Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Dr. Harvey, a pediatrician, has $2$ annual checkups and $3$ sick visits scheduled next Tuesday, which will fill a total of $190$ minutes on her schedule. Next Wednesday, she has $1$ annual checkup and $3$ sick visits on the schedule, which should take $140$ minutes. How much time is allotted for each type of appointment? $\newline$ The time allotted is $\_$ minutes for an annual checkup and $\_$ minutes for a sick visit.

Full solution

Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

\newline

Dr. Harvey, a pediatrician, has

2

annual checkups and

3

sick visits scheduled next Tuesday, which will fill a total of

190

minutes on her schedule. Next Wednesday, she has

1

annual checkup and

3

sick visits on the schedule, which should take

140

minutes. How much time is allotted for each type of appointment?

\newline

The time allotted is

\_

minutes for an annual checkup and

\_

minutes for a sick visit.

Define Variables: Let's denote the time for an annual checkup as $a$ minutes and the time for a sick visit as $s$ minutes. Dr. Harvey has $2$ annual checkups and $3$ sick visits scheduled next Tuesday, taking up $190$ minutes. This gives us the equation $2a + 3s = 190$ .
Equations for Tuesday: For next Wednesday, Dr. Harvey has $1$ annual checkup and $3$ sick visits scheduled, taking up $140$ minutes. This gives us the equation $a + 3s = 140$ .
Equations for Wednesday: We now have a system of two equations. To use elimination, we need to eliminate one of the variables, $a$ or $s$ . We choose to eliminate $a$ by multiplying the second equation by $2$ , the coefficient of $a$ in the first equation.
Elimination Method: Multiplying the second equation by $2$ gives us $2a + 6s = 280$ .
Multiply Second Equation: We now subtract the first equation from the new second equation to eliminate $a$ . This gives us $3s - 6s = 280 - 190$ , which simplifies to $-3s = -90$ .
Subtract Equations: Dividing both sides of $-3s = -90$ by $-3$ gives us $s = 30$ . This means each sick visit is allotted $30$ minutes.
Find Sick Visit Time: We substitute $s = 30$ into the first equation $2a + 3s = 190$ and solve for $a$ . This gives us $2a + 3(30) = 190$ , which simplifies to $2a + 90 = 190$ .
Substitute and Solve: Subtracting $90$ from both sides gives us $2a = 100$ . Dividing both sides by $2$ gives us $a = 50$ . This means each annual checkup is allotted $50$ minutes.