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, Jamal shipped $7$ small boxes and $4$ large boxes, which cost him $\$68$ to ship. Meanwhile, Brian shipped $1$ small box and $5$ large boxes, and paid $\$54$ . 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, Jamal shipped

7

small boxes and

4

large boxes, which cost him

\$68

to ship. Meanwhile, Brian shipped

1

small box and

5

large boxes, and paid

\$54

. 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 Equations: Let's denote the cost to ship a small box as $x$ dollars and the cost to ship a large box as $y$ dollars. We can write two equations based on the information given: $\newline$ For Jamal: $7$ small boxes + $4$ large boxes = $\$68$ $\newline$ For Brian: $1$ small box + $5$ large boxes = $\$54$ $\newline$ This translates to the system of equations: $\newline$ $7x + 4y = 68$ $\newline$ $x + 5y = 54$
Use Elimination Method: To use elimination, we need to make the coefficients of one of the variables the same in both equations. We can multiply the second equation by $7$ to match the coefficient of $x$ in the first equation: $\newline$ $7(1x + 5y) = 7(54)$ $\newline$ This gives us: $\newline$ $7x + 35y = 378$
Perform Subtraction: Now we have the system of equations: $\newline$ $7x + 4y = 68$ $\newline$ $7x + 35y = 378$ $\newline$ We can eliminate $x$ by subtracting the first equation from the second equation: $\newline$ $(7x + 35y) - (7x + 4y) = 378 - 68$
Solve for y: Perform the subtraction to solve for y: $\newline$ $7x + 35y - 7x - 4y = 378 - 68$ $\newline$ $31y = 310$ $\newline$ Now, divide both sides by $31$ to find y: $\newline$ $y = \frac{310}{31}$ $\newline$ $y = 10$
Substitute to Find $x$ : Now that we have the value of $y$ , we can substitute it back into one of the original equations to find $x$ . Let's use the second equation: $\newline$ $x + 5y = 54$ $\newline$ $x + 5(10) = 54$ $\newline$ $x + 50 = 54$ $\newline$ Subtract $50$ from both sides to solve for $x$ : $\newline$ $x = 54 - 50$ $\newline$ $x = 4$
Final Cost Solution: We have found the values of $x$ and $y$ : $\newline$ $x = 4$ (cost to ship a small box) $\newline$ $y = 10$ (cost to ship a large box) $\newline$ This means it costs $\$4$ to ship a small box and $\$10$ to ship a large box.