Lesson 3. 5 Identifying Solutions

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Lesson 3. 5 Identifying Solutions. Concept: Identifying Solutions EQ: How do we identify and interpret the solutions of an equation f(x) = g(x)? Standard: REI.10-11 Vocabulary: Expenses, Income, Profit, Break-even point.

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Lesson 3. 5

Identifying Solutions

Concept:Identifying Solutions

EQ:How do we identify and interpret the solutions of an equation f(x) = g(x)?

Standard:REI.10-11

Vocabulary:Expenses, Income, Profit, Break-even point

A solutionto a system of equations is a value that makes both equations true.

(-1,-3)

y=-x -4 y=2x -1

-3=-(-1) -4

-3=2(-1) -1

-3= -2 -1

-3=-3

-3= -3

In real world problems, we are often only concerned with the x-coordinate.

Remember that in real-world problems, the slope of the equation is the amount that describes the rate of change, and the y-intercept is the amount that represents the initial value.

For business problems that deal with making a profit, the break-even point is when the expenses and the income are equal. In other words you don’t make money nor lose money…your profit is \$0.

Words to know for any business problems:

• Expenses- the money spent to purchase your product or equipment
• Income- the total money obtained from selling your product.
• Profit - the expenses subtracted from the income.
• Break-even point- the point where the expenses and the income are equal. In other words you don’t make money nor lose money…your profit is \$0.

In this lesson you will learn to find the x-coordinate of the intersection of two linear functions in three different ways:

By observing their graphs

Making a table

Setting the functions equal to each other (algebraically)

Example 1

Aly and Dwayne work at a water park and have to drain the water from the small pool at the bottom of their ride at the end of the month. Each uses a pump to remove the water.

Aly’s pool has 35,000 gallons of water in it and drains at a rate of 1,750 gallons a minute.

Dwayne’s pool has 30,000 gallons of water in it and drains at a rate of 1,000 gallons a minute.

After approximately how many minutes will Aly and Dwayne’s pools have the same amount of water in them?

Example 1

Aly’s pool has 35,000 gallons of water in it and drains at a rate of 1,750 gallons a minute.

Dwayne’s pool has 30,000 gallons of water in it and drains at a rate of 1,000 gallons a minute.

After approximately how many minutes will Aly and Dwayne’s pools have the same amount of water in them?

Example 1

slope

We need to write 2 equations!

First we can identify our slope and y-intercept.

y-intercept

Aly’s pool has 35,000 gallons of water in it and drains at a rate of 1,750 gallons a minute.

Dwayne’s pool has 30,000 gallons of water in it and drains at a rate of 1,000 gallons a minute.

Both of the slopes will be negative because the water is leaving the pools.

y-intercept

slope

x=# of minutes; a(x) & d(x)=amount of water left in pool

Aly’s a(x)= -1,750x + 35,000

Dwayne’s d(x)= -1,000x + 30,000

Example 1

The graph below represents the amount of water in Aly’s pool, a(x), and Dwayne’s pool, d(x), over time. After how many minutes will Aly’s pool and Dwayne’s pool have the same amount of water?

Approximate the x-coordinate.

Aly’s pool and Dwayne’s pool will have an equal amount of water after 10 minutes.

In a problem like this, we are only concerned with the x-coordinate.

Find the point of intersection.

Aly’s pool

Dwayne’s pool

Example 1

Can you think of a problem where an approximation might be sufficient?

Here, the graph helps us solve, but graphing can also help us to estimate the solution.

Aly’s Pool

Dwayne’s Pool

Example 2

slope

We need to write 2 equations!

First we can identify our slope and y-intercept.

y-intercept

A large cheese pizza at Paradise Pizzeria costs \$6.80 plus \$0.90 for each topping.

The cost of a large cheese pizza at Geno’sPizza is \$7.30 plus \$0.65 for each topping.

How many toppings need to be added to a large cheese pizza from Paradise and Geno’s in order for the pizzas to cost the same, not including tax?

slope

y-intercept

Geno’s g(x)=.65x + 7.30

Example 2

We need one chart but 3 columns for two equations!

Now we make a chart to organize our data!

Geno’s g(x)=.65x + 7.30

x=# of toppings; p(x) & g(x)=total cost

x p(x)=.90x + 6.80

g(x)=.65x + 7.30

0 .90(0) + 6.80 = 6.80

.65(0) + 7.30 = 7.30

.65(1) + 7.30 = 7.95

1 .90(10 + 6.80 = 7.70

2 .90(2) + 6.80 = 8.60

.65(2) + 7.30 = 8.60

The pizzas cost the same!

the pizzas will cost the same!

Example 3

Eric sells model cars from a booth at a local flea market. He purchases each model car from a distributor for \$12, and the flea market charges him a booth fee of \$50. Eric sells each model car for \$20. How many model cars must Eric sell in order to reach the break-even point?

Example 3

We need to write 2 equations!

slope

First we can identify our slope and y-intercept.

Eric sells model cars from a booth at a local flea market. He purchases each model car from a distributor for \$12, and the flea market charges him a booth fee of \$50. Eric sells each model car for \$20.

slope

y-intercept

x=# of model cars; e(x)=Eric’s expenses; f(x)= Eric’s Income

e(x)= 12x + 50

f(x)= 20x

Example 3

Eric’s Expenses e(x)=12x + 50

Eric’s Income f(x)=20x

Since both e(x) and f(x) are are equal to “y”, you can set the equations equal to each other and solve for “x”.

Eric needs to sell more than 6 model cars to break even!

e(x) = f(x)

12x + 50 = 20x

50 = 8x

6.25 = x

Example 3

Eric’s Expenses e(x)=12x + 50

Eric’s Income f(x)=20x

How can we write a function to represent Eric’s Profit?

Profit = Income – Expenses

So take the two given functions and subtract them.

P(x) = f(x) – e(x)

P(x) = 20x – (12x + 50)

P(x) = 8x - 50

You Try 1 – Solve using graphing

Chen starts his own lawn mowing business. He initially spends \$180 on a new lawnmower. For each yard he mows, he receives \$20 and spends \$4 on gas.

If x represents the # of lawns, then let Chen’s expenses be modeled by the function m(x)=4x + 180 and his income be modeled by the function p(x) = 20x

How many lawns must Chen mow to break-even?

You Try 2 – Solve using a table

Olivia is building birdhouses to raise money for a trip to Hawaii. She spends a total of \$30 on the tools needed to build the houses. The material to build each birdhouse costs \$3.25. Olivia sells each birdhouse for \$10.

If x represents the # of birdhouses, then let Olivia’s expenses be modeled by the function b(x)=3.25x + 30 and her income be modeled by the function p(x) = 10x

How many birdhouses must Olivia sell to break-even?

You Try 3 – Solve using algebra

Text Away cell phone company charges a flat rate of \$30 per month plus \$0.20 per text.

It’s Your Dime cell phone company charges a flat rate of \$20 per month plus \$0.40 per text.

If x represents the # of texts, then let your Text Away bill be modeled by the function t(x)=.20x + 30 and Your Dime bill be modeled by the function d(x) = .40x + 20

How many texts must you send before your bill for each company will be the same?