Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Kira, an office manager, needs to find a courier to deliver a package. The first courier she is considering charges a fee of $\$16$ plus $\$6$ per kilogram. The second charges $\$12$ plus $\$7$ per kilogram. Kira 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 $\_\_$ kilograms, the two couriers both cost $\$\_\_\_\_\_$ .

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Solve a system of equations using any method: word problems

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

\newline

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

\$16

plus

\$6

per kilogram. The second charges

\$12

plus

\$7

per kilogram. Kira 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

\_\_

kilograms, the two couriers both cost

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

Define Weight and Costs: Let's denote the weight of the package as ' extit{w}' kilograms. The cost for the first courier is $\$16$ plus $\$6$ per kilogram, and the cost for the second courier is $\$12$ plus $\$7$ per kilogram. We can write two equations to represent the costs for each courier. $\newline$ First courier: Cost = $16 + 6w$ $\newline$ Second courier: Cost = $12 + 7w$ $\newline$ Since the costs are equivalent, we can set these two expressions equal to each other to find the weight '\textit{w}'.
Set Equation for Costs: Set up the equation based on the costs being equal. $16 + 6w = 12 + 7w$
Solve for Weight: Solve for $w$ by rearranging the equation. $\newline$ Subtract $6w$ from both sides: $\newline$ $16 = 12 + w$ $\newline$ Subtract $12$ from both sides: $\newline$ $4 = w$
Calculate Cost for First Courier: Now that we have the weight of the package, we can find out how much it will cost for either courier. $\newline$ Using the first courier's cost equation: $\newline$ Cost = $16 + 6w$ $\newline$ Cost = $16 + 6(4)$ $\newline$ Cost = $16 + 24$ $\newline$ Cost = $40$
Check Cost with Second Courier: To ensure we did not make a mistake, we can check the cost with the second courier's cost equation. $\newline$ $\text{Cost} = 12 + 7w$ $\newline$ $\text{Cost} = 12 + 7(4)$ $\newline$ $\text{Cost} = 12 + 28$ $\newline$ $\text{Cost} = 40$