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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Duncan is going to ship some gifts to family members, and he is considering two shipping companies. The first shipping company charges a fee of $\$12$ to ship a medium box, plus an additional $\$3$ per pound. A second shipping company charges $\$5$ for the same size of box, plus an additional $\$4$ per pound. At a certain weight, the two shipping methods will cost the same amount. What is that weight? How much will it cost? $\newline$ At a weight of $\underline{\quad}$ pounds, the two shipping methods both cost $\$\underline{\quad}$ .

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Solve a system of equations using substitution: word problems

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

\newline

Duncan is going to ship some gifts to family members, and he is considering two shipping companies. The first shipping company charges a fee of

\$12

to ship a medium box, plus an additional

\$3

per pound. A second shipping company charges

\$5

for the same size of box, plus an additional

\$4

per pound. At a certain weight, the two shipping methods will cost the same amount. What is that weight? How much will it cost?

\newline

At a weight of

\underline{\quad}

pounds, the two shipping methods both cost

\$\underline{\quad}

.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the weight of the gifts in pounds. $\newline$ Let $y$ be the total cost of shipping. $\newline$ For the first shipping company: $\newline$ The cost is $\$12$ plus $\$3$ per pound. $\newline$ So the equation for the first company is: $\newline$ $y = 3x + 12$
First Shipping Company: For the second shipping company: $\newline$ The cost is $\$5$ plus $\$4$ per pound. $\newline$ So the equation for the second company is: $\newline$ $y = 4x + 5$
Second Shipping Company: Now we need to find the weight at which the cost of shipping is the same for both companies. This means we need to set the two equations equal to each other and solve for $x$ : $3x + 12 = 4x + 5$
Set Equations Equal: Subtract $3x$ from both sides to get: $\newline$ $12 = x + 5$
Solve for x: Now subtract $5$ from both sides to solve for $x$ : $\newline$ $x = 12 - 5$ $\newline$ $x = 7$
Find Total Cost: Now that we have the weight $x$ , we need to find the total cost $y$ . We can substitute $x$ back into either of the original equations. Let's use the first company's equation: $\newline$ $y = 3x + 12$ $\newline$ $y = 3(7) + 12$ $\newline$ $y = 21 + 12$ $\newline$ $y = 33$

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