Bezout's Identity: Finding A Pair of Integers 
There is only one integer $x$, between 100 and 200 such that the integer pair $(x, y)$ satisfies the equation $42x + 55y = 1$. What's the value of $x$ in this integer pair?

We know that
$$\begin{align} x &= x_0 + 5t \\ y &= y_0 - 4t \end{align}$$
But we need to know what $x_0, y_0$ are. By applying the GCD algorithm we can get the answer to be $x_0 = 17$ and $y_0 = 13$. So we need to find $100 \leq 17 + 5t \leq 200$. But treating this parametrically yields too many solutions. How do I discover the one solution?

First you need to find all integer solutions of $42x+55y=1$.[See this post]
Here $\gcd(42,55)=1$, so by Euclidean algorithm, $$1=42(-17)+55(13)\;[\text{check!}]$$ 
So integer solutions are $$x=-17+55r$$ $$y=13-42r$$ where $r \in \Bbb{Z}$ 
For your task, you need to find one $r$ so that $100 \leq x \leq 200$  and $r$ satisfies this.
Clearly for $r=1,2$, $x <100$. For $r=3$, we get $x=3(55)-17=\color{red}{148}$, which is the required  number. For $r>3$ we get $x >200$ and if $r <0$ then $x<0$. Hence  the required pair is $$(\color{red}{148},-113)$$ 
