Answer
$x=200$, $y=150$, $z=50$
Work Step by Step
We denote the number of 8usd ticket by $x$, the number of 10usd ticket sold by $y$ and the number of 12usd ticket sold by $z$.
We are given the system:
$\begin{cases}
x+y+z=400\\
8x+10y+12z=3700\\
x+y=7z
\end{cases}$
We will use the addition method.Add Equation1 and Equation 3 after sustracting $7z$ from both sides of Equation 3 to get the variables on one side of the equation and multiplying Equation 3 by $-1$.
$\begin{cases}
x+y+z+-1(x+y-7z)=400+0,\\
8z=400,\\
z=50\\
\end{cases}$
now substitute $z$ result we got into Equation 1 and Equation 2. Multiply Equation 1 by -8 and add it to Equation 2.
$\begin{cases}
-8x-8y+(8x+10y)=-2800+3100,\\
2y=300\\
y=150
\end{cases}$
substituting the answer we got for $y$ and $z$ into Equation 3 we can get an answer for $x$
$\begin{cases}
x+150=350,\\
x=200\\
\end{cases}$
$x=200$, $y=150$, $z=50$
