −1
1
A−1 =  0
−1 2

2
1
1

−1
−1
1


0 2 1
A = 1 7 −2
1 6 −2


2 −10 11
1
−1
A−1 = 0
1 −2
2

2 2
A= 0 1
−1 0

1 −2
A−1 = −1 3
1 −2

1
1
1

1
−2
2
18. Find all solutions to the system:
2x2 + y − 2x = 19.
2x + y = 3.
Check your work by substituting your answers in the original system.
solution.
x = −2, y = 7.
x = 4, y = −5.
19. Student tickets to a concert were sold for $3 each. Non-student tickets were sold for $5 each. If
900 tickets were sold for $3300, how many each of student and non-student tickets were sold?
solution. 600 student tickets and 300 non-student tickets.
20. Solve the system
x + 3y − 2z = 3.
3x + 10y − 7z = 6.
x + 7y − 5z = 4.
Check your work by substitution.
3
solution. x = −1, y = 10, z = 13.
21.
2 1
A=
3 0
−1 2
B=
1 2
Compute AB and BA.
solution.
−1 6
AB =
−3 6
4 −1
BA =
8 1
22. Find the inverse of

1
A = 0
0
solution.
2
1
3

3
−1 .
−2

13 −5
−2 1 
−3 1

1
0
0
23. Simplify
1
−3
solution.
0
2
−5
1
−3
−9
12
3
4
−15
.
−19
24. Simplify
7!
.
3!
solution. 840.
25. Compute the determinant of

−1
−3
1
2
1
−1

0
2
1
solution. 7
26. Compute
5
X
3i−1
i=1
solution. 121.
27. Solve the following exponential equation. Remember to check for extraneous solutions.
3e2x − 3ex − 10 = 2e2x .
4
solution. x = ln 5.
28. Solve the following logarithmic equation. Remember to check for extraneous solutions.
log7 x + log7 (5x − 2) = log7 (x + 2).
solution. x = 1.
29. Solve the given equations. Where needed, express your answers in terms of logarithms.
(1)
(2)
(3)
(4)
(5)
2
ex −x = ex+15
e2x + ex = 2
2−x
7·5
√ = 13
ln( x − 8) = 5
log x + log(x + 2) = log(x + 6)
solution.
(1) x = 5, −3.
(2) x = 0.
(3) x = 2 − log5
(4) x = e10 + 8.
(5) x = 2.
13
.
7
5