1. Exercise. A. Perform the following operations..
Systems of Linear Equations I
8x
–9x+)
1. 8
–9–)
2. –8y
–9y–)
3. –8
9–)
4. –4y
–5y+)
5. +4x
–5x–)
6.
10.{–x + 2y = –12
2x + y = 4
Exercise B. Solve by the elimination method.
7. {x + y = 3
2x + y = 4
8. 9. {x + 2y = 3
2x – y = 6
{x + y = 3
2x – y = 6
11. {3x + 4y = 3
x – 2y = 6
12. { x + 3y = 3
2x – 9y = –4
13.{–3x + 2y = –1
2x + 3y = 5
14. {2x + 3y = –1
3x + 4y = 2
15. {4x – 3y = 3
3x – 2y = –4
16.{5x + 3y = 2
2x + 4y = –2
17. {3x + 4y = –10
–5x + 3y = 7
18. {–4x + 9y = 1
5x – 2y = 8
{
x – y = 3
x – y = –1
3
2
2
3
1
2
1
4
19. {
x + y = 1
x – y = –1
1
2
1
5
3
4
1
6
20.

2. 21.{ x + 3y = 4
2x + 6y = 8
Exercise C. Which is inconsistent and which is dependent?
22.{2x – y = 2
8x – 4y = 6
23.{ x + 3y = 4
2x + 6y = 25
24 {2x – y = 2
8x – 4y = 8
25. Two hamburgers and one order of fries cost $7.
One hamburger and one order of fries cost $4.
Given the following information, find the price of each item if
possible, if not, explain why not.
26. Three hamburgers and one order of fries cost $7.
One hamburger and one order of fries cost $3.
27. Two hamburgers and one order of fries cost $7.
Four hamburgers and two orders of fries cost $14.
28. Two hamburgers and one order of fries cost $7.
Four hamburgers and two orders of fries cost $15.

3. Systems of Linear Equations II
Solve the following using matrix notation.
1.
x + y + z = 3
x – y – z = –1
2x + y – z = 4
2. 3.
x + y – z = 3
x – y + z = –1
2x + y – z = 4
x + 2y – z = 2
x – y + z = 1
2x – y + z = 2
4.
x + 2y + z = 5
x – y – z = –1
2x + y – z = 4
5. 6.
x + y – z = 2
2x – 2y + z = –1
2x + y – z = 3
x + 2y – z = 1
x – y + 2z = 4
2x – y + z = 3
7.
x + y + 2z = 1
2x – y – z = 0
2x + y – z = –2
8. 9.
x + 2y – z = 2
–x – y + 2z = 0
2x + y – z = 3
x + 2y – z = –2
–x – y + z = 2
2x – y + 2z = 0
10.
x – y + z = –1
x – y – 2z = –4
2x + y – 3z = –4
11. 12.
x + y – 2z = 3
2x – 2y + z = 6
2x + y – z = 6
x + 3y – z = 4
x – y – 2z = –1
2x – y + z = 4

4. 13. We bought the following items:
2 hamburgers, 3 orders of fries and 3 sodas cost $13.
1 hamburger, 2 orders of fries and 2 sodas cost $8.
3 hamburgers, 2 fries, 3 sodas cost $13.
Find the price of each item.
14. We bought the following items:
2 triangles, 2 circles, and 1 squares at a cost of $10.
3 triangles, 2 circles of fries and 2 squares at a cost of $11.
1 triangles, 3 circles, 2 squares at a cost of $14.
Find the price of each item.
15. Farmer’s Andy’s store sells three types of fruit baskets.
Red basket has 2 apples, 2 bananas and 1 cantaloupe.
Green basket has 3 apples, 1 bananas and 1 cantaloupe.
Blue basket has 2 apples, no bananas and 2 cantaloupes.
We bought some of each type of the baskets, all together we
have 16 apples, 8 bananas and 9 cantalopes. How many of
each type of baskets did we purchased?

