Question

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

Dr. Stewart, a pediatrician, has 3 annual checkups and 1 sick visit scheduled next Tuesday, which will fill a total of 174 minutes on his schedule. Next Wednesday, he has 2 annual checkups and 2 sick visits on the schedule, which should take 152 minutes. How much time is allotted for each type of appointment?

The time allotted is how many minutes for an annual checkup and how many minutes for a sick visit.

Answer 1

Let \( x \) represent the time allotted for an annual checkup (in minutes) and \( y \) represent the time allotted for a sick visit (in minutes).

Based on the information given, we can set up the following system of equations:

1. For Tuesday's appointments:
\[
3x + 1y = 174
\]

2. For Wednesday's appointments:
\[
2x + 2y = 152
\]

Now we can solve this system of equations. The second equation can be simplified by dividing everything by 2:

\[
x + y = 76
\]

Now we have the simplified system:

1. \( 3x + y = 174 \) (Equation 1)
2. \( x + y = 76 \) (Equation 2)

Next, we can solve for \( y \) in terms of \( x \) using Equation 2:

\[
y = 76 - x
\]

Now, we can substitute this expression for \( y \) into Equation 1:

\[
3x + (76 - x) = 174
\]

Combine like terms:

\[
3x - x + 76 = 174
\]
\[
2x + 76 = 174
\]

Now, subtract 76 from both sides:

\[
2x = 174 - 76
\]
\[
2x = 98
\]

Now, divide by 2:

\[
x = 49
\]

Now that we have \( x \), we can find \( y \) by substituting \( x \) back into Equation 2:

\[
y = 76 - x
\]
\[
y = 76 - 49
\]
\[
y = 27
\]

Thus, the time allotted for each type of appointment is:

- Annual checkup: **49 minutes**
- Sick visit: **27 minutes**

Final answers:

- The time allotted for an annual checkup is **49** minutes.
- The time allotted for a sick visit is **27** minutes.