UNIT 7
Algebra and Equations
Topics Covered in this Unit Include: collecting like terms, distributive property and solving
equations
Evaluations Given this Unit (Record Your Marks Here)
Mastery Test – Algebra
Mastery Test – Equations
Assignment – Pool Pass
Unit Test
151
Substituting and Collecting Like Terms
1. Substitute and simplify if a = 4, b = 2 and c = 3.
a)
a+b
b)
b–c
c)
ac
d)
bc – a
e)
ab ÷ c
2. Substitute and simplify if m = -1, n = 5 and p = 3.
a) m + n
b)
p–m
c) mn
d) np – m
e) np ÷ m
3. Organize the following terms into like-term groups.
a) a, 2b, 3a, 4b, -a, -6b
b) 2x, -3y, -2y, -4x, -5y
c) 2x2, -2x2, -4x, x2, 5x
d) a2, 2a, -3a, 4a2, -5a
4. Simplify each of the following polynomials by collecting like terms.
a) a + b + a + b + a + b + a + b
b) m + 3m + n + 2n
c) 2x + 3y + 2x + 3y + x + y
d) 4a2 + 2a + a2 + 4a + 3a2
152
4. (continued) Simplify each of the following by collecting like terms.
e) 5m + 7m2 + 8m2 + 3m + m
f) 9x + x2 + 4x + 3x2 + 6x2
5. Simplify each of the following by collecting like terms (Watch the Signs!!!)
a) 8x – 7x + 5y – 3y
b) 4m + 6n – m – 3n
c) 5a – 6b – 7a – 2b
d) -x – y – x – y – x – y
e) -7m2 – 5m + m2 – 5m
f) -5a2 + 8a + 3a2 – 10a
6. Simplify and then evaluate using m = -2 and n = 4.
a) 3m + 4n + n + 5n
b) 6m + 7n – 8m + 3n
c) -4m – 3n – 2m + 5n
d) 3m + 8n – m – 4n – m - 2n
e) 7m2 + 2m2 – 12m2
f) 5n + 7n2 + 2n -5n2 – 4n – n2
153
Algebra Puzzle Worksheet
154
Multiplication and Division of Monomials Worksheet
1. Find each product
a) (3r)(5s)
b) (6)(5x)
c) (-4x)(3y)
d) (3x)(2y)(4z)
e) (4x)(5x)
f) (-3x)(10x2)
g) (-6c2)(4c3)
h) (5m)(4mn)
i) (xy)(x2y3)
j) 3(2xy)(4x)
k) (9xy2)(x2y)(2x5y3) l) (7abc)(-abc)(2abc)
2. Find each quotient.
a)
8n
8
b)
e)
m4
m2
f)
16x
4x
3xy 2
24xy
c)
12
24a
d)
g)
12x 5
13x 2
h)
12x 3
2x
 10x 5 y 6
2x 3 y 2
155
2. (continued)
i)
(3x )(4y )
2x
j)
7x
24y
k)
(2x 2 y 3 )(xy 5 )
x 3y 6
l)
( 3x 2 yz 5 )(4xy 4 )
(2xyz )(3yz 3 )
3. State the missing term
a) (3x)( ? ) = 9x2
b) ( ? )(4d2) = 8d4
c) (8y)( ? ) = -40x2y2
d) ( ? ) ÷ (2r) = 4rs
e) (27p6q2) ÷ ( ? ) = 9p4q
f) ( ? ) ÷ (-5abc) = 4a2b3c4
156
Distributive Property
1. Simplify.
a) 3(a + 2)
b) 4(m – 5)
c) 8(x2 + 7)
d) -6(2y + 4)
e) -7(3b – 8)
f) 5(7r – 9)
g) m(p + q)
h) x(y – z)
i) a(2b + c)
j) 3n(n2 + 4)
k) 2y2(y – 5)
l) -6x2(4x – 3)
2. Simplify completely.
a) 2(x + 3) + 3(x – 5)
d) (x + 4y – 3) + (3x – 2y + 7)
g) 2(4x + 5) – 3(2x – 4)
b) 3a(2a + b – 4) + 3(a + 4b + 6)
e) (3x + 4y) – (x + 2y)
h) 3x + 2(x – 4) – 5(2x + 1)
c) 9a(a2 + a + 3)
f) 3(x – y) – (4x + y)
i) (-6ab + a – 4b) – (4ab + 8b – 7a)
157
Solving One-Step Equations
1.
Solve the following equations.
(a)
p + 17 = 28
(b)
r – 22 = 47
(c)
9k = 189
(d)
x + 42 = -13
(e)
e – 76 = -55
(f)
-12a = 72
(g)
24 + y = 8
(h)
-17 + w = 43
(i)
19d = 114
(j)
q + 59 = -11
(k)
h – 16 = -35
(l)
-17w = -187
(m)
85 + a = 27
(n)
17 = b – 85
(o)
144 = 4p
(p)
44 = g + 19
(q)
-73 = d – 38
(r)
-87 = e + 66
158
Solving Two-Step Equations
1.
Solve the following equations.