5. A. Given the following matrix, perform the indicated row–
operations.
Matrix Notation
1 4 -7
2 -3 8
–2R21. 2. 3. 3R1R1 R2 4. –2R2 add  R1
5. 3R1 add  2R2 6. –3R2 add  –2R1
2 3 3
1 2 2
3 2 3
7. –3R2 add  R3
9. –3R1 add  –2R3
then
8. –4R2 add  R1
10. –3R3 add  –2R2
then –2R1 add  –1R3
B. Given the following matrix, perform the indicated row–
operations.
R3 R2

6. 14.{–x + 2y = –12
2x + y = 4
Solve by sing the matrix noation. Write down each of the row
operations performed.
11.{x + y = 3
2x + y = 4
12. 13 {x + 2y = 3
2x – y = 6{x + y = 3
2x – y = 6
15. {3x + 4y = 3
x – 2y = 6
16 { x + 3y = 3
2x – 9y = –4
17.{–3x + 2y = –1
2x + 3y = 5
18. {2x + 3y = –1
3x + 4y = 2
19. {4x – 3y = 3
3x – 2y = –4
20.{5x + 3y = 2
2x + 4y = –2
21. {3x + 4y = –10
–5x + 3y = 7
22. {
–4x + 9y = 1
5x – 2y = 8

7. Systems of Linear Equations II
Solve the following using matrix notation.
23.
x + y + z = 3
x – y – z = –1
2x + y – z = 4
24. 25.
x + y – z = 3
x – y + z = –1
2x + y – z = 4
x + 2y – z = 2
x – y + z = 1
2x – y + z = 2
26.
x + 2y + z = 5
x – y – z = –1
2x + y – z = 4
27. 29.
x + y – z = 2
2x – 2y + z = –1
2x + y – z = 3
x + 2y – z = 1
x – y + 2z = 4
2x – y + z = 3
30
x + y + 2z = 1
2x – y – z = 0
2x + y – z = –2
31. 32.
x + 2y – z = 2
–x – y + 2z = 0
2x + y – z = 3
x + 2y – z = –2
–x – y + z = 2
2x – y + 2z = 0
33.
x – y + z = –1
x – y – 2z = –4
2x + y – 3z = –4
34. 35.
x + y – 2z = 3
2x – 2y + z = 6
2x + y – z = 6
x + 3y – z = 4
x – y – 2z = –1
2x – y + z = 4

8. Sequences
Exercise A.
1. Given the sequence of odd numbers an: 1, 3, 5,…
what is a4? a7?
2. Given the sequence of square numbers sn: 1, 4, 9,
… what is s4? s7?
3. Given the sequence of powers of two: rn: 2, 4, 8, …
what is r4? r7?
4. Given the sequence of prime numbers pn: 2, 3, 5,…
what is p4? p7?
5. Given that ak is 181 in the sequence of odd
numbers what is ak –1? ak + 1? ak + 2?
6. Given that sk is 100 in the sequence of square
numbers what is sk – 2? sk + 2?
7. Given that rk is 256 in the sequence of powers of
two, what is rk – 2? rk + 2?

9. Sequences
8. Given the sequence of odd numbers an: 1, 3, 5,…
9. Given the sequence of square numbers sn: 1, 4, 9, ..
10. Given the sequence of powers of two: rn: 2, 4, 8,…
11. Given the sequence of prime numbers pn: 2, 3, 5,…
what is Σ an?n=2
4
what is Σ sn?n=3
6
what is Σ ?n=2 rn
1
4
what is Σ ?n=2 pn
1
4
Exercise B. Expand each of the following Σ–notation
into an explicit symbolic sum.
12. Σ gnn=2
4
13. Σ hi
i =3
5
14. Σ (hk + gk)k=3
5
15.Σ xjyj
j =2
4
16. Σn=3
5
17. Σ (hk – gk)2
i =1
4
18. Σ√(xj + yj)
j=1
4
bn
an 2 2

10. Sequences
Exercise C. Calculate the following sums.
19. Σ 3n=2
4
22. Σ (i2 + 1)
5
26. Σ
j=0
4
27. Σk =2
5
(–1)j 2j
1
20. Σ (2n + 1)k=5
8
21.
Σ (–1)j
j=1
4500
23. Σ (i + 1)2
i = 3
5
24. Σ k(k – 1)k= 5
8
Σ (3n – 1)n=2
4
30.
k – 1
(–1)kk
28. Σk =2
19
k – 1
1
k
1( )
Σ
k= 1
4
25.
k
1
29. Σk =3
19
k + 1
1
k + 2
1( )

