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Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks.\newlineLola received some gift cards for music and movie downloads for her birthday. Using one of them, she downloaded 11 movie, which cost a total of $15\$15. Using another, she purchased 1717 songs and 77 movies, which cost a total of $139\$139. How much does each download cost?\newlineDownloads cost $\$_____ for a song and $\$_____ for a movie.

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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.\newlineLola received some gift cards for music and movie downloads for her birthday. Using one of them, she downloaded 11 movie, which cost a total of $15\$15. Using another, she purchased 1717 songs and 77 movies, which cost a total of $139\$139. How much does each download cost?\newlineDownloads cost $\$_____ for a song and $\$_____ for a movie.
  1. Question Prompt: question_prompt: How much does each song and movie download cost?
  2. Equations Setup: Let's call the cost of a song xx and the cost of a movie yy. We have two transactions:\newline11 movie = $15\$15, so y=15y = 15.\newline1717 songs and 77 movies = $139\$139, so 17x+7y=13917x + 7y = 139.
  3. Standard Form: Now we write the system of equations:\newliney=15y = 15\newline17x+7y=13917x + 7y = 139
  4. Augmented Matrix: To solve using an augmented matrix, we'll rewrite the equations in standard form:\newline0x+1y=150x + 1y = 15\newline17x+7y=13917x + 7y = 139
  5. Leading 11: The augmented matrix is:\newline \begin{array}{cc|c} 0 & 1 & 15 \ 17 & 7 & 139 \end{array}
  6. Row Operations: We need to get a leading 11 in the top left, but since the first row is already solved for yy, we can swap rows:\newline \begin{array}{cc|c} 17 & 7 & 139 \ 0 & 1 & 15 \end{array}
  7. New Matrix: Now we'll use row operations to make the element below the leading 11 in the first column to be 00: We can multiply the second row by 1717 and subtract it from the first row: 17∗[01∣15]=[017∣255]17*[0 1 | 15] = [0 17 | 255] [177∣139]−[017∣255]=[17−10∣−116][17 7 | 139] - [0 17 | 255] = [17 -10 | -116]
  8. Divide First Row: The new matrix is:\newline \begin{array}{cc|c} 17 & -10 & -116 \ 0 & 1 & 15 \end{array}
  9. Simplify First Row: Now we divide the first row by 1717 to get the leading 11: \newline1amp;−1017amp;∣−11617\begin{array}{ccc} 1 & -\frac{10}{17} & \left| -\frac{116}{17} \right.\end{array}
  10. Simplify First Row: Now we divide the first row by 1717 to get the leading 11:\newline[1−1017∣−11617][1 -\frac{10}{17} | -\frac{116}{17}] Simplify the first row:\newline[1−1017∣−6.8235… ][1 -\frac{10}{17} | -6.8235\dots]\newlineOops, there's a mistake here. The division should be exact, not a decimal.

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