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?

$$42x+55y=1\implies 42x\equiv 1\pmod {55}$$ $$\implies -13x\equiv  (-55+42)x\equiv 42x\equiv 1 \pmod {55}$$ $$ \implies -52x\equiv 4(-13x)\equiv 4(1)\equiv 4 \pmod {55}$$ $$\implies 3x\equiv (55-52)x \equiv -52 x\equiv 4 \pmod {55}$$ $$\implies  54x\equiv (18)(3x)\equiv (18)(4)\equiv 72\equiv 17 \pmod {55}$$ $$ \implies -1x\equiv (54-55)x\equiv 54x \equiv 17 \pmod {55}$$ $$\implies x\equiv -17 \pmod {55}.$$
So $x=55z-17$ with $z\in \Bbb Z.$ And $100<x<200.$
So $100<  55z-17<200.$ So $117<55z<217$ so (obviously) $z=3$ and $x=55(3)-17=148.$
EXPLANATION: For the 2nd line we take the nearest multiple of