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Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks.\newlineAt a candy store, Miranda bought 22 kilograms of jelly beans and 11 kilogram of gummy worms for $13\$13. Meanwhile, Vicky bought 11 kilogram of gummy worms for $5\$5. How much does the candy cost?\newlineA kilogram of jelly beans costs $\$_____, and a kilogram of gummy worms costs $\$_____.

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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.\newlineAt a candy store, Miranda bought 22 kilograms of jelly beans and 11 kilogram of gummy worms for $13\$13. Meanwhile, Vicky bought 11 kilogram of gummy worms for $5\$5. How much does the candy cost?\newlineA kilogram of jelly beans costs $\$_____, and a kilogram of gummy worms costs $\$_____.
  1. Define Variables: Let xx be the cost per kilogram of jelly beans and yy be the cost per kilogram of gummy worms. We can write two equations based on the information given:\newlineEquation 11: 2x+y=132x + y = 13 (Miranda's purchase)\newlineEquation 22: y=5y = 5 (Vicky's purchase)
  2. Create Augmented Matrix: To solve using an augmented matrix, we first write the coefficients of the variables and the constants in a matrix form: [2amp;1amp;∣amp;13 0amp;1amp;∣amp;5]\begin{bmatrix} 2 & 1 & \vert & 13 \ 0 & 1 & \vert & 5 \end{bmatrix}
  3. Perform Row Operations: Since the second equation already has yy isolated, we can use it to substitute into the first equation. But let's stick to the matrix method and perform row operations to solve for xx and yy.
  4. Solve for xx: We can use the second row to make the yy coefficient in the first row zero. To do this, we multiply the second row by −1-1 and add it to the first row:\newline(-1)\cdot\begin{bmatrix}0 & 1 | & 5\end{bmatrix} + \begin{bmatrix}2 & 1 | & 13\end{bmatrix} = \begin{bmatrix}2 & 0 | & 8\end{bmatrix}\(\newlineNow the matrix looks like:\newline\$\begin{bmatrix}2 & 0 | & 8\0 & 1 | & 5\end{bmatrix}\)
  5. Solve for y: From the first row of the new matrix, we have \(2x = 8\). Dividing both sides by \(2\) gives us \(x = 4\).
  6. Final Cost Calculation: From the second row of the matrix, we have \(y = 5\), which we already knew from Vicky's purchase.
  7. Final Cost Calculation: From the second row of the matrix, we have \(y = 5\), which we already knew from Vicky's purchase.So, a kilogram of jelly beans costs \(\$4\), and a kilogram of gummy worms costs \(\$5\).

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