11. Arithmetic Sequences
Exercise B. Answer each of the following questions assuming
all sequences are arithmetic.
2. Given a1 = –4 and d = –2, find the specific formula an and a100.
3. Given a1 = –6 and d = 3, find the specific formula an and a100.
4. Given a1 = 9 and d = –2, find the specific formula an and a100.
5. Given a1 = 3½ and d = 2½, find an. Find k if aK = 61.
6. Given a1 = –4 and a4 = –4, find d, an, and a100.
7. Given a2 = –4 and a12 = 4 , find d and an. Find k if aK = 0.
8. Given a2 = 6 and a4 = –2, find d and an. Find k if aK = –32.
9. Given a5 = 3½ and a10 = 2½, find d and an. Find k if aK = –25.
Exercise A.
1. Which of the following sequences are arithmetic?
a. 2, 4, 6,… b. 2, 4, 8,… c. 5, 4, 2, 1, 0.. d. 1, 5, 11, 15…
e. 12, –4, –20, –36,.. f. e, 2e, 3e, .. g. x, x + 3, x + 6, x + 9,..

12. Arithmetic Sequences
10. Given a1 = –11 and a400 = –4001, find d and an.
Find k if aK = –3871.
11. Given a5 = 3½ and a9 = 5½, find an and a387..
Find k if aK = 387.
12. Given a5 = 3½ and a9 = –2½, find an. Find k if aK = –62½ .
13. Given a6 = 1½ and a14 = 3½, find an and a100.
Find k if aK = 100.
14. Given a5 = ½ and a15 = 1½, find the specific formula an,
and a100. Find k if aK = 100.
16. The 6th house on Forest Drive is No. 248 and the 11th house
is No. 284. Let be an be the house number of the nth house on
Forrest Drive, find d, a1 and an. Which house is No. 890?
15. The price of renting a bus consists of a base price plus the
number of passengers for the bus. Let cn be the cost for n
passengers and that c5 = $135 and c11 = $207. Find d, c1and cn.

13. Arithmetic Sequences
17. Farmer Andy wants to plan a
row of the Andy–berry trees 120 ft
from the river and the trees are to
be spaced 15 ft apart as shown.
120 ft 15 ft
…
Let dn be the distance from the river to the nth tree, find d1 and
the specific formula dn. How far is the 25th tree from the river?
18. At 1 pm, Farmer Andy turned on
the hose to refill his water tank.
At 4 pm, the depth of the tank was 42”
and at 8 pm the depth was 48”.
Let d1 be the depth of the water at 1pm,
and dn be the depth of the water
at “n” pm. Find d1and dn. What would
the depth be at 11pm? Next day 10 am?
dn

14. Exercise C. Find the following arithmetic sums.
Sum of Arithmetic Sequences
19. How may terms are there in 1 + 2 + 3 + .. + 1000
and find their sum.
20. How may terms are there in 4 + 7 + 10 + .. + 130
and find their sum.
21. How may terms are there in –12 – 16 – 20 – 24 .. – 1000
and find their sum.
22. How may terms are there in 4 + 7 + 10 + .. + 130
and find their sum.
23. How may terms are there in
1 + 1¼ + 1½ + 1¾ + 2 + 2¼ + …+ 100 and find their sum.
24. Find the sum of the first 1000 terms of 4 + 7 + 10 + ..
25. Find the sum of the first 500 terms of –12 – 16 – 20 – 24 ...
27. Find the sum of the first one million terms of
1 + 1.1 + 1.2 + 1.3 + 1.4 + 1.5 + …
26. Find the sum of the first 100 terms of 1 + 1¼ + 1½ + 1¾ +…

15. Geometric Sequences
Exercise B. Answer each of the following questions assuming
all sequences are geometric.
2. Given a1 = –4 and r = 2, find the specific formula an and a10.
3. Given a1 = –6 and r = 3, find the specific formula an and a6.
4. Given a1 = 9 and r = –2, find the specific formula an and a7.
5. Given a3 = 3 and r = 1/2, find a1 and an. Find k if aK = 3/64.
6. Given a4 = 4 and a5 = 2, find r, a1 , an, and a10.
7. Given a4 = 20 and a7 = –5/2, find r and an. Find k if aK = –5/32.
8. Given a3 = 2 and a6 = –54, find r and an. Find k if aK = 972.
9. Given a6 = 1.43 and a9 = 0.00143, find r, a1 and an.
Exercise A.
1. Which of the following sequences are arithmetic?
a. 2, 4, 6,… b. 2, 4, 8,… c. 3, 6, 12, 24,.. d. 1, 11, 111,…
e. 1, –1, 1, –1,.. f. 2e, 4e, 8e,16e, .. g. x, x3, x6, x9,..