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Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks.\newlineSandra is working furiously to knit scarves and beanies for a craft fair next weekend. Yesterday she completed 11 scarf, using a total of 88 yards of yarn. The day before, she used 3030 yards to knit 33 scarves and 33 beanies. Assuming Sandra is using the same pattern and type of yarn for each scarf and beanie, how much yarn does each project require?\newlineEach scarf requires ____\_\_\_\_ yards of yarn, and each beanie requires ____\_\_\_\_ yards.

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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.\newlineSandra is working furiously to knit scarves and beanies for a craft fair next weekend. Yesterday she completed 11 scarf, using a total of 88 yards of yarn. The day before, she used 3030 yards to knit 33 scarves and 33 beanies. Assuming Sandra is using the same pattern and type of yarn for each scarf and beanie, how much yarn does each project require?\newlineEach scarf requires ____\_\_\_\_ yards of yarn, and each beanie requires ____\_\_\_\_ yards.
  1. Define Variables: Let xx be the amount of yarn for a scarf and yy be the amount of yarn for a beanie.\newlineFrom the first day: 11 scarf = 88 yards, so we have the equation 1x+0y=81x + 0y = 8.
  2. Form Equations: From the second day: 33 scarves and 33 beanies = 3030 yards, so we have the equation 3x+3y=303x + 3y = 30.
  3. Create Augmented Matrix: We now have the system of equations:\newline1x+0y=81x + 0y = 8\newline3x+3y=303x + 3y = 30\newlineWe can represent this system as an augmented matrix.
  4. Row Operations: The augmented matrix is:\newline \begin{array}{cc|c} 1 & 0 & 8 \ 3 & 3 & 30 \end{array} \newlineWe will use row operations to solve for xx and yy.
  5. Eliminate Variables: First, we'll make the leading coefficient of the second row a 11 by dividing the entire row by 33.\newlineNew matrix:\newline\begin{matrix}1 & 0 & | & 8\1 & 1 & | & 10\end{matrix}
  6. Solve for yy: Subtract the first row from the second row to eliminate xx from the second row.\newlineNew matrix:\newline1amp;0∣amp;8 0amp;1∣amp;2\begin{matrix} 1 & 0 | & 8 \ 0 & 1 | & 2 \end{matrix}
  7. Substitute and Find xx: Now we can see from the second row that 1y=21y = 2, so y=2y = 2.
  8. Final Results: Substitute y=2y = 2 into the first equation, 1x+0y=81x + 0y = 8, to find xx.\newline1x+0(2)=81x + 0(2) = 8\newlinex=8x = 8
  9. Final Results: Substitute y=2y = 2 into the first equation, 1x+0y=81x + 0y = 8, to find xx.
    1x+0(2)=81x + 0(2) = 8
    x=8x = 8We found x=8x = 8 and y=2y = 2.
    Each scarf requires 88 yards of yarn, and each beanie requires 22 yards.

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