Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Jeanette just became a personal trainer and is finalizing her pricing plans. One plan is to charge $\$49$ for the initial consultation and then $\$68$ per session. Another plan is to charge $\$38$ for the consultation and $\$79$ per session. Jeanette 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.

Full solution

Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

\newline

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

\$49

for the initial consultation and then

\$68

per session. Another plan is to charge

\$38

for the consultation and

\$79

per session. Jeanette 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.
Write Equations: We can write two equations to represent each plan: $\newline$ Plan $1$ : $C = 49 + 68x$ $\newline$ Plan $2$ : $C = 38 + 79x$
Set Equations Equal: Since Jeanette realizes that the two plans have the same cost for a certain number of sessions, we can set the two equations equal to each other to find the number of sessions where the cost is the same: $49 + 68x = 38 + 79x$
Solve for x: Now, we will solve for x using substitution or elimination. In this case, we will isolate x by subtracting $68x$ from both sides of the equation: $\newline$ $49 + 68x - 68x = 38 + 79x - 68x$ $\newline$ $49 = 38 + 11x$
Substitute $x$ : Next, we will subtract $38$ from both sides to solve for $x$ : $49 - 38 = 11x$ $11 = 11x$
Find Cost: Now, we divide both sides by $11$ to find the value of $x$ : $\frac{11}{11} = x$ $1 = x$
Substitute $x$ into Plan $1$ : We have found that the number of sessions for which the cost is the same is $1$ session. Now we need to find the cost for that session. We can substitute $x = 1$ into either of the original equations. Let's use Plan $1$ : $\newline$ $C = 49 + 68(1)$ $\newline$ $C = 49 + 68$
Calculate Total Cost: Now, we add $49$ and $68$ to find the total cost $C$ : $C = 117$