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
Top Related