Transcript Applications of Linear Systems

Applications of Linear
Systems
Now that you know how to solve a
linear system, you can use it to
solve real-life problems.
Methods we can use…

Graphing- Use this method when after
both equations are in slope-intercept form.

Substitution-Use this method when one of
the variables is isolated.

Elimination-Use this method when both
equations are in Standard Form
Selling Shoes

A store sold 28 pairs of cross-trainer shoes
for a total of $2200. Style A sold for $70
per pair and Style B sold for $90 per pair.
How many of each style were sold?

Keep in mind that a system has two
equations. We need an equation for the
quantity of shoes sold and one for the
total price.
Given Information
Total number of shoes
28
 Total receipts
2220
 Price of Style A
$70
 Price of Style B
$90
 Assign variables to unknowns

– Number of style A
– Number of style B
x
y
Equation 1: Number of style A + Number
of style B = Total number sold
 x + y = 28

Equation 2: Price A*Quantity A + Price
B*Quantity B = Total Price
 70x + 90y = 2220

Choose a method & solve
I will use substitution…
 x + y = 28
 y = 28 – x
 Substitute into 2nd equation
 70x + 90*(28 – x) = 2220
 70x + 2520-90x = 2220
 -20x = -300
 x = 15 pairs of Style A

continued
y = 28 – x
 Substitute x = 15 to find y
 y = 28 – 15
 y = 13 pairs of Style B


Solution (15 pairs of Style A, 13 pairs of
Style B)
Mixture Problem
Your car’s manual recommends that you use at
least 89-octane gasoline. Your car’s 16-gallon gas
tank is almost empty. How much regular
gasoline (87-octane) do you need to mix with
premium gasoline (92-octane) to produce 16
gallons of 89-octane gasoline?
 You need to know that an octane rating is the
percent of isooctane in the gasoline, so 16
gallons of 89-octane gasoline contains 89% of
16, or 14.24, gallons of isooctane.

Given information

Unknowns
– Volume of regular gas
– Volume of premium gas
Volume of 89-octane
 Isooctane in regular
 Isooctane in premium
 Isooctane in 89-octane

x
y
16 gallons
.87x
.92y
16*.89 = 14.24
Equations

Volume of regular + volume of premium =
total volume
x + y = 16

Isooctane in regular + isooctane in
premium = Isooctane in 89-octane.
.87x + .92y = 14.24
Solve the system
x + y = 16
 y = 16 – x
 Substitute into 2nd equation
 0.87x + 0.92*(16 – x) = 14.24
 0.87x + 14.72 – 0.92x = 14.24
 -.05x = -0.48
 x = 9.6 gallons of 87 octane
 y = 16 – 9.6 = 6.4 gallons of 92 octane

Making a decision
You are offered two different jobs. Job A offers
an annual salary of $30,000 plus a year-end
bonus of 1% of your total sales. Job B offers an
annual salary of $24,000 plus a year-end bonus
of 2% of your total sales.
 How much would you have to sell to earn the
same amount in each job?
 If you believe you can sell between $500,000
and $800,000 of merchandise per year, which
job should you choose?

Given information

If you pay attention to the wording, the
problems gives you an initial amount (b)
and percent of sales (m). Both equations
can be written in y = mx + b form.

Job 1: y = 0.01x + 30,000

Job 2: y = 0.02x + 24,000
Solve the system

Use your graphing calculator to solve the
system since both equations are in slopeintercept form.
Solution (break even point)
 (x = 600,000, y = 36,000)
 This represents the break even point.

continued

If you believe you can sell between
$500,000 and $800,000 of merchandise
per year, which job should you choose?

If you looked at the graph of the linear
system, you can see that if your sales are
greater than $600,000, Job B would pay
you better than Job A.
Assignment
Algebra book
 Page 422
 Problems 31 – 45 odd, 46 - 56
