Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ The postal service offers flat-rate shipping for priority mail in special boxes. Today, Pam shipped $9$ small boxes and $6$ large boxes, which cost her $\$138$ to ship. Meanwhile, Donald shipped $3$ small boxes and $7$ large boxes, and paid $\$116$ . How much does it cost to ship these two sizes of box? $\newline$ Shipping costs $\$\_\_\_\_$ for a small box and $\$\_\_\_\_$ for a large box.

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Algebra 1

Solve a system of equations using elimination: word problems

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

\newline

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

9

small boxes and

6

large boxes, which cost her

\$138

to ship. Meanwhile, Donald shipped

3

small boxes and

7

large boxes, and paid

\$116

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

\newline

Shipping costs

\$\_\_\_\_

for a small box and

\$\_\_\_\_

for a large box.

Define variables: Define the variables for the cost of shipping a small box and a large box. $\newline$ Let $x$ be the cost to ship a small box, and $y$ be the cost to ship a large box.
Write Pam's equation: Write the equation for Pam's shipment. $\newline$ Pam shipped $9$ small boxes and $6$ large boxes for $\$138$ . $\newline$ The equation is: $9x + 6y = 138$ .
Write Donald's equation: Write the equation for Donald's shipment. $\newline$ Donald shipped $3$ small boxes and $7$ large boxes for $\$116$ . $\newline$ The equation is: $3x + 7y = 116$ .
Eliminate variable $x$ : Choose which variable to eliminate. $\newline$ We will eliminate $x$ by multiplying the second equation by $3$ to match the coefficient of $x$ in the first equation.
Multiply second equation: Multiply the second equation by $3$ . $\newline$ $3(3x + 7y) = 3(116)$ $\newline$ $9x + 21y = 348$
Subtract equations: Subtract the first equation from the modified second equation to eliminate $x$ . $\$9x + 21y$ - $9x + 6y$ = $348$ - $138$ \) $9x + 21y - 9x - 6y = 348 - 138$ $15y = 210$
Solve for y: Solve for y. $\newline$ $15y = 210$ $\newline$ $y = \frac{210}{15}$ $\newline$ $y = 14$
Substitute y value: Substitute the value of y into one of the original equations to solve for x. $\newline$ Using the first equation: $9x + 6(14) = 138$ $\newline$ $9x + 84 = 138$ $\newline$ $9x = 138 - 84$ $\newline$ $9x = 54$
Solve for x: Solve for x. $\newline$ $9x = 54$ $\newline$ $x = \frac{54}{9}$ $\newline$ $x = 6$