Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Mrs. Hutchinson is a dance teacher whose students are going to have a recital. She is comparing two companies' prices for filming the recital and producing DVDs. Kevin's Productions charges $\$3$ per DVD, plus $\$99$ to come and film the recital. Somerville Media charges $\$54$ to film, plus $\$4$ per DVD. To decide which company to use, Mrs. Hutchinson determines the number of DVDs that would make the two options equivalent in terms of cost. How much would each option cost? How many DVDs would that be? $\newline$ The cost would be $\$$ _ if Mrs. Hutchinson ordered _ DVDs.

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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

Mrs. Hutchinson is a dance teacher whose students are going to have a recital. She is comparing two companies' prices for filming the recital and producing DVDs. Kevin's Productions charges

\$3

per DVD, plus

\$99

to come and film the recital. Somerville Media charges

\$54

to film, plus

\$4

per DVD. To decide which company to use, Mrs. Hutchinson determines the number of DVDs that would make the two options equivalent in terms of cost. How much would each option cost? How many DVDs would that be?

\newline

The cost would be

\$

_____ if Mrs. Hutchinson ordered _____ DVDs.

Define Variables: Define the variables for the system of equations. $\newline$ Let $x$ represent the number of DVDs and $y$ represent the total cost. $\newline$ For Kevin's Productions: Cost = \$($3$) per DVD + \$($99$) filming fee.$\newline$For Somerville Media: Cost = \$($4$) per DVD + \$($54$) filming fee.
Write Equations: Write the system of equations based on the given information.$\newline$For Kevin's Productions: $y = 3x + 99$$\newline$For Somerville Media: $y = 4x + 54$
Solve Using Substitution: Solve the system of equations using substitution.$\newline$Set the two equations equal to each other since we want to find the number of DVDs that would make the costs equivalent.$\newline$$3x + 99 = 4x + 54$
Isolate Variable: Isolate the variable $x$ by moving terms involving $x$ to one side and constants to the other side.$\newline$Subtract $3x$ from both sides: $99 = x + 54$$\newline$Subtract $54$ from both sides: $45 = x$
Solve for x: Solve for x.$\newline$$x = 45$$\newline$This means that if Mrs. Hutchinson orders $45$ DVDs, the cost from both companies will be equivalent.
Substitute $x$: Substitute the value of $x$ back into one of the original equations to find the total cost $y$.$\newline$Using Kevin's Productions equation: $y = 3(45) + 99$$\newline$Calculate the cost: $y = 135 + 99$$\newline$$y = 234$
Verify Solution: Verify the solution by substituting $x = 45$ into the Somerville Media equation to ensure the cost is the same.\[y = 4(45) + 54\]Calculate the cost: \[y = 180 + 54\]\[y = 234\]