Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Mrs. Jefferson wants to get family portraits taken and is comparing prices between two different photography studios. Garrett Photography charges $\$14$ per portrait sheet, plus $\$70$ for the session fee. Lasting Memories Company charges $\$30$ for the session fee and $\$18$ per portrait sheet. If Mrs. Jefferson plans to purchase a certain number of portrait sheets, the cost will be the same at either studio. What would the total cost be? How many portrait sheets would that be? $\newline$ The cost of the portraits will be $\$$ _ if Mrs. Jefferson orders _ portrait sheets.

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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. Jefferson wants to get family portraits taken and is comparing prices between two different photography studios. Garrett Photography charges

\$14

per portrait sheet, plus

\$70

for the session fee. Lasting Memories Company charges

\$30

for the session fee and

\$18

per portrait sheet. If Mrs. Jefferson plans to purchase a certain number of portrait sheets, the cost will be the same at either studio. What would the total cost be? How many portrait sheets would that be?

\newline

The cost of the portraits will be

\$

_____ if Mrs. Jefferson orders _____ portrait sheets.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of portrait sheets Mrs. Jefferson plans to purchase. $\newline$ Let $y$ be the total cost for the portraits. $\newline$ Now, we can write the equations for each photography studio based on the given information. $\newline$ For Garrett Photography: $\newline$ The total cost is $\$14$ per portrait sheet plus a $\$70$ session fee. $\newline$ So, the equation is: $y = 14x + 70$
Write Equations - Garrett Photography: For Lasting Memories Company: The total cost is $\$18$ per portrait sheet plus a $\$30$ session fee. So, the equation is: $y = 18x + 30$
Write Equations - Lasting Memories Company: Since the cost will be the same at either studio for a certain number of portrait sheets, we can set the two equations equal to each other to find the value of $x$ . $14x + 70 = 18x + 30$
Set Equations Equal: Now, we solve for $x$ by subtracting $14x$ from both sides of the equation: $\newline$ $14x + 70 - 14x = 18x + 30 - 14x$ $\newline$ $70 = 4x + 30$
Solve for x: Next, we subtract $30$ from both sides of the equation to isolate the term with $x$ : $\newline$ $70 - 30 = 4x + 30 - 30$ $\newline$ $40 = 4x$
Isolate x Term: Now, we divide both sides by $4$ to solve for x: $\newline$ $\frac{40}{4} = \frac{4x}{4}$ $\newline$ $10 = x$
Solve for x: We have found that $x = 10$ , which means Mrs. Jefferson plans to purchase $10$ portrait sheets. Now we need to find the total cost $y$ . We can substitute $x = 10$ into either of the original equations. Let's use the Garrett Photography equation: $y = 14x + 70$ $y = 14(10) + 70$
Substitute $x$ into Equation: Now, we calculate the total cost: $y = 140 + 70$ $y = 210$