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Math Coach Amy
Amy Ferguson Moncure

H Algebra 2/Trig · Unit 1

Linear Word Problems

Write and solve a linear equation from a story: rentals, rectangle perimeters, and commission after expenses.

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Key Ideas Easy Medium Hard
What to Remember

Name the variable

Say what the letter stands for, with units, before you write the equation.

Perimeter

P = 2ℓ + 2w. If the length is written in terms of the width, substitute, then solve.

Commission after expenses

(rate)(sales) − (expenses) = (amount left). Solve for the sales price.

Check the story

Plug the answer back in. The length should match the words, and money left should match the target.

Level 1

Easy

One unknown. Translate, then solve.

Example
A bike rental costs $12 plus $5 per hour. The total is $37. How many hours h were rented?

Let h = the number of hours.

12 + 5h = 37

5h = 25

h = 5

Answer: 5 hours

1
A number n plus 9 is 31. Find n.
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n + 9 = 31

n = 22

Answer: 22

2
Four times a number n is 28. Find n.
Show Answer

4n = 28

n = 7

Answer: 7

3
A gym charges $8 per visit. Maya spent $56. How many visits v did she buy?
Show Answer

8v = 56

v = 7

Answer: 7 visits

4
A number n is 15 less than 40. Find n.
Show Answer

n = 40 − 15

n = 25

Answer: 25

5
Tickets cost $9 each plus a $3 service fee for the order. The total is $48. How many tickets t?
Show Answer

9t + 3 = 48

9t = 45

t = 5

Answer: 5 tickets

Level 2

Medium

A rectangle: length in terms of width, then perimeter.

Example
The length of a rectangle is 3 feet more than twice the width. The perimeter is 54 feet. Find the dimensions.

Let w = the width in feet.

Length = 2w + 3

The formula is

P = 2ℓ + 2w

Substitute.

54 = 2(2w + 3) + 2w

54 = 4w + 6 + 2w

54 = 6w + 6

48 = 6w

w = 8

Length = 2(8) + 3

Length = 19

Answer: width 8 ft; length 19 ft

1
The length of a rectangular garden is 5 meters less than twice the width. The perimeter is 50 meters. Find the dimensions.
Show Answer

Let w = the width in meters.

Length = 2w − 5

The formula is

P = 2ℓ + 2w

Substitute.

50 = 2(2w − 5) + 2w

50 = 4w − 10 + 2w

50 = 6w − 10

60 = 6w

w = 10

Length = 2(10) − 5

Length = 15

Answer: width 10 m; length 15 m

2
The length of a frame is 4 inches more than the width. The perimeter is 56 inches. Find the dimensions.
Show Answer

Let w = the width in inches.

Length = w + 4

The formula is

P = 2ℓ + 2w

Substitute.

56 = 2(w + 4) + 2w

56 = 2w + 8 + 2w

56 = 4w + 8

48 = 4w

w = 12

Length = 16

Answer: width 12 in; length 16 in

3
The length of a patio is triple the width. The perimeter is 96 feet. Find the dimensions.
Show Answer

Let w = the width in feet.

Length = 3w

The formula is

P = 2ℓ + 2w

Substitute.

96 = 2(3w) + 2w

96 = 6w + 2w

96 = 8w

w = 12

Length = 36

Answer: width 12 ft; length 36 ft

4
The length of a rectangular lot is 8 yards more than twice the width. The perimeter is 76 yards. Find the dimensions.
Show Answer

Let w = the width in yards.

Length = 2w + 8

The formula is

P = 2ℓ + 2w

Substitute.

76 = 2(2w + 8) + 2w

76 = 4w + 16 + 2w

76 = 6w + 16

60 = 6w

w = 10

Length = 28

Answer: width 10 yd; length 28 yd

5
The length of a banner is 2 feet less than four times the width. The perimeter is 46 feet. Find the dimensions.
Show Answer

Let w = the width in feet.

Length = 4w − 2

The formula is

P = 2ℓ + 2w

Substitute.

46 = 2(4w − 2) + 2w

46 = 8w − 4 + 2w

46 = 10w − 4

50 = 10w

w = 5

Length = 18

Answer: width 5 ft; length 18 ft

Level 3

Hard

Percent commission after expenses.

Example
A realtor receives 5% commission on the sales price of a house. She has $1500 in expenses. She wants $8500 left after expenses. What sales price is needed?

Let s = the sales price in dollars.

The formula is

(rate)(sales) − (expenses) = (amount left)

Substitute.

0.05s − 1500 = 8500

0.05s = 10000

s = 200000

Answer: $200,000

1
An agent receives 4% commission on a sale. Expenses are $1800. He wants $6200 left after expenses. What sales price is needed?
Show Answer

Let s = the sales price in dollars.

The formula is

(rate)(sales) − (expenses) = (amount left)

Substitute.

0.04s − 1800 = 6200

0.04s = 8000

s = 200000

Answer: $200,000

2
An agent receives 2% commission. Expenses are $900. She wants $4100 left after expenses. What sales price is needed?
Show Answer

Let s = the sales price in dollars.

The formula is

(rate)(sales) − (expenses) = (amount left)

Substitute.

0.02s − 900 = 4100

0.02s = 5000

s = 250000

Answer: $250,000

3
An agent receives 6% commission. Expenses are $2400. He wants $9600 left after expenses. What sales price is needed?
Show Answer

Let s = the sales price in dollars.

The formula is

(rate)(sales) − (expenses) = (amount left)

Substitute.

0.06s − 2400 = 9600

0.06s = 12000

s = 200000

Answer: $200,000

4
An agent receives 8% commission. Expenses are $500. She wants $3500 left after expenses. What sales price is needed?
Show Answer

Let s = the sales price in dollars.

The formula is

(rate)(sales) − (expenses) = (amount left)

Substitute.

0.08s − 500 = 3500

0.08s = 4000

s = 50000

Answer: $50,000

5
An agent receives 2.5% commission. Expenses are $1250. He wants $3750 left after expenses. What sales price is needed?
Show Answer

Let s = the sales price in dollars.

The formula is

(rate)(sales) − (expenses) = (amount left)

Substitute.

0.025s − 1250 = 3750

0.025s = 5000

s = 200000

Answer: $200,000

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