Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. $\newline$ The postal service offers flat-rate shipping for priority mail in special boxes. Today, Irma shipped $1$ large box, which cost her $\$14$ to ship. Meanwhile, Britney shipped $6$ small boxes and $7$ large boxes, and paid $\$140$ . How much does it cost to ship these two sizes of box? $\newline$ Shipping costs $\$$ _ for a small box and $\$$ _ for a large box.

Full solution

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

\newline

The postal service offers flat-rate shipping for priority mail in special boxes. Today, Irma shipped

1

large box, which cost her

\$14

to ship. Meanwhile, Britney shipped

6

small boxes and

7

large boxes, and paid

\$140

. How much does it cost to ship these two sizes of box?

\newline

Shipping costs

\$

_____ for a small box and

\$

_____ for a large box.

Set up equations: Let's set up two equations to represent the costs. Let $x$ be the cost of a small box and $y$ be the cost of a large box. Irma's shipment gives us the equation $y = 14$ . Britney's shipment gives us the equation $6x + 7y = 140$ .
Write system of equations: Now we'll write the system of equations: $\newline$ $1$ . $y = 14$ $\newline$ $2$ . $6x + 7y = 140$
Convert to matrix form: To solve using an augmented matrix, we'll convert the system of equations into matrix form. The matrix will look like this: $\newline$ \begin{array}{cc|c} 0 & 1 & 14 \ 6 & 7 & 140 \end{array}
Eliminate variable $y$ : We need to use the first equation to eliminate the $y$ variable from the second equation in the matrix. But since the first equation already has $y$ isolated and the coefficient is $1$ , we can just multiply the first row by $-7$ and add it to the second row to eliminate $y$ from the second equation.
Perform row operation: After performing the row operation, our matrix should now look like this: $\newline$ $\begin{matrix} 0 & 1 & | & 14 \ 6 & 0 & | & 140 - (7 \times 14) \end{matrix}$