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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineA daycare center in Dover currently has 33 assistant caregivers and 1111 senior caregivers. Since demand is high, the owner is going to be hiring 33 assistant caregivers per month and 22 senior caregivers per month. Her goal is to have a larger staff, including an equal number of assistant caregivers and senior caregivers. How many of each type will there be? How long will that take?\newlineThere will be _\_ of each type of caregiver in _\_ months.

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineA daycare center in Dover currently has 33 assistant caregivers and 1111 senior caregivers. Since demand is high, the owner is going to be hiring 33 assistant caregivers per month and 22 senior caregivers per month. Her goal is to have a larger staff, including an equal number of assistant caregivers and senior caregivers. How many of each type will there be? How long will that take?\newlineThere will be _\_ of each type of caregiver in _\_ months.
  1. Define variables: Let's define the variables.\newlineLet xx be the number of months after which there will be an equal number of assistant and senior caregivers.\newlineLet AA be the number of assistant caregivers after xx months.\newlineLet SS be the number of senior caregivers after xx months.
  2. Write equations: Write the equations based on the given information.\newlineInitially, there are 33 assistant caregivers and 1111 senior caregivers.\newlineThe owner hires 33 assistant caregivers per month and 22 senior caregivers per month.\newlineAfter xx months, the number of assistant caregivers will be A=3+3xA = 3 + 3x.\newlineAfter xx months, the number of senior caregivers will be S=11+2xS = 11 + 2x.\newlineThe goal is to have A=SA = S.
  3. Set up equation: Set up the equation A=SA = S to find the value of xx.3+3x=11+2x3 + 3x = 11 + 2x
  4. Solve for x: Solve the equation for x.\newlineSubtract 2x2x from both sides:\newline3x−2x=11−33x - 2x = 11 - 3\newlinex=8x = 8
  5. Calculate caregivers after xx months: Calculate the number of each type of caregiver after xx months.\newlineA=3+3x=3+3(8)=3+24=27A = 3 + 3x = 3 + 3(8) = 3 + 24 = 27\newlineS=11+2x=11+2(8)=11+16=27S = 11 + 2x = 11 + 2(8) = 11 + 16 = 27
  6. Verify equality: Verify that the numbers of assistant and senior caregivers are equal. A=SA = S 27=2727 = 27

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