Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Lola just became a personal trainer and is finalizing her pricing plans. One plan is to charge $\$17$ for the initial consultation and then $\$38$ per session. Another plan is to charge $\$19$ for the consultation and $\$36$ per session. Lola 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

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

\$17

for the initial consultation and then

\$38

per session. Another plan is to charge

\$19

for the consultation and

\$36

per session. Lola 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 for the number of sessions as $n$ and the total cost for each plan as $C$ . We can write two equations to represent each plan: $\newline$ Plan $1$ : $C = 17 + 38n$ $\newline$ Plan $2$ : $C = 19 + 36n$ $\newline$ We are looking for the number of sessions $n$ where the cost $C$ is the same for both plans.
Use substitution to solve: To solve for $n$ , we can use substitution by setting the two equations equal to each other since they both equal $C$ at the point where the plans cost the same. $\newline$ $17 + 38n = 19 + 36n$
Subtract to isolate variable: Now, we will solve for $n$ by subtracting $36n$ from both sides of the equation: $\newline$ $17 + 38n - 36n = 19 + 36n - 36n$ $\newline$ $17 + 2n = 19$
Divide to solve for n: Next, we subtract $17$ from both sides to isolate the variable ' $n$ ': $\newline$ $17 + 2n - 17 = 19 - 17$ $\newline$ $2n = 2$
Substitute $n$ back in equation: Now, we divide both sides by $2$ to solve for ' $n$ ': $\newline$ $\frac{2n}{2} = \frac{2}{2}$ $\newline$ $n = 1$
Calculate total cost: We have found that the number of sessions ' $n$ ' where the cost is the same for both plans is $1$ . Now we need to find out what that cost is. We can substitute ' $n$ ' back into either of the original equations. Let's use Plan $1$ : $\newline$ $C = 17 + 38n$ $\newline$ $C = 17 + 38(1)$
Calculate total cost: We have found that the number of sessions 'n' where the cost is the same for both plans is $1$ . Now we need to find out what that cost is. We can substitute 'n' back into either of the original equations. Let's use Plan $1$ : $\newline$ $C = 17 + 38n$ $\newline$ $C = 17 + 38(1)$ Now, we calculate the total cost ' $C$ ' for $1$ session: $\newline$ $C = 17 + 38$ $\newline$ $C = 55$