Answer :
EXPLANATION
Let's see the facts:
One month:
Rented movies = 5
Rented video games = 3
Total = $34
Next month:
Rented movies = 2
Rented video games = 12
Total = $73
Let's call x to the rental cost for each movie and y to the rental cost for each game, representing the given data on a system of equations give us the following expressions:
(1) 5x + 3y = 34
(2) 2x + 12y = 73
Now, we need to solve this system of equations:
Isolate x for 5x + 3y = 34:
[tex]\mathrm{Subtract\: }3y\mathrm{\: from\: both\: sides}[/tex][tex]5x+3y-3y=34-3y[/tex]Simplify:
[tex]5x=34-3y[/tex][tex]\mathrm{Divide\: both\: sides\: by\: }5[/tex][tex]\frac{5x}{5}=\frac{34}{5}-\frac{3y}{5}[/tex][tex]Simplify\colon[/tex][tex]x=\frac{34-3y}{5}[/tex][tex]\mathrm{Substitute\: }x=\frac{34-3y}{5}[/tex][tex]\begin{bmatrix}2\cdot\frac{34-3y}{5}+12y=73\end{bmatrix}[/tex][tex]Simplify\colon[/tex][tex]\begin{bmatrix}\frac{68+54y}{5}=73\end{bmatrix}[/tex][tex]\mathrm{Multiply\: both\: sides\: by\: }5[/tex][tex]\frac{5\left(68+54y\right)}{5}=73\cdot\: 5[/tex][tex]\text{Simplify:}[/tex][tex]68+54y=365[/tex][tex]\mathrm{Subtract\: }68\mathrm{\: from\: both\: sides}[/tex][tex]68+54y-68=365-68[/tex][tex]Simplify\colon[/tex][tex]54y=297[/tex][tex]\mathrm{Divide\: both\: sides\: by\: }54[/tex][tex]\frac{54y}{54}=\frac{297}{54}[/tex][tex]Simplify\colon[/tex][tex]y=\frac{11}{2}[/tex][tex]\mathrm{For\: }x=\frac{34-3y}{5}[/tex][tex]\mathrm{Substitute\: }y=\frac{11}{2}[/tex][tex]x=\frac{34-3\cdot\frac{11}{2}}{5}[/tex][tex]x=\frac{34-\frac{33}{2}}{5}[/tex][tex]x=\frac{\frac{35}{2}}{5}[/tex][tex]x=\frac{35}{2\cdot\:5}[/tex][tex]x=\frac{35}{10}[/tex][tex]\mathrm{Cancel\: the\: common\: factor\colon}\: 5[/tex][tex]x=\frac{7}{2}[/tex][tex]\mathrm{The\: solutions\: to\: the\: system\: of\: equations\: are\colon}[/tex][tex]x=\frac{7}{2},\: y=\frac{11}{2}[/tex]Now, as x=rental cost for each movie and y= rental cost for each video game, we can conclude:
Rental cost for each movie = $3.5
Rental cost for each video game = $5.5