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Lora has a gift card where all downloads for music cost the same amount. after downloading 2 songs, the balance on her card was \$18.10. after downloading a total of 5 songs, the balance was \$15.25

1. write an equation in slope-intercept form that represents the amount in dollars remaining on the card as a function of songs downloaded.
2. identify the slope of the line and tell what the slope represents.
3.identify the y intercept of the line and tell what it represents.
4. how many additional songs can Lora download when there is \$15.25 left on the card?
in Mathematics by Platinum (91,198 points) | 7 views

1 Answer

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let \(x=\) number of songs that are downloaded.
when \(x=2\), the remaining balance is \(18.10\)
when \(x=5\), the remaining balance is \(15.25\)
the slope intercept form of the equation for a straight line is \(y=m x+b\)
\(m\) is the slope
\(b\) is the \(y\)-intercept.
the formula for the slope is:
\[
\begin{aligned}
&m=(y 2-y 1) /(x 2-x 1) \\
&(x 1, y 1)=(2,18.10) \\
&\left(x^{2}, y^{2}\right)=(5,15.25)
\end{aligned}
\]
the \(x\)-value of the coordinate pair is the number of songs that were downloaded. the \(y\)-value of the coordinate pair is the remaining balance.
slope \(=m=(y 2-y 1) /(x 2-x 1)=(15.25-18.10) /(5-2)=-2.85 / 3=-.95\)
replace \(m\) with - \(.95\) in the slope intercept form of the equation to get:

\[
y=-.95 * x+b
\]
now you want to solve for \(b\), which is the \(y\)-intercept.
take any one of the point pairs on the line and replace y with the y-coordinate and replace \(x\) with the \(x\)-coordinate and solve for \(b\).
we'll use the point \((2,18.1)\)
\(y=-.95 * x+b\) becomes:
\(18.1=-.95 * 2+b\)
simplify to get:
\(18.1=-1.9+b\)
add \(1.9\) to both sides of the equation to get:
\(18.1+1.9=-1.9+1.9+b\)
combine like terms to get:
\[
20=b
\]
that's your \(y\)-intercept.

the slope intercept form of the equation becomes:
\[
y=-.95^{*} x+20
\]

answer to question 1:
the equation is \(y=-.95 * x+20\)
answer to question 2:
the \(y\)-intercept is 20
it represents the remaining balance of the account before any songs were downloaded.
answer to question 3:

find out the \(x\)-intercept.
the \(x\)-intercept is the value of \(x\) when \(y\) is equal to 0 .
this will be when there is no more money remaining in the account.
find this by setting \(y=0\) and solving for \(x\)
equation of \(y=-.95 * x+20\) becomes:
\(0=-.95 * x+20\)
subtract 20 from both sides of this equation to get:
\(-20=-.95 * x+20-20\)
combine like terms to get:
\[
-20=-.95 * x
\]
divide both sides of this equation by \(-.95\) to get:
\[
-20 /-.95=x
\]

solve for \(x\) to get:
\(x=21.0505263158\) which we can rounded off to \(21.05\)
when she bought \(21 \mathrm{~cd}\), she had a small balance left.
that balance is \(y=-.95 * 21+20=.05\) dollars which is equal to 5 cents.
there's not enough money left in the account to buy another song, so this is her limit.
when she had \(15.25\) remaining balance, she had already bought 5 songs.
she will run out of money when she buys 21 songs.
she can buy an additional \(21-5=16\) songs before she runs out of money.
by Platinum (91,198 points)

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