Question

modeling real life there are a total of 64 students in a filmmaking club and a yearbook club. the filmmaking club has 14 more students than the yearbook club. a. write a system of linear equations that represents this situation. let x represent the number of students in the filmmaking club and y represent the number of students in the yearbook club. system of equations: x = y + 14 \square x + \square y = 64 b. how many students are in the filmmaking club? the yearbook club? there are \square students in the filmmaking club and \square students in the yearbook club.

Explanation:

Part a

Step1: Analyze total students

The total number of students in both clubs is 64. Since \( x \) is filmmaking and \( y \) is yearbook, the equation is \( x + y = 64 \). So the coefficients for \( x \) and \( y \) are both 1.

Step2: Confirm the system

We already have \( x = y + 14 \) from the problem, and now we have \( 1x + 1y = 64 \).

Step1: Substitute \( x \) into total equation

We know \( x = y + 14 \) and \( x + y = 64 \). Substitute \( x \) in the second equation: \( (y + 14) + y = 64 \).

Step2: Solve for \( y \)

Simplify: \( 2y + 14 = 64 \). Subtract 14: \( 2y = 64 - 14 = 50 \). Divide by 2: \( y = \frac{50}{2} = 25 \).

Step3: Solve for \( x \)

Use \( x = y + 14 \), so \( x = 25 + 14 = 39 \).

Answer:

For part a, the system of equations is \( x = y + 14 \) and \( \boldsymbol{1}x + \boldsymbol{1}y = 64 \).

Part b