Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Fred, an office manager, needs to find a courier to deliver a package. The first courier he is considering charges a fee of $\$10$ plus $\$3$ per pound. The second charges $\$15$ plus $\$2$ per pound. Fred determines that, given his package's weight, the two courier services are equivalent in terms of cost. What is the weight? How much will it cost? $\newline$ At a package weight of $\underline{\quad}$ pounds, the two couriers both cost $\$\underline{\quad}$ .

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Grade 8

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

Fred, an office manager, needs to find a courier to deliver a package. The first courier he is considering charges a fee of

\$10

plus

\$3

per pound. The second charges

\$15

plus

\$2

per pound. Fred determines that, given his package's weight, the two courier services are equivalent in terms of cost. What is the weight? How much will it cost?

\newline

At a package weight of

\underline{\quad}

pounds, the two couriers both cost

\$\underline{\quad}

Define Variables: Let's define the variables. Let $x$ represent the weight of the package in pounds, and let $y$ represent the total cost for the courier service.
Write Equations: Write the equations based on the given information. For the first courier, the cost is $\$10$ plus $\$3$ per pound, which gives us the equation $y = 3x + 10$ . For the second courier, the cost is $\$15$ plus $\$2$ per pound, which gives us the equation $y = 2x + 15$ .
Set Equations Equal: Since the costs are equivalent, we can set the two equations equal to each other to find the weight of the package. So, we have $3x + 10 = 2x + 15$ .
Solve for x: Solve for x by subtracting $2x$ from both sides of the equation to isolate $x$ on one side. This gives us $x = 15 - 10$ .
Calculate x: Calculate the value of x. $x = 15 - 10 = 5$ . So, the weight of the package is $5$ pounds.
Substitute $x$ : Substitute the value of $x$ back into either of the original equations to find the total cost $y$ . Using the first courier's equation: $y = 3(5) + 10$ .
Calculate Total Cost: Calculate the total cost $y$ . $y = 3(5) + 10 = 15 + 10 = 25$ . So, the cost for both couriers is $\$25$ .