Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. $\newline$ Lola received some gift cards for music and movie downloads for her birthday. Using one of them, she downloaded $1$ movie, which cost a total of $\$15$ . Using another, she purchased $17$ songs and $7$ movies, which cost a total of $\$139$ . How much does each download cost? $\newline$ Downloads cost $\$$ _ for a song and $\$$ _ for a movie.

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Solve a system of equations using any method: word problems

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Q. Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks.

\newline

Lola received some gift cards for music and movie downloads for her birthday. Using one of them, she downloaded

1

movie, which cost a total of

\$15

. Using another, she purchased

17

songs and

7

movies, which cost a total of

\$139

. How much does each download cost?

\newline

Downloads cost

\$

_____ for a song and

\$

_____ for a movie.

Question Prompt: question_prompt: How much does each song and movie download cost?
Equations Setup: Let's call the cost of a song $x$ and the cost of a movie $y$ . We have two transactions: $\newline$ $1$ movie = $\$15$ , so $y = 15$ . $\newline$ $17$ songs and $7$ movies = $\$139$ , so $17x + 7y = 139$ .
Standard Form: Now we write the system of equations: $\newline$ $y = 15$ $\newline$ $17x + 7y = 139$
Augmented Matrix: To solve using an augmented matrix, we'll rewrite the equations in standard form: $\newline$ $0x + 1y = 15$ $\newline$ $17x + 7y = 139$
Leading $1$ : The augmented matrix is: $\newline$ \begin{array}{cc|c} 0 & 1 & 15 \ 17 & 7 & 139 \end{array}
Row Operations: We need to get a leading $1$ in the top left, but since the first row is already solved for $y$ , we can swap rows: $\newline$ \begin{array}{cc|c} 17 & 7 & 139 \ 0 & 1 & 15 \end{array}
New Matrix: Now we'll use row operations to make the element below the leading $1$ in the first column to be $0$ : We can multiply the second row by $17$ and subtract it from the first row: $17*[0 1 | 15] = [0 17 | 255]$ $[17 7 | 139] - [0 17 | 255] = [17 -10 | -116]$
Divide First Row: The new matrix is: $\newline$ \begin{array}{cc|c} 17 & -10 & -116 \ 0 & 1 & 15 \end{array}
Simplify First Row: Now we divide the first row by $17$ to get the leading $1$ : $\newline$ $\begin{array}{ccc} 1 & -\frac{10}{17} & \left| -\frac{116}{17} \right.\end{array}$
Simplify First Row: Now we divide the first row by $17$ to get the leading $1$ : $\newline$ $[1 -\frac{10}{17} | -\frac{116}{17}]$ Simplify the first row: $\newline$ $[1 -\frac{10}{17} | -6.8235\dots]$ $\newline$ Oops, there's a mistake here. The division should be exact, not a decimal.