1. Lesson 6-4 Warm-Up

2. “Applications of Linear Systems” (6-4)
When should you use a particular method to solve real world problems using systems of equations?
Summary: There are three methods for solving system of equation problems. Graphing (6-1): Use this method if both equations are easy to graph and the point of intersection (solution) contains only integers for coordinates (in other words, no fraction / decimals) Substitution (6-2): Use this method if the point of intersection (solution) does not contain integers or if at least one of the equations has a variable isolated (for easy substitution). Elimination (6-3): Use this method if one of the variables can be eliminated by adding or subtracting the equations (you may have to multiply both sides to create “opposite terms” first)

3. A chemist has one solution that is 50% acid. She has another
Applications of Linear Systems
LESSON 6-4
Additional Examples
A chemist has one solution that is 50% acid. She has another
solution that is 25% acid. How many liters of each type of acid solution
should she combine to get 10 liters of a 40% acid solution?
Word Method:
Words: amount of acid in 50% solutions + amount of acid in 25% solutions
= the amount of acid in mixture
Define: Let x = volume of the Let y = volume of the
50% solution. 25% solution.
Amount of Solution Equation: x y =
Amount of Acid Equation: 50% of x % of y = 40% of 10
 x  y =  10
Solve using substitution.
Step 1: Choose one of the equations and solve for a variable.
x + y = 10 Solve for a.
-y y Subtract y from both sides.
x = 10 – y Subtract b from each side.

4. Step 2: Create a one variable equation.
Applications of Linear Systems
LESSON 6-4
Additional Examples
(continued)
Step 2: Create a one variable equation.
0.50(10 – y) y = 0.4  Substitute: x = 10 - y
0.50  10 – 0.5  y y = Use the Distributive Property.
5 – y y = 4 Combine like terms.
y = 4
Subtract 5 from each side.
–0.25y = -1
Divide each side by –0.25.
y = 4
Step 3: Find x. To do this, substitute 4 for y in either equation.
x + (4) = 10
Subtract 4 from each side x = 6
x = 6
To make 10 L of 40% acid solution, you need 6 L of 50% solution and 4 L of 25% solution.

5. Amount of Acid in Solution
Applications of Linear Systems
LESSON 6-4
(continued)
Table Method:
Define: Let x = amount of the 50% solution.
% Acid
(in decimal form)
Amount of Solution
Amount of Acid in Solution
50% Solution
.50 x
.50x
25% Solution
.25
10 - x
.25 (10 - x)
Mixture
.40 10
.40 (10)
Amount of Acid Equation: x (10 – x) = .40 (10)
.50x (10) - .25(x) = .40(10) Distributive Property
.50x (x) = Simplify
.25x = Combine Like Terms
Subtract 2.5 from both sides.
.25x = Simplify

6. the amount of the 25% solution is 10 – 6, or 4 L.
Applications of Linear Systems
LESSON 6-4
(continued)
.25x = 1.5
.25x = Divide both sides by .25
x = 6
Since x is the amount of the 50% solution, she will need 6 L of the 50% solution.
Since the 50% solution and the 25% solution equal 10 L when mixed, then
the amount of the 25% solution is 10 – 6, or 4 L.

7. “Applications of Linear Systems” (6-4)
What is the “break-even point”?
Break-Even Point: In business, this is the point at which income (how much you’re making) equals expenses (how much you’re spending). In the graph below, the red line represents expenses (number of dollars spent for each item sold) and the blue line represents income (number of dollars made for each item sold). Notice that the slope of the blue income line is “steeper” than the red expenses line, which means that the business sells each item for a higher price (profit) than they buy or make it for. Since there is usually a startup cost (cost to start a business), it will take some time to “break even”.

8. “Applications of Linear Systems” (6-4)
How do you find the break-even point?
To find the break even point, write two equations, one for the expenses and one for the income. Both equations should contain two variables, x for the number of items, and y for the total amount in dollars. Example: Suppose a club publishes a newsletter. Expenses are $0.90 for printing and mailing each copy, plus $600 for 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? Define: Let x = the number of copies Let y = the amount of dollars of expenses and income Words: Expenses are printing costs Income is price time copies sold plus research and writing. Equation: y = 0.9x y = 1.50x or y = 1.5x y = 0.9x Given (1.5x) = 0.9x y = 1.5x (Substitute) -0.9x -0.9x Subtract 0.9x from both sides. 0.6x = 600 Simplify Divide both sides by 0.6 x = 1000 Simplify To break even, the club must sell 1000 copies of the newsletter.

