Question: 3. Volunteer drivers are needed to bring 80 students to the championship baseball game. Drivers…

3. Volunteer drivers are needed to bring 80 students to the championship baseball game. Drivers either have cars, which can seat 4 students, or vans, which can seat 6 students. The equation 4c + 6v = 80 describes the relationship between the number of cars, c, and number of vans, v, that can transport exactly 80 students. Select all statements that are true about the situation A. If 12 cars go, the 2 vans are needed B. c = 14 and v = 4 are a pair of solutions to the equation.

Solution

The problem involves a linear equation that describes the transportation situation for 80 students using cars and vans. The equation given is: \[ 4c + 6v = 80 \] Let’s analyze each statement: A. If 12 cars go, 2 vans are needed. Substitute \( c = 12 \) into the equation: \[ 4(12) + 6v = 80 \] Calculate: \[ 48 + 6v = 80 \] Subtract 48 from both sides: \[ 6v = 32 \] Divide both sides by 6: \[ v = \frac{32}{6} = \frac{16}{3} \] Since \( v \) is not an integer, this statement is false. B. \( c = 14 \) and \( v = 4 \) are a pair of solutions to the equation. Substitute \( c = 14 \) and \( v = 4 \) into the equation: \[ 4(14) + 6(4) = 80 \] Calculate: \[ 56 + 24 = 80 \] Since the equation holds true, this statement is true. Therefore, only statement B is true.