Question

Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

Hayley received some gift cards for music and movie downloads for her birthday. Using one of them, she downloaded 20 songs and 20 movies, which cost a total of $240. Using another, she purchased 19 songs and 13 movies, which cost a total of $168. How much does each download cost?

Downloads cost $
for a song and $
for a movie.

Answer 1

Let's denote the cost of a song as \(x\) and the cost of a movie as \(y\). Based on the information provided in the problem, we can formulate the following system of equations:

1. From the first gift card:
\[
20x + 20y = 240
\]

2. From the second gift card:
\[
19x + 13y = 168
\]

Now we will simplify the first equation. We can divide everything by 20:

\[
x + y = 12 \quad \text{(Equation 1)}
\]

Now we have the system of equations:

1. \(x + y = 12\)
2. \(19x + 13y = 168\)

Next, we will use the elimination method to solve for \(x\) and \(y\). We can express \(y\) in terms of \(x\) from Equation 1:

\[
y = 12 - x
\]

Now, substitute \(y\) into the second equation:

\[
19x + 13(12 - x) = 168
\]

Distributing \(13\):

\[
19x + 156 - 13x = 168
\]

Combine like terms:

\[
(19x - 13x) + 156 = 168
\]

This simplifies to:

\[
6x + 156 = 168
\]

Next, we isolate \(x\):

\[
6x = 168 - 156
\]
\[
6x = 12
\]
\[
x = 2
\]

Now we can find \(y\) using Equation 1:

\[
y = 12 - x = 12 - 2 = 10
\]

Thus, the cost of each download is:

- Downloads cost **$2** for a song
- Downloads cost **$10** for a movie

Final answer:

- Downloads cost $**2** for a song and $**10** for a movie.