Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Allie is going to ship some gifts to family members, and she is considering two shipping companies. The first shipping company charges a fee of $\$16$ to ship a medium box, plus an additional $\$6$ per kilogram. A second shipping company charges $\$12$ for the same size of box, plus an additional $\$8$ per kilogram. At a certain weight, the two shipping methods will cost the same amount. How much will it cost? What is that weight? $\newline$ The two shipping methods both cost $\$$ _ at a weight of _ kilograms.

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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

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

\$16

to ship a medium box, plus an additional

\$6

per kilogram. A second shipping company charges

\$12

for the same size of box, plus an additional

\$8

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

\newline

The two shipping methods both cost

\$

_____ at a weight of _____ kilograms.

Representation of Variables: Let $x$ represent the weight in kilograms and $y$ represent the total cost in dollars. $\newline$ For the first shipping company: $\newline$ Base fee: $\$16$ $\newline$ Cost per kilogram: $\$6$ $\newline$ Equation based on the given information: $\newline$ Total cost = base fee + (cost per kilogram * weight) $\newline$ $y = 6x + 16$ $\newline$ For the first company, the equation is: $y = 6x + 16$
First Shipping Company: For the second shipping company: $\newline$ Base fee: $\$12$ $\newline$ Cost per kilogram: $\$8$ $\newline$ Equation based on the given information: $\newline$ Total cost = base fee + (cost per kilogram $*$ weight) $\newline$ $y = 8x + 12$ $\newline$ For the second company, the equation is: $y = 8x + 12$
Second Shipping Company: System of equations: $\newline$ $y = 6x + 16$ $\newline$ $y = 8x + 12$ $\newline$ To find the weight at which the cost is the same, set the two equations equal to each other. $\newline$ $6x + 16 = 8x + 12$
System of Equations: Solve for $x$ : $\newline$ Subtract $6x$ from both sides of the equation: $\newline$ $6x + 16 - 6x = 8x + 12 - 6x$ $\newline$ $16 = 2x + 12$
Solving for x: Subtract $12$ from both sides of the equation: $\newline$ $16 - 12 = 2x + 12 - 12$ $\newline$ $4 = 2x$
Substituting $x$ into Equation: Divide both sides by $2$ to solve for $x$ : $\newline$ $\frac{4}{2} = \frac{2x}{2}$ $\newline$ $2 = x$ $\newline$ So, $x = 2$
Final Result: We found $x = 2$ . Now, find the value of $y$ by substituting $x$ into one of the original equations. $\newline$ Substitute $2$ for $x$ in $y = 6x + 16$ : $\newline$ $y = 6(2) + 16$ $\newline$ $y = 12 + 16$ $\newline$ $y = 28$ $\newline$ So, $y = 28$
Final Result: We found $x = 2$ . Now, find the value of $y$ by substituting $x$ into one of the original equations. $\newline$ Substitute $2$ for $x$ in $y = 6x + 16$ : $\newline$ $y = 6(2) + 16$ $\newline$ $y = 12 + 16$ $\newline$ $y = 28$ $\newline$ So, $y = 28$ We found: $\newline$ $x = 2$ $\newline$ $y = 28$ $\newline$ The weight at which both shipping methods cost the same is $2$ kilograms, and the cost at that weight is $y$ $3$ .