Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. $\newline$ Manuel often orders party trays from his favorite Mexican food restaurant for company events. For a recent company party, he spent $\$34$ on $1$ burrito platter. For a company meeting, he spent $\$123$ on $3$ burrito platters and $1$ taco platter. How much does each type of platter cost? $\newline$ Each burrito platter costs $\$\_\_\_\_$ and each taco platter costs $\$\_\_\_\_$ .

Full solution

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

\newline

Manuel often orders party trays from his favorite Mexican food restaurant for company events. For a recent company party, he spent

\$34

1

burrito platter. For a company meeting, he spent

\$123

3

burrito platters and

1

taco platter. How much does each type of platter cost?

\newline

Each burrito platter costs

\$\_\_\_\_

and each taco platter costs

\$\_\_\_\_

Define Variables: Let's call the cost of a burrito platter $b$ and the cost of a taco platter $t$ . We can write two equations based on the given information: $\newline$ $1$ burrito platter = $\$34$ , so $b = 34$ . $\newline$ $3$ burrito platters + $1$ taco platter = $\$123$ , so $3b + t = 123$ .
Write Equations: Now we'll write the system of equations as an augmented matrix: $\newline$ \begin{array}{cc|c} 1 & 0 & 34 \ 3 & 1 & 123 \ \end{array}
Create Augmented Matrix: To solve the system using an augmented matrix, we'll use row operations to get the matrix into reduced row-echelon form (RREF). First, we'll make sure the first row is our pivot row since it already has a leading $1$ .
Perform Row Operations: Next, we'll use the pivot row to eliminate the $3$ in the second row. We'll multiply the first row by $-3$ and add it to the second row: $\newline$ $-3 \times \begin{pmatrix} 1 & 0 \ 34 & \end{pmatrix} + \begin{pmatrix} 3 & 1 \ 123 & \end{pmatrix} = \begin{pmatrix} 0 & 1 \ 21 & \end{pmatrix}$ $\newline$ This gives us the new second row.