Question

write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. a new hospital in bloomington starts out with 10 junior residents and 6 senior residents on its staff. management plans to hire additional personnel at a rate of 1 junior resident per month and 2 senior residents per month. eventually, there will be an equal number of each on the hospital staff. how long will that take? how many of each type will there be? after \\(\square\\) months, there will be \\(\square\\) of each on staff.

Explanation:

Step1: Define variables

Let $m$ = number of months, $J$ = total junior residents, $S$ = total senior residents.

Step2: Write equations for each group

For junior residents: $J = 10 + 1m$
For senior residents: $S = 6 + 2m$
Set $J=S$ (equal numbers):
$$10 + m = 6 + 2m$$

Step3: Solve for $m$

Subtract $m$ and 6 from both sides:
$10 - 6 = 2m - m$
$4 = m$

Step4: Find total of each group

Substitute $m=4$ into $J = 10 + m$:
$J = 10 + 4 = 14$
(Check $S = 6 + 2(4) = 14$, matches)

Answer:

After $\boldsymbol{4}$ months, there will be $\boldsymbol{14}$ of each on staff.