Exercise 17 Page 599

We will start by determining on what day(s) the company sold the same number of each player. Then we will be able to calculate how many players of each type were sold.

On what day did the company sell the same number of each player?

We are given two equations that model the numbers y of two portable music players sold x days after both players were introduced. Music Player A: & y =191x-32 Music Player B: & y =- x^2+200x+20 To determine on what day(s) the company sold the same number of each player, we have to find out for which x-value the y-values in both equations are equal. This is equivalent to solving the system of equations. y=191x-32 & (I) y=- x^2+200x+20 & (II) We will solve the system using the Substitution Method. Since y is isolated in both equations, we will substitute for this variable.

y=191x-32 y=- x^2+200x+20

Substitute

(II):y= 191x-32

y=191x-32 191x-32=- x^2+200x+20

▼

(II):Simplify

AddEqn

(II):LHS+x^2=RHS+x^2

y=191x-32 x^2+191x-32=200x+20

SubEqn

(II):LHS-200x=RHS-200x

y=191x-32 x^2-9x-32=20

SubEqn

(II):LHS-20=RHS-20

y=191x-32 x^2-9x-52=0

We want to solve the second equation for x. To do so we will factor it and then use the Zero-Product Property. x^2-9x-52=0 ⇔ x^2+(- 9)x+(- 52)=0 To factor the expression on the left-hand side, we have to find a pair of factors of - 52 that has a sum of - 9. Since the sum has to be negative, we will only consider the pairs of factors where the absolute value of the negative factor is greater than the positive factor.

Factors of - 52	Sum of Factors
1 and - 52	- 51
2 and - 26	- 24
4 and - 13	- 9 ✓

Now, we can rewrite the left-hand side of the equation in the factored form. x^2-9x-52=0 ⇔ (x+4)(x-13)=0 Finally, we will use the Zero-Product Property.

(x+4)(x-13)=0

ZeroProdProp

Use the Zero Product Property

lcx+4=0 & (I) x-13=0 & (II)

▼

(I), (II):Simplify

SubEqn

(I):LHS-4=RHS-4

lx=- 4 x-13=0

AddEqn

(II):LHS+13=RHS+13

lx=- 4 x=13

We have that x=- 4 or x=13. Since x stands for the number of days, it cannot be negative. Therefore, the only solution to the equation is x= 13. This means that on day 13 the company sold the same number of each player.

How many players of each type were sold?

To determine how many players of each type were sold, we have to substitute x= 13 into one of the original equations. Let's use the first one!

y=191x-32

Substitute

x= 13

y=191( 13)-32

Multiply

Multiply

y=2483-32

SubTerm

Subtract term

y=2451

2451 players of each type were sold.