Summative Assessment Review: Equations and Problem Solving
1. The equation “3x – 4 = 11” is solved as shown. Describe the inverse operations used in each step.
Step 1:
Step 2:
3x – 4 = 11
3x – 4 + 4 = 11 + 4
3x = 15
3𝑥
15
=
3
3
x=5
Step 1: Used the Addition Property of Equality to collect like terms.
Step 2: Used the Division Property of Equality to isolate variables.
Solve each equation.
2
5
3. [ x – 4 = ] 12
4
3
2. 2x – 7 = 19
+7 =+7
8x – 48 =
2x = 26
2
2
x = 13
15
+ 48 = + 48
8x
= 63
8
8
63
7
x
= 8 = 78
4. Determine if there is one solution, no solution, or an infinite number of solutions.
2(3x + 4) – (x – 8) = 3(4x + 2) – 7x + 10
6x + 8 – x + 8 = 12x + 6 – 7x + 10
5x + 16 = 5x + 16
Identity  Infinite number of solutions
5. Monica bought 3 types of fruit for a fruit salad. She paid twice as much for blueberries as for
oranges, and $1.50 less for strawberries than for blueberries.
a. Define a variable and write algebraic expressions to represent the amount she spent on
each type of fruit.
b  blueberries
1
b  oranges
2
𝑏 – 1.50  strawberries
b. If the total cost was $12.25, how much did Monica spend on each type of fruit?
1
b + 2 b + b – 1.50 = 12.25
2.5b – 1.50 = 12.25
+ 1.50 = + 1.50
2.5b = 13.75
B = 5.50
Blueberries: $5.50; Oranges $2.75; Strawberries $4.00
Solve and check each equation.
6.
[
6(2𝑥−1)
5
]
= −18 5
7. 3[
−2(5𝑥+4)
7
= −3(3x + 2) − ]
3
3
6(2x – 1) = -90
-2 (5x + 4) = -9(3x + 2) - 7
12x – 6 = -90
-10x – 8 = -27x – 18 - 7
+6=+6
12x
x
-10x – 8 = -27x - 25
= -84
17x – 8 = -25
= -7
17x = -17
x = -1
Check:
6(2(−7) −1)
5
6(−15)
5
−2(5(−1)+4))
= -18
3
−2(−1)
3
= -18
7
3
7
= 3(-1) - 3
2
3
-18 = -18
= -3(3x + 2) -
=
2
3
8. Consider the equation 3(x + 1) + x + 2 = 2(2x + 1) + 3.
a. Solve the equation.
3(x + 1) + x + 2 = 2(2x + 1) + 3
3x + 3 + x + 2 = 4x + 2 + 3
4x + 5 = 4x + 5
Identity  infinite solutions
b. Is the resulting equation from part (a) always true, sometimes true, or never true? Explain
your reasoning.
The equation is always true for every value of x.
9. A supertanker leaves port traveling north at an average speed of 10 knots. Two hours later a
cruise ship leaves the same port heading south at an average speed of 18 knots. How many hours
after the cruise ship sails will the two ships be 209 nautical miles apart? (Hint: 1 knot = a nautical
mile per hour)
r
10
18
supertanker
Cruise ship
The cruise ship will have traveled 8
3
4
t
t
(t – 2)
10t + 18(t – 2) = 209
10t + 18t – 36 = 209
28t – 36 = 208
+ 36 = + 36
28t = 245
3
t = 84
3
4
– 2, or 6 hours.
d
10t
18(t – 2)
10. The sum of three consecutive even integers is – 198. Find the three integers.
1: n
-68
2: n + 2
-66
3: n + 4
-64
3n + 6 = -198
- 6
- 6
3n = -204
n = -68
11. The length of a rectangle is 4 in. greater than the width. The perimeter of the rectangle is 24 in.
Find the dimensions of the rectangle.
w  width
P = 2ℓ + 2w
w + 4  length
24 = 2(w + 4) + 2w
P = 24 in
24 = 2w + 8 + 2w
Width = 4 in
24 = 4w + 8
Length = 8 in
-8
- 8
16 = 4w
4 =w
The formula “A = 2h(ℓ + w)” gives the lateral area A of a rectangular solid with length ℓ, width w, and
height h.
12. Solve the formula for h.
A
2 (ℓ + w)
A
2 (ℓ + w)
13. Solve the formula for ℓ.
=
2h(ℓ + w)
2(ℓ + w)
=
h
A
2h
=
A
= ℓ+w
2h
w = -w
A - w = ℓ
2h
14. Find h if A = 144 cm2, ℓ = 7 cm, and w = 5 cm.
h=
A___
=
144__ = 144 = 144
2 (ℓ + w)
2(7 + 5)
2(12)
24
15. Find ℓ if A = 9338 m2, h = 29 m, and w = 52 m.
ℓ= A
2h
- w = 9338
2(29)
2h(ℓ + w)
2h
- 52 = 9338 - 52 = 161 – 52 = 109 m
58
= 6 cm