(a) The system that models the situation is: $m+r=20 \\2m+6r=60$ (b) There are $15$ multiple choice questions and $5$ extended response questions.
Work Step by Step
(a) Let $m =$ number of multiple choice questions $r =$ number of extended response questions. Knowing a total of $20$ questions: $m + r = 20$ (Equation 1) Knowing 60 minutes taken, with each multiple choice taking 2 minutes, and extended response taking 6 minutes. $2m + 6r = 60$ (Equation 2) Thus, the system of equations that model the relationship between the number of multiple choice questions and the number of extended-response questions is: $m+r=20 \\2m+6r=60$ Solve for $m$ in Equation 1 by subtracting $r$ to both sides: $m = 20 - r$ Substitute $20 - r$ to $m$ in Equation 2: $2m + 6r=60 \\2(20-r) + 6r = 60$ Distribute 2: $40 - 2r + 6r = 60$ Collect like terms: $40 + 4r = 60$ Subtract $40$ to both sides: $4r = 20$ Divide both sides by $4$ : $r = 5$ Substitute $r = 5$ into Equation 1: $m+r=20 \\m +5 = 20 \\m=20-5 \\m = 15$ Thus, $m = 15$ , $r = 5$ (b) Therefore, there are: $15$ multiple choice questions $5$ extended response questions.