Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Shannon, an office manager, needs to find a courier to deliver a package. The first courier she is considering charges a fee of $\$15$ plus $\$1$ per pound. The second charges $\$6$ plus $\$4$ per pound. Shannon determines that, given her package's weight, the two courier services are equivalent in terms of cost. How much will it cost? What is the weight? $\newline$ The two couriers both cost $\$$ _ at a package weight of _ pounds.

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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

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

\$15

plus

\$1

per pound. The second charges

\$6

plus

\$4

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

\newline

The two couriers both cost

\$

_____ at a package weight of _____ pounds.

Define Variables: Let $x$ represent the weight of the package in pounds and $y$ represent the cost to deliver the package. $\newline$ For the first courier: $\newline$ Cost = $\$15$ + $\$1$ per pound $\newline$ The equation based on the given information is: $\newline$ $y = 1x + 15$
First Courier: For the second courier: $\newline$ Cost = $\$6$ + $\$4$ per pound $\newline$ The equation based on the given information is: $\newline$ y = $4$ x + $6$
Second Courier: System of equations: $\newline$ $y = 1x + 15$ $\newline$ $y = 4x + 6$ $\newline$ To find the weight $x$ at which the cost $y$ is the same for both couriers, we set the two equations equal to each other: $\newline$ $1x + 15 = 4x + 6$
System of Equations: Solve for $x$ by isolating the variable. $\newline$ Subtract $1x$ from both sides: $\newline$ $1x + 15 - 1x = 4x + 6 - 1x$ $\newline$ $15 = 3x + 6$ $\newline$ Now, subtract $6$ from both sides: $\newline$ $15 - 6 = 3x + 6 - 6$ $\newline$ $9 = 3x$
Solve for x: Divide both sides by $3$ to solve for $x$ : $\frac{9}{3} = \frac{3x}{3}$ $3 = x$ So, the weight of the package is $3$ pounds.
Substitute x: Substitute $x = 3$ into one of the original equations to find the cost $y$ . $\newline$ Using the first courier's equation: $\newline$ $y = 1x + 15$ $\newline$ $y = 1(3) + 15$ $\newline$ $y = 3 + 15$ $\newline$ $y = 18$ $\newline$ So, the cost to deliver the package is $\$18$ .