Applications of Linear Equations
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School
Purdue University *
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Course
15300Y
Subject
Mathematics
Date
Jan 9, 2024
Type
Pages
7
Uploaded by Bhshaffer03
16-week Lesson 10 (8-week Lesson 8)
Applications of Linear Equations
1
Remember that when working with equations that whatever you do to one
side, you must do to the other side.
Also, when working with unknown quantities,
use variables.
When solving applied problems, it might be helpful to ask yourself
questions to help set-up equations that you can solve.
Examples of
questions you could ask yourself are given in red below.
Keep in mind that
the questions below in red will not appear in the homework, or on quizzes
and exams; these are simply questions that you should be asking yourself
when you see problems like these in order to help get equations that you can
solve.
Examples:
1.
An employee makes
$14
an hour.
The employee gets paid double for
any hours (or fraction of an hour) worked over
40
hours in a week.
One week the employee made
$840
before taxes; how many
total
hours did the employee work?
How much does the employee make for working
40
hours? (hourly
wage times
40
hours)
How many hours over
40
did the employee work?
(if you don’t know,
assign a variable to represent this value)
How much did the employee make for the hours over
40
? (overtime
hourly wage times overtime hours)
How much did the employee make total?
(write an equation;
$ for
40
hours + $ for overtime = total $)
16-week Lesson 10 (8-week Lesson 8)
Applications of Linear Equations
2
2.
A couple has budgeted
$4,000
for the rehearsal dinner for their
upcoming wedding.
If sales tax is
5%
and they tip
20%
, how much
can they spend on the meal only, before tax and tip, if they tip on the
meal only and do include the tax on the meal when calculating the tip?
Round your answer to the nearest penny.
What is the cost of the meal only?
(if you don’t know, assign a variable
to represent this value)
How much sales tax will the couple pay? (cost of the meal only times
0.05
)
How much will the couple pay for the tip? (cost of the meal only times
0.2
, since they’re tipping on the meal only)
How much will the couple pay in total?
(write an equation;
$ for the meal only + $ for tax + $ for tip = total $)
16-week Lesson 10 (8-week Lesson 8)
Applications of Linear Equations
3
3.
A couple has budgeted
$4,000
for the rehearsal dinner for their
upcoming wedding.
If sales tax is
5%
and they tip
20%
, how much
can they spend on the meal only, before tax and tip, if they tip on the
total bill (dinner + tax)?
Round your answer to the nearest penny.
What is the cost of the meal only?
(if you don’t know, assign a variable
to represent this value)
?
How much sales tax will the couple pay? (cost of the meal only times
0.05
)
0.05?
How much will the couple pay for the tip? (
0.2
times (cost of the meal
only
+ tax), since they’re tipping on the total bill)
0.2(? + 0.05?)
How much will the couple pay in total?
(write an equation;
$ for the meal only + $ for tax + $ for tip = total $)
? + 0.05? + 0.2(? + 0.05?) = 4,000
? + 0.05? + 0.2(1.05?) = 4,000
? + 0.05? + 0.21? = 4,000
1.26? = 4,000
? =
4,000
1.26
? = 3,174.60
So the couple can spend
$3,174.60
of their
$4,000
budget on just the
meal, and they’ll still have enough to cover the tax on the meal, and the
tip.
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16-week Lesson 10 (8-week Lesson 8)
Applications of Linear Equations
4
4.
An electrician and his apprentice both work
10
hours on a particular
job.
If they bill a customer
$735
for labor on a job, and the apprentice
pay rate is
1
4
that of the electrician, how much does
each
make per
hour?
How much does the electrician make per hour?
How many hours did the electrician work?
How much does the apprentice make per hour?
How many hours did the apprentice work?
How much did the pair make together?
(write an equation;
$ for electrician + $ for apprentice = total $)
16-week Lesson 10 (8-week Lesson 8)
Applications of Linear Equations
5
5.
When a popular band played at Elliott Hall, the box-office receipts
totaled
$120,000
.
Non-student tickets cost
3
times more than student
tickets.
If
4,000
students attended the concert, as well as
2,000
non-
students, what was the ticket charge for
each
?
How much did student tickets cost ?
? dollars
How many students attended the concert?
4,000 students
How much did non-student tickets cost ?
3? dollars
How many non-students attended the concert?
2,000 students
How much total box-office revenue was generated?
(write an
equation; $ from student tickets + $ from non-student tickets = total $)
$ from students + $ from non − students = Total $
4000? dollars + 6000? dollars = $120000
$10000? = $120000
? = 12
(?
dollars
student
) ∙ (4000 students) + (3?
dollars
non − student
) (2000 non − students) = $120000
So student tickets cost
$12
each while non-student tickets cost
$36
each.
16-week Lesson 10 (8-week Lesson 8)
Applications of Linear Equations
6
6.
A salesperson purchased a car that is advertised as getting half as many
miles per gallon of gasoline in the city than it does on the highway. On
a recent sales trip that covered
1200
miles in total, the car used up
25
gallons on the highway and
10
gallons in the city. Assuming that the
advertised mileage estimates were correct, what is the mileage of the
car in the city and on the highway?
How many miles per gallon does the car get on the highway?
?
miles
gallon
How many gallons of gas were used on the highway?
25 gallons
How many miles per gallon does the car get in the city?
1
2
?
miles
gallon
How many gallons of gas were used in the city?
10 gallons
How many total miles were driven?
(write an equation; miles driven
on the highway + miles driven in the city = total miles)
Highway miles + City miles = Total miles
(?
miles
gallon
) ∙ (25 gallons) + (
1
2
?
miles
gallon
) (10 gallons) = 1200 miles
25? miles + 5? miles = 1200 miles
30? miles = 1200 miles
? = 40
So the car gets
40
miles
gallon
on the highway and
20
miles
gallon
in the city.
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16-week Lesson 10 (8-week Lesson 8)
Applications of Linear Equations
7
Answers to Examples:
1.
50 ℎ????
;
2.
$3,200
; 3.
$3,174.60
;
4.
𝐸?????𝑖?𝑖𝑎? ?𝑎??? $58.80 ??? ℎ???;
𝐴??????𝑖?? ?𝑎??? $14.70 ??? ℎ???
;
5.
𝑆?????? ?𝑖????? ???? $12 ; 𝑁?? − ??????? ?𝑖????? ???? $36
;
6.
40 ??𝑔 ?? ?ℎ? 𝐻𝑖𝑔ℎ?𝑎? ; 20 ??𝑔 𝑖? ?ℎ? 𝐶𝑖??
;