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

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineLola just became a personal trainer and is finalizing her pricing plans. One plan is to charge $17\$17 for the initial consultation and then $38\$38 per session. Another plan is to charge $19\$19 for the consultation and $36\$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?\newlineFor _____ sessions, the cost is $\$_____ on either plan.
  1. Define variables: Let's define the variables for the number of sessions as nn and the total cost for each plan as CC. We can write two equations to represent each plan:\newlinePlan 11: C=17+38nC = 17 + 38n\newlinePlan 22: C=19+36nC = 19 + 36n\newlineWe are looking for the number of sessions nn where the cost CC is the same for both plans.
  2. Use substitution to solve: To solve for nn, we can use substitution by setting the two equations equal to each other since they both equal CC at the point where the plans cost the same.\newline17+38n=19+36n17 + 38n = 19 + 36n
  3. Subtract to isolate variable: Now, we will solve for nn by subtracting 36n36n from both sides of the equation:\newline17+38n−36n=19+36n−36n17 + 38n - 36n = 19 + 36n - 36n\newline17+2n=1917 + 2n = 19
  4. Divide to solve for n: Next, we subtract 1717 from both sides to isolate the variable 'nn':\newline17+2n−17=19−1717 + 2n - 17 = 19 - 17\newline2n=22n = 2
  5. Substitute nn back in equation: Now, we divide both sides by 22 to solve for 'nn':\newline2n2=22\frac{2n}{2} = \frac{2}{2}\newlinen=1n = 1
  6. Calculate total cost: We have found that the number of sessions 'nn' where the cost is the same for both plans is 11. Now we need to find out what that cost is. We can substitute 'nn' back into either of the original equations. Let's use Plan 11:\newlineC=17+38nC = 17 + 38n\newlineC=17+38(1)C = 17 + 38(1)
  7. Calculate total cost: We have found that the number of sessions 'n' where the cost is the same for both plans is 11. 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 11:\newlineC=17+38nC = 17 + 38n\newlineC=17+38(1)C = 17 + 38(1)Now, we calculate the total cost 'CC' for 11 session:\newlineC=17+38C = 17 + 38\newlineC=55C = 55

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