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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineDr. Harvey, a pediatrician, has 22 annual checkups and 33 sick visits scheduled next Tuesday, which will fill a total of 190190 minutes on her schedule. Next Wednesday, she has 11 annual checkup and 33 sick visits on the schedule, which should take 140140 minutes. How much time is allotted for each type of appointment?\newlineThe time allotted is _\_ minutes for an annual checkup and _\_ minutes for a sick visit.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineDr. Harvey, a pediatrician, has 22 annual checkups and 33 sick visits scheduled next Tuesday, which will fill a total of 190190 minutes on her schedule. Next Wednesday, she has 11 annual checkup and 33 sick visits on the schedule, which should take 140140 minutes. How much time is allotted for each type of appointment?\newlineThe time allotted is _\_ minutes for an annual checkup and _\_ minutes for a sick visit.
  1. Define Variables: Let's denote the time for an annual checkup as aa minutes and the time for a sick visit as ss minutes. Dr. Harvey has 22 annual checkups and 33 sick visits scheduled next Tuesday, taking up 190190 minutes. This gives us the equation 2a+3s=1902a + 3s = 190.
  2. Equations for Tuesday: For next Wednesday, Dr. Harvey has 11 annual checkup and 33 sick visits scheduled, taking up 140140 minutes. This gives us the equation a+3s=140a + 3s = 140.
  3. Equations for Wednesday: We now have a system of two equations. To use elimination, we need to eliminate one of the variables, aa or ss. We choose to eliminate aa by multiplying the second equation by 22, the coefficient of aa in the first equation.
  4. Elimination Method: Multiplying the second equation by 22 gives us 2a+6s=2802a + 6s = 280.
  5. Multiply Second Equation: We now subtract the first equation from the new second equation to eliminate aa. This gives us 3s−6s=280−1903s - 6s = 280 - 190, which simplifies to −3s=−90-3s = -90.
  6. Subtract Equations: Dividing both sides of −3s=−90-3s = -90 by −3-3 gives us s=30s = 30. This means each sick visit is allotted 3030 minutes.
  7. Find Sick Visit Time: We substitute s=30s = 30 into the first equation 2a+3s=1902a + 3s = 190 and solve for aa. This gives us 2a+3(30)=1902a + 3(30) = 190, which simplifies to 2a+90=1902a + 90 = 190.
  8. Substitute and Solve: Subtracting 9090 from both sides gives us 2a=1002a = 100. Dividing both sides by 22 gives us a=50a = 50. This means each annual checkup is allotted 5050 minutes.

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