Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ A new hospital in Silvergrove starts out with $10$ junior residents and $9$ senior residents on its staff. Management plans to hire additional personnel at a rate of $3$ junior residents per month and $4$ senior residents per month. Eventually, there will be an equal number of each on the hospital staff. How many of each type will there be? How long will that take? $\newline$ There will be _ of each on staff in _ months.

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Solve a system of equations using substitution: word problems

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

\newline

A new hospital in Silvergrove starts out with

10

junior residents and

9

senior residents on its staff. Management plans to hire additional personnel at a rate of

3

junior residents per month and

4

senior residents per month. Eventually, there will be an equal number of each on the hospital staff. How many of each type will there be? How long will that take?

\newline

There will be _____ of each on staff in _____ months.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of months after which there will be an equal number of junior and senior residents. $\newline$ Let $J$ be the number of junior residents. $\newline$ Let $S$ be the number of senior residents. $\newline$ We start with the following initial conditions: $\newline$ $J = 10$ (initial number of junior residents) $\newline$ $S = 9$ (initial number of senior residents) $\newline$ The rate of hiring is: $\newline$ $3$ junior residents per month $\newline$ $4$ senior residents per month $\newline$ We can write two equations to represent the situation: $\newline$ For junior residents: $J = 10 + 3x$ $\newline$ For senior residents: $S = 9 + 4x$ $\newline$ We want to find the point at which $J = S$ .
Initial Conditions: Now we set the two equations equal to each other to find the value of $x$ when the number of junior and senior residents is the same: $10 + 3x = 9 + 4x$
Rate of Hiring: We solve for $x$ by subtracting $3x$ from both sides of the equation: $\newline$ $10 + 3x - 3x = 9 + 4x - 3x$ $\newline$ $10 = 9 + x$
Write Equations: Now we subtract $9$ from both sides to isolate $x$ : $\newline$ $10 - 9 = 9 + x - 9$ $\newline$ $1 = x$
Set Equations Equal: We have found that $x = 1$ , which means that after $1$ month, there will be an equal number of junior and senior residents. Now we need to find out how many of each there will be. We can substitute $x$ back into either the junior or senior resident equation: $\newline$ $J = 10 + 3(1) = 10 + 3 = 13$
Solve for x: We can also check this result by substituting $x$ into the senior resident equation: $S = 9 + 4(1) = 9 + 4 = 13$
Isolate $x$ : We have confirmed that after $1$ month, there will be $13$ junior residents and $13$ senior residents, making the total number of each equal.