Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Dr. Thompson, a pediatrician, has $3$ annual checkups and $2$ sick visits scheduled next Tuesday, which will fill a total of $220$ minutes on his schedule. Next Wednesday, he has $3$ annual checkups and $3$ sick visits on the schedule, which should take $246$ 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.

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Solve a system of equations using any method: word problems

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

\newline

Dr. Thompson, a pediatrician, has

3

annual checkups and

2

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

220

minutes on his schedule. Next Wednesday, he has

3

annual checkups and

3

sick visits on the schedule, which should take

246

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 Equations: Let's denote the time for an annual checkup as $x$ minutes and the time for a sick visit as $y$ minutes. $\newline$ We need to set up two equations based on the information given.
Set Up Equations: For next Tuesday, the equation based on the information given is: $\newline$ $3x$ (time for annual checkups) + $2y$ (time for sick visits) = $220$ minutes. $\newline$ So, we have the equation: $3x + 2y = 220$ .
Solve for Variables: For next Wednesday, the equation based on the information given is: $\newline$ $3x$ (time for annual checkups) + $3y$ (time for sick visits) = $246$ minutes. $\newline$ So, we have the equation: $3x + 3y = 246$ .
Substitute and Solve: We now have a system of equations: $\newline$ $3x + 2y = 220$ $\newline$ $3x + 3y = 246$ $\newline$ We can solve this system using the elimination method by subtracting the first equation from the second to eliminate $x$ .
Final Results: Subtracting the first equation from the second gives us: $\newline$ $(3x + 3y) - (3x + 2y) = 246 - 220$ $\newline$ This simplifies to: $\newline$ $3y - 2y = 26$ $\newline$ $y = 26$
Final Results: Subtracting the first equation from the second gives us: $\newline$ $(3x + 3y) - (3x + 2y) = 246 - 220$ $\newline$ This simplifies to: $\newline$ $3y - 2y = 26$ $\newline$ $y = 26$ Now that we have the value for $y$ , we can substitute it back into one of the original equations to solve for $x$ . Let's use the first equation: $\newline$ $3x + 2(26) = 220$
Final Results: Subtracting the first equation from the second gives us: $\newline$ $(3x + 3y) - (3x + 2y) = 246 - 220$ $\newline$ This simplifies to: $\newline$ $3y - 2y = 26$ $\newline$ $y = 26$ Now that we have the value for $y$ , we can substitute it back into one of the original equations to solve for $x$ . Let's use the first equation: $\newline$ $3x + 2(26) = 220$ Simplify and solve for $x$ : $\newline$ $3x + 52 = 220$ $\newline$ $3x = 220 - 52$ $\newline$ $3x = 168$ $\newline$ $3y - 2y = 26$ $0$ $\newline$ $3y - 2y = 26$ $1$
Final Results: Subtracting the first equation from the second gives us: $\newline$ $(3x + 3y) - (3x + 2y) = 246 - 220$ $\newline$ This simplifies to: $\newline$ $3y - 2y = 26$ $\newline$ $y = 26$ Now that we have the value for $y$ , we can substitute it back into one of the original equations to solve for $x$ . Let's use the first equation: $\newline$ $3x + 2(26) = 220$ Simplify and solve for $x$ : $\newline$ $3x + 52 = 220$ $\newline$ $3x = 220 - 52$ $\newline$ $3x = 168$ $\newline$ $3y - 2y = 26$ $0$ $\newline$ $3y - 2y = 26$ $1$ We have found the values for $x$ and $y$ : $\newline$ $3y - 2y = 26$ $1$ minutes for an annual checkup $\newline$ $y = 26$ minutes for a sick visit $\newline$ These are the times allotted for each type of appointment.