Question: (\[ \begin{array}{l} \text{3. Jen joined the Fan Favorite Movie Club at the local movie…

\(\[ \begin{array}{l} \text{3. Jen joined the Fan Favorite Movie Club at the local movie theater. At this theater, the cost of admission in May} \\ \text{and June remains the same. In May, she saw 2 matinees and 3 regular-priced shows and spent \$38.50. In June, she} \\ \text{went to 6 matinees and one regular-priced show and spent \$47.50.} \end{array} \]\)

Solution

To find the cost of a matinee and regular-priced show, let’s set up the equations based on the given information: Let \( x \) be the price of a matinee, and \( y \) be the price of a regular-priced show. In May, Jen saw 2 matinees and 3 regular-priced shows, spending $38.50: \[ 2x + 3y = 38.50 \] In June, she saw 6 matinees and 1 regular-priced show, spending $47.50: \[ 6x + y = 47.50 \] Now, solve the system of equations. First, solve the second equation for \( y \): \[ 6x + y = 47.50 \] \[ y = 47.50 - 6x \] Substitute \( y = 47.50 - 6x \) into the first equation: \[ 2x + 3(47.50 - 6x) = 38.50 \] Simplify and solve for \( x \): \[ 2x + 142.50 - 18x = 38.50 \] \[ -16x + 142.50 = 38.50 \] \[ -16x = 38.50 - 142.50 \] \[ -16x = -104 \] \[ x = 6.50 \] Now substitute \( x = 6.50 \) back into the equation for \( y \): \[ y = 47.50 - 6(6.50) \] \[ y = 47.50 - 39 \] \[ y = 8.50 \] Thus, the cost of a matinee is $6.50, and the cost of a regular-priced show is $8.50.