Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Kate, an office manager, needs to find a courier to deliver a package. The first courier she is considering charges a fee of $\$14$ plus $\$1$ per pound. The second charges $\$13$ plus $\$2$ per pound. Kate determines that, given her 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 $\_\_$ pounds, the two couriers both cost $\$\_\_\_\_\_\_$ .

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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

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

\$14

plus

\$1

per pound. The second charges

\$13

plus

\$2

per pound. Kate determines that, given her 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

\_\_

pounds, the two couriers both cost

\$\_\_\_\_\_\_

Define Variables: Let's define the variables: $\newline$ Let $x$ be the weight of the package in pounds. $\newline$ Let $y$ be the total cost for the courier service. $\newline$ The first courier's cost can be represented by the equation: $y = 14 + 1 \times x$ . $\newline$ The second courier's cost can be represented by the equation: $y = 13 + 2 \times x$ . $\newline$ We need to write a system of equations to represent the situation.
Write Equations: Now we have the system of equations: $\newline$ $1$ ) $y = 14 + x$ $\newline$ $2$ ) $y = 13 + 2x$ $\newline$ We will use substitution to solve this system. Since both equations equal $y$ , we can set them equal to each other to find the value of $x$ . $\newline$ $14 + x = 13 + 2x$
Use Substitution: Next, we solve for $x$ : $14 + x = 13 + 2x$ Subtract $x$ from both sides: $14 = 13 + x$ Subtract $13$ from both sides: $1 = x$ So, the weight of the package is $1$ pound.
Solve for x: Now that we have the value of x, we can substitute it back into either of the original equations to find the cost y. We'll use the first equation: $\newline$ y = $14$ + x $\newline$ y = $14$ + $1$ $\newline$ y = $15$ $\newline$ So, the cost for both couriers is $\$15$ .