Answer: The price of a ticket for the gold section exceeds the price for a ticket for the silver section by 8 dollars.
Step-by-step explanation: Given that the following system of equations relates the ticket price , in dollars, for seats in the silver (x) and gold (y) sections at a rock concert :
![x+2y=82~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\3x+y=96~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)](https://tex.z-dn.net/?f=x%2B2y%3D82~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%28i%29%5C%5C%5C%5C3x%2By%3D96~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%28ii%29)
We are to find by how much the price of a ticket for the gold section exceed the price for a ticket for the silver section.
First, we will solve the given system of equations as follows :
From equation (i), we have
![x+2y=82\\\\\Rightarrow x=82-2y~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(iii)](https://tex.z-dn.net/?f=x%2B2y%3D82%5C%5C%5C%5C%5CRightarrow%20x%3D82-2y~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%28iii%29)
Substituting the value of x from equation (iii) in equation (ii), we get
![3x+y=96\\\\\Rightarrow 3(82-2y)+y=96\\\\\Rightarrow 246-6y+y=96\\\\\Rightarrow 246-5y=96\\\\\Rightarrow 5y=246-96\\\\\Rightarrow 5y=150\\\\\Rightarrow y=\dfrac{150}{5}\\\\\Rightarrow y=30.](https://tex.z-dn.net/?f=3x%2By%3D96%5C%5C%5C%5C%5CRightarrow%203%2882-2y%29%2By%3D96%5C%5C%5C%5C%5CRightarrow%20246-6y%2By%3D96%5C%5C%5C%5C%5CRightarrow%20246-5y%3D96%5C%5C%5C%5C%5CRightarrow%205y%3D246-96%5C%5C%5C%5C%5CRightarrow%205y%3D150%5C%5C%5C%5C%5CRightarrow%20y%3D%5Cdfrac%7B150%7D%7B5%7D%5C%5C%5C%5C%5CRightarrow%20y%3D30.)
From equation (ii), we get
![x=82-2\times30=82-60=22.](https://tex.z-dn.net/?f=x%3D82-2%5Ctimes30%3D82-60%3D22.)
So, the price of a ticket for a ticket for the silver section = 22 dollars
and
the price of a ticket for a ticket for the gold section = 30 dollars.
Therefore, we get
![y-x=30-22=8.](https://tex.z-dn.net/?f=y-x%3D30-22%3D8.)
Thus, the price of a ticket for the gold section exceeds the price for a ticket for the silver section by 8 dollars.