1. Section 7-4: Applications of Linear Systems
Objective:
To model rate-time-distance problems using two variables
To use systems of equations to solve problems graphically

2. Applications of Linear Systems
We have solved systems by graphing, substitution and elimination.
We are now going to apply these skills toward work problems (real world problems)
Let’s look at the following examples

3. Definitions Ingot Alloy Kg
A metal melted down and cast into a bar or other shape
Alloy
Any combination of metals fused together Kg
Abbreviation of Kilograms grams.
A gram is about the weight of a paperclip
A kilogram is about 2.2 lbs.

4. Example #1
A metalworker has some ingots of metal alloy that are 20% copper and others that are 60% copper. How many kilograms of each type of ingot should the metalworker combine to create 80kg of a 52% copper alloy?

5. Example #1 Let x = the mass of the 20% alloy
Let y = the mass of the 60% alloy
What is the mass of the alloys?
- x + y = 80
What is the mass of the copper?
x + 0.6y = 0.52(80)
What does that leave us with?

6. Example #1
x + y = x y = 0.52(80) x + y = 80 - y -y x = 80 – y 0.2(80-y) y =0.52(80)
A SYSTEM OF EQUATIONS!!! : )
Think of which method would be easiest…
Would it be easier to solve for x or y OR would be easier to multiply either or both equations to eliminate one of the variables?
Substitution is probably easier.

7. Example #1
0.2(80-y) + 0.6y =0.52(80) 16 – 0.2y + 0.6y = y = y = y = 64
Solve for y
Now plug in y in either equation to find x

8. Example #1
x = 80 – y x = 80 – (64) x = 16 Mass of 20% alloy needed is 16kg. Mass of 60% alloy needed is 64kg.
I chose this equation
Now, we have an answer for both x and y.
Remember what x and y stood for

9. Example #2: Finding a break even point
Suppose a gum chewing club publishes a newsletter. Expenses are $0.90 for printing and mailing each copy, plus $600 total for gum (both chewing and bubble), research, and writing. The price of the newsletter is $1.50 per copy. How many copies of the newsletter must the club sell to break even?

10. Example #2: Finding a break even point
Let x = the number of copies Let y = the amount of dollars of expenses or income. Expenses = y = 0.9x Income = y = 1.5x
We need to create two equations.
One for expenses
One for income

11. Example #2: Finding a break even point
y = 0.9x y = 1.5x 1.5 x = 0.9 x x -0.9x 0.6 x = x = 1000
This gives us …
Another system
Since y = y always
The club must sell 1000 copies of just to break even

12. Example #3
Suppose you fly from Miami to San Francisco. It takes 6.5 hours to fly 2600 mile against a head wind. At the same time, your friends flies from San Francisco to Miami. Her plane travels at the same average airspeed, but her flight only takes 5.2 hour. Find the average airspeed of the planes. Find the average wind speed.

13. Example #3
Let A = Air speed Let W = Wind speed Remember: Rate x Time = Distance (A + W)(Time) = Distance (A – W)(Time) = Distance
We need to create two equations
Which formula do we need?
This is the formula with the tailwind
This is the formula with the head wind
Now, plug in values

14. Example #3 (A + W)5.2 = 2600 (A – W)6.5 = 2600 5.2 5.2 A + W = 500
Plug in the values that we know
Take the 1st equation and divide by 5.2

15. Example #3 (A – W)6.5 = 2600 6.5 6.5 A – W = 400 A + W = 500
Now divide the 2nd equation by 6.5
This leaves us with…
… an easier system to work with

16. Example #3
A + W = 500 A – W = 400 2A = A = W = W = 50
Solve this by elimination
Airspeed = 450 mph
Now, solve for W
Wind speed = 50 mph