Q41. - Algebra 2

Question

Campus Rentals rents 2- and 3-bedroom apartments for $\backslash (700$ and $\backslash )900$ per month, respectively. Last month they had six vacant apartments and reported $\$4600$ in the lost rent. How many of each type of apartment were vacant? 

Step-by-Step Solution

Answer

The number of vacant rooms of 2-bedroom apartments are $4$ and number of vacant rooms of 3-bedroom apartments are $2$.

Step-1 – Form the equations

Let us suppose the number of vacant rooms of 2-bedroom apartments be x and number of vacant rooms of 3-bedroom apartments be y.

Now according to the question, there were six vacant bedrooms in total.

The equation so formed is below.

$x+y=6$

Consider the second statement.

Due to vacant apartments, they had lost rent of $\$4600$ which is expressed as.

$\begin{array}{c}700x+900y=4600\\ 7x+9y=46\end{array}$

So, both the equations so formed for the given question are

$\begin{array}{c}x+y=6\\ 7x+9y=46\end{array}$

Step-2 – Apply the substitution method of solving equations

The algebraic method of substitution involves solving the one of the two equations for one variable in terms of other variable and then substituting the expression so formed for the variable in the second equation.

Step-3 – Solve for x in terms of y

To solve the equations, express $x+y=6$ in terms of y by subtracting y from both sides as shown below. 

$\begin{array}{c}x+y=6\\ x+y-y=6-y\\ x=6-y\end{array}$

Step-4 – Substitute the value of expression

Now, substitute $x=6-y$ in the equation $7x+9y=46$ and then solve as shown.

$\begin{array}{c}7x+9y=46\\ 7\left(6-y\right)+9y=46\\ 42-7y+9y=46\\ 2y=4\end{array}$

Simplify it further by dividing 2 on both sides.

$\begin{array}{c}2y=4\\ y=2\end{array}$

Thus, the value of y is $2$.

Step-5 – Substitute the value of variable

Now, substitute $y=2$ in the equation $x+y=6$ and then solve as shown.

$\begin{array}{c}x+y=6\\ x+2=6\\ x=6-2\\ x=4\end{array}$

Thus, the value of x is $4$.

Hence, the number of vacant rooms of 2-bedroom apartments are 4 and number of vacant rooms of 3-bedroom apartments are 2.