Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Brittany just became a personal trainer and is finalizing her pricing plans. One plan is to charge $\$41$ for the initial consultation and then $\$96$ per session. Another plan is to charge $\$13$ for the consultation and then $\$98$ per session. Brittany realizes that the two plans have the same cost for a certain number of sessions. How many sessions is that? What is that cost? $\newline$ For _ sessions, the cost is $\$$ _ on either plan.

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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

Brittany just became a personal trainer and is finalizing her pricing plans. One plan is to charge

\$41

for the initial consultation and then

\$96

per session. Another plan is to charge

\$13

for the consultation and then

\$98

per session. Brittany realizes that the two plans have the same cost for a certain number of sessions. How many sessions is that? What is that cost?

\newline

For _____ sessions, the cost is

\$

_____ on either plan.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of sessions. $\newline$ Let $C$ be the total cost for the sessions. $\newline$ We can write two equations to represent each plan: $\newline$ Plan $1$ : $C = 41 + 96x$ $\newline$ Plan $2$ : $C = 13 + 98x$ $\newline$ We are looking for the number of sessions ( $x$ ) where the cost ( $C$ ) is the same for both plans.
Set Equations Equal: To solve the system using substitution, we set the two equations equal to each other because they both equal $C$ at the point where the plans cost the same: $\newline$ $41 + 96x = 13 + 98x$
Solve for x: Now, we solve for x: $\newline$ Subtract $96x$ from both sides: $\newline$ $41 + 96x - 96x = 13 + 98x - 96x$ $\newline$ $41 = 13 + 2x$
Isolate $x$ : Subtract $13$ from both sides to isolate the term with $x$ : $41 - 13 = 13 - 13 + 2x$ $28 = 2x$
Calculate Total Cost: Divide both sides by $2$ to solve for $x$ : $\frac{28}{2} = \frac{2x}{2}$ $14 = x$
Final Cost Calculation: Now that we have the number of sessions $x = 14$ , we can find the total cost $C$ for that number of sessions using either of the original equations. Let's use Plan $1$ 's equation: $\newline$ $C = 41 + 96x$ $\newline$ $C = 41 + 96(14)$
Final Cost Calculation: Now that we have the number of sessions $x = 14$ , we can find the total cost $C$ for that number of sessions using either of the original equations. Let's use Plan $1$ 's equation: $\newline$ $C = 41 + 96x$ $\newline$ $C = 41 + 96(14)$ Calculate the total cost: $\newline$ $C = 41 + 96(14)$ $\newline$ $C = 41 + 1344$ $\newline$ $C = 1385$