Question

jen joined the fan favorite movie club at the local movie theater. at this theater, the cost of admission in may and june remains the same. in may, she saw 2 matinees and 3 regular - priced shows and spent $38.50. in june, she went to 6 matinees and one regular - priced show and spent $47.50.
write a system of equations to represent the cost, m, of a matinee ticket and the cost, r, of a regular - priced ticket.
jen said she spent $5.75 on each matinee and $9 on each regular show. is jen correct? justify your answer.
use your system of equations to algebraically determine both the actual cost of each matinee ticket and the actual cost of each regular ticket.

Explanation:

Part 1: Write the system of equations

Step1: Define variables and May's cost

Let \( m \) be the cost of a matinee ticket and \( r \) be the cost of a regular - priced ticket. In May, Jen saw 2 matinees and 3 regular - priced shows and spent $38.50. So the equation for May is \( 2m + 3r=38.50 \).

Step2: Define June's cost equation

In June, she went to 6 matinees and 1 regular - priced show and spent $47.50. So the equation for June is \( 6m + r = 47.50 \).

Step1: Substitute Jen's values into May's equation

Jen claims \( m = 5.75 \) and \( r = 9 \). Substitute into the first equation \( 2m+3r \):
\( 2(5.75)+3(9)=11.5 + 27=38.5 \)

Step2: Substitute Jen's values into June's equation

Substitute \( m = 5.75 \) and \( r = 9 \) into the second equation \( 6m + r \):
\( 6(5.75)+9=34.5+9 = 43.5
eq47.5 \)
Since the second equation is not satisfied, Jen is not correct.

Step1: Solve the second equation for \( r \)

From the equation \( 6m + r=47.50 \), we can express \( r \) in terms of \( m \) as \( r=47.50 - 6m \).

Step2: Substitute \( r \) into the first equation

Substitute \( r = 47.50-6m \) into \( 2m + 3r=38.50 \):
\( 2m+3(47.50 - 6m)=38.50 \)
Expand the left - hand side: \( 2m+142.5-18m = 38.50 \)
Combine like terms: \( 2m-18m=38.50 - 142.5 \)
\( - 16m=-104 \)

Step3: Solve for \( m \)

Divide both sides of the equation \( - 16m=-104 \) by \(-16\):
\( m=\frac{- 104}{-16}=6.5 \)

Step4: Solve for \( r \)

Substitute \( m = 6.5 \) into \( r = 47.50-6m \):
\( r=47.50-6\times6.5=47.50 - 39 = 8.5 \)

Answer:

The system of equations is \(

$$\begin{cases}2m + 3r=38.50\\6m + r=47.50\end{cases}$$

\)

Part 2: Is Jen correct?