Lesson 3. 5 Identifying Solutions

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

Let’s Review

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

y=-x -4 y=2x -1(-1,-3)

-3=-(-1) -4

-3=-3 ✓-3=2(-1) -1-3= -2 -1-3= -3✓

Let’s Review

0 1 2 3 4 5 60

5

10

15

20

25

The point where two lines intersect is a solution to both equations.

In real world problems, we are often only

concerned with the x-coordinate.

Let’s Review

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.

Let’s Review

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:

1. By observing their graphs2. Making a table3. Setting the functions equal to

each other (algebraically)

Core Lesson

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

Core Lesson

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

Core LessonWe need to

write 2 equations!

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.

First we can identify our slope and y-

intercept.

slope

slopey-intercept

y-intercept

Aly’s a(x)= -1,750x + 35,000 Dwayne’s d(x)= -1,000x + 30,000

Example 1

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

Both of the slopes will be negative

because the water is leaving the

pools.

Core Lesson

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? Aly’s pool

Dwayne’s pool

Find the point of intersection.

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.

Example 1

Core Lesson

Aly’s Pool

Dwayne’s Pool

Here, the graph helps us solve, but graphing can also

help us to estimate the solution.

Can you think of a problem where an

approximation might be sufficient?

Example 1

Core Lesson

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’s Pizza 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?

First we can identify our slope and y-

intercept.

slope

slopey-intercept

y-interceptWe need to

write 2 equations!

Paradise p(x)=.90x + 6.80 Geno’s g(x)=.65x + 7.30

Example 2

Core Lesson

Geno’s g(x)=.65x + 7.30Paradise p(x)=.90x + 6.80

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

The pizzas cost the same!After adding two toppings,the pizzas will cost the same!

g(x)=.65x + 7.30

.65(0) + 7.30 = 7.30

.65(1) + 7.30 = 7.95

.65(2) + 7.30 = 8.60

x p(x)=.90x + 6.80

0 .90(0) + 6.80 = 6.80

1 .90(10 + 6.80 = 7.70

2 .90(2) + 6.80 = 8.60

Example 2Now we make a

chart to organize our

data!

We need one chart but 3 columns for

two equations!

Core Lesson

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

Core Lesson

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.

First we can identify our slope and y-

intercept.

slopey-intercept

slopeWe need to

write 2 equations!

e(x)= 12x + 50

Example 3

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

f(x)= 20x

Core Lesson

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’s Expenses e(x)=12x + 50 Eric’s Income f(x)=20x

e(x) = f(x)12x + 50 = 20x

50 = 8x6.25 = x

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

Example 3

Core Lesson

Profit = Income – ExpensesSo take the two given functions and subtract them.

Eric’s Expenses e(x)=12x + 50

Example 3

How can we write a function to

represent Eric’s Profit?

Eric’s Income f(x)=20x

P(x) = f(x) – e(x)P(x) = 20x – (12x + 50)

P(x) = 8x - 50

Core Lesson 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?

Core Lesson 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?

Core Lesson

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?

You Try 3 – Solve using algebra