9. Define: Let x = the number of pages.
Applications of Linear Systems
LESSON 6-4
Additional Examples
Suppose you have a typing service. You buy a personal computer for $1750 on which to do your typing. You charge $5.50 per page for typing. Expenses are $.50 per page for ink, paper, electricity, and other expenses. How many pages must you type to break even?
Define: Let x = the number of pages.
Let y = the amount of dollars of expenses or income.
Words: Expenses are per-page Income is price
expenses plus times pages typed.
computer purchase.
Equations: y = x y = x

10. y = 0.50x + 1750 Start with one equation.
Applications of Linear Systems
LESSON 6-4
Additional Examples
(continued)
Choose a method to solve this system. Use substitution since it is easy to substitute for d with these equations.
y = 0.50x Start with one equation.
(5.50x) = 0.50x Substitute 5.50y for x.
-0.50x x Subtract 5 from both sides.
5.00x = Solve for p.
Divide both sides by 5.
x =
To break even, you must type 350 pages.

11. “Applications of Linear Systems” (6-4)
What is airspeed, groundspeed, and windspeed? How do you solve a problem involving airspeed, groundspeed, and windspeed?
Windspeed: the speed of the wind Airspeed: the speed an airplane goes without the help or hindrance of the wind Groundspeed: the speed an airplane goes with the help or hindrance of the wind If the wind is going in the direction of the plane (called a tailwind), add the wind speed to the plane’s airspeed to find the plane’s grounspeed (in other words, the wind helps the plane to go faster than it would normally go). If the wind is going in the direction of the plane (called a headwind), subtract the wind speed from the plane’s airspeed to find the plane’s groundspeed (in other words, the wind pushes against the it go faster than it would normally go).

12. Define: Let A = the airspeed. Let W = the wind speed.
Applications of Linear Systems
LESSON 6-4
Additional Examples
Suppose it takes you 6.8 hours to fly about 2800 miles from Miami, Florida to Seattle, Washington. At the same time, your friend flies from Seattle to Miami. His plane travels with the same average airspeed, but his flight takes only 5.6 hours. Find the average airspeed of the planes. Find the average wind speed.
Define: Let A = the airspeed. Let W = the wind speed.
Words: with tail wind with head wind
(rate)(time) = distance (rate)(time) = distance
(A + W) (time) = distance (A  W) (time) = distance
Equations: (A + W) = (A + W) = 2800

13. (A + W)5.6 = 2800 A + W = 500 Divide each side by 5.6 5.6 5.6
Applications of Linear Systems
LESSON 6-4
Additional Examples
(continued)
Solve by elimination. First divide to get the variables on the left side of each equation with coefficients of 1 or –1.
(A + W)5.6 = A + W = 500 Divide each side by 5.6
(A – W)6.8 = A – W = Divide each side by 6.8.
Step 1: Eliminate W.
A + W = 500
A – W = Add the equations to eliminate W.
2A =
Step 2: Solve for A.
A = Divide each side by 2.
Step 3: Solve for W using either of the original equations.
A + W = Use the first equation.
W = Substitute for A.
W = Solve for W.
The average airspeed of the planes is mi/h. The average wind speed is 44.1 mi/h.

14. One antifreeze solution is 10% alcohol. Another antifreeze solution
Applications of Linear Systems
LESSON 6-4
Lesson Quiz
One antifreeze solution is 10% alcohol. Another antifreeze solution
is 18% alcohol. How many liters of each antifreeze solution should be combined to create 20 liters of antifreeze solution that is 15% alcohol?
2. A local band is planning to make a compact disk. It will cost $12,500 to record and produce a master copy, and an additional $2.50 to make each sale copy. If they plan to sell the final product for $7.50, how many disks must they sell to break even?
3. Suppose it takes you and a friend 3.2 hours to canoe 12 miles downstream (with the current). During the return trip, it takes you and your friend 4.8 hours to paddle upstream (against the current) to the original starting point. Find the average paddling speed in still water of you and your friend and the average speed of the current of the river. Round answers to the nearest tenth.
still water: 3.1 mi/h; current: 0.6 mi/h