(a)
3x – 7 = 2
(b)
5y + 11 = -14
(c)
-7z + 2 = -12
(d)
9p – 17 = 1
(e)
11b + 6 = -71
(f)
-v – 19 = 37
(g)
14r + 9 = 121
(h)
-32c – 55 = -183
(i)
74k + 30 = -44
(j)
81x + 9 = -234
(k)
-45u – 51 = 219
(m)
31p – 89 = 438
(n)
-27q + 19 = -143
(l)
24y + 18 = -102
159
Solving Multi-Step Equations
Part A: Collecting Like Terms First
a) 2x + 5 + 7x = -4
b) 11a – 8 = 50 – 14
c) 19 + 5 – 8 = 3p – 11
d) 4c – 16 – 3c + 5 = 9 + 7
e) 38r – 56 + r = -17
f) m – 3m + 2 + 5m = 14
g) 23 – 4z + 16 + z – 38 = 4
h) 25b – b + 13b – 3 = 7 + 64
i) e + e + e + 9 – 16 – 22 = 1
j) 17n + 39 – 16 – 8n = -4 + 9
k) 5q + 7 + 11q – 13 = -29 – 41 l) 15v + 2v + 18 – 21 = 31
Part B: Solving Equations With Distributive Law
a) 3(x + 4) = 24
b) 5(a – 6) = 45
c) -2(4p + 6) = 36
d) -6 = -6(3c – 2)
e) 8(9 – 2r) = 184
f) 66 = 6(-3y – 1)
160
Part C: Variables on Both Sides of the Equation
a) 6x + 7 = 4x – 1
b) 3a + 2 = -a – 6
c) -5p – 7 = -3p – 9
d) 4c + c = 6 = 9c – 10
e) 6 + 8r – 2r = 4 + r + 22
f) m + 3m – 2 + 2m = 5m + 13
161
Sample EQAO Question – Equations, Algebra and Measurement
The permimeter of the following figure is 38 cm, what is the length of each of the sides?
Show your work.
x + 10
3x - 6
x+3
2x - 4
example 2: If the perimeter of the following triangle is 27 cm, what type of triangle is it?
Justify your answer.
3x + 3
2x + 5
x+7
162
Algebra and Equations Assignment - Pool Pass
The local swimming pool is open 5 days a week for 8 weeks during the summer holidays. The
admission prices are displayed at the entrance.
Splash World Swim Park
Price List
Full Season Pass...................$120
Partial Season Pass...............$60 plus $2 per day
Daily Swim Pass...................$5
How much will it cost one person to go to the pool every day the pool is open?
a)
with a full season's pass?
b)
with a partial season's pass?
c)
with a daily pass?
Is the full season's pass the best deal if you attend for 20 days? Give reasons.
163
Complete the following tables.
Full Season's Pass
Days
Attended
0
5
10
15
20
25
30
35
40
Cost
Partial Season's Pass
Days
Attended
0
5
10
15
20
25
30
35
40
Cost
Daily Pass
Days
Attended
0
5
10
15
20
25
30
35
40
Cost
If C is the total cost and d is the number of days attended, write an equation for:
a) the full season pass
b) the partial season pass
c) the daily pass
164
Graph all 3 lines on the same set of axes using the tables or equations. Clearly label each of
the lines.
According to the graph, how many days would you need to attend for:
a) the full season pass to cost the same amount as the partial season pass?
b) the full season pass to cost the same amount as the daily pass?
165
Investigation
Patty Poolhog, Tom Thong, and Cindy Sunburn are hoping to spend some time cooling off at the
pool this summer. Each person investigated the cost of each plan based on the number of days
they plan to go to the pool.
Patty determined that the daily pass was cheapest for her.
Tom has chosen the full season pass.
Cindy purchased the partial season pass.
Determine the number of days Patty, Tom and Cindy plan to attend the pool. Included details
to explain how you came up with your solution.
166
Algebra and Equations Review
1. Substitute and evaluate if y = 3 and x = -1.
a) 5x
b) x + y
c) xy
d) 4x + 2y
2. Collect like terms to simplify the following.
a) 5m + 7m
b) 3x + 4x – 2x
c) 4x – 6 + 2x + 13
d) 6x2 + 5 – 2x + 3x + 8 + x2
e) 4a + 7b – 3 – 3b + 5 + 6a
f) x + y – x + 3y + 5y2
3. Simplify each of the following.
a) 2(x + 4) + 3
b) 5(m – 2) – 1 + m
c) 4y + 7(2y + 5) – 3
d) 3(4a + 2) + (2a – 1)
e) 3(2x + 1) – (4x + 3)
f) (3x + 4y) – (2x – 5y – 7)
g) (x – 4)(x + 3)
h) (x + 5)(x + 7)
167
4. Solve each of the following equations.
a) 2x = 14
b) x – 9 = 13
c)
3x – 5 = 23
d) 2x + 4x – 8 = 28
e) 189 = -9(-3n – 6)
f) (7e – 2) – (3e + 6) = 8
g) 11z – 8 – 9 = 4z – 3z – 27
h) 3(2b + 6) + 4 = b – 3
i) 4(8n – 1) = 5(5n + 3) + 2
168