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, Dakota shipped $1$ small box and $9$ large boxes, which cost her $\$121$ to ship. Meanwhile, Abby shipped $7$ small boxes and $5$ large boxes, and paid $\$93$ . 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, Dakota shipped

1

small box and

9

large boxes, which cost her

\$121

to ship. Meanwhile, Abby shipped

7

small boxes and

5

large boxes, and paid

\$93

. 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: Let's define the variables for the cost to ship a small box as $x$ and the cost to ship a large box as $y$ . Dakota's shipment of $1$ small box and $9$ large boxes costing $\$121$ can be represented by the equation: $\newline$ $1x + 9y = 121$
Represent Dakota's shipment: Similarly, Abby's shipment of $7$ small boxes and $5$ large boxes costing $\$93$ can be represented by the equation: $\newline$ $7x + 5y = 93$
Create system of equations: We now have a system of two equations: $\newline$ $1x + 9y = 121$ $\newline$ $7x + 5y = 93$ $\newline$ To use elimination, we need to make the coefficients of one of the variables the same in both equations. We can multiply the first equation by $7$ to match the coefficient of $x$ in the second equation.
Use elimination method: Multiplying the first equation by $7$ gives us: $\newline$ $7x + 63y = 847$ $\newline$ Now we have the system: $\newline$ $7x + 63y = 847$ $\newline$ $7x + 5y = 93$
Eliminate $x$ : To eliminate $x$ , we subtract the second equation from the first: $\newline$ $(7x + 63y) - (7x + 5y) = 847 - 93$ $\newline$ $7x + 63y - 7x - 5y = 847 - 93$ $\newline$ $58y = 754$
Solve for y: Solving for y, we divide both sides by $58$ : $\newline$ $y = \frac{754}{58}$ $\newline$ $y = 13$
Substitute back to solve for $x$ : Now that we have the value for $y$ , we can substitute it back into one of the original equations to solve for $x$ . Let's use the first equation: $\newline$ $1x + 9(13) = 121$
Final shipping costs: Simplify and solve for $x$ : $x + 117 = 121$ $x = 121 - 117$ $x = 4$
Final shipping costs: Simplify and solve for $x$ : $x + 117 = 121$ $x = 121 - 117$ $x = 4$ We have found the values for $x$ and $y$ : $x = 4$ (cost to ship a small box) $y = 13$ (cost to ship a large box)The shipping costs $\$4$ for a small box and $\$13$ for a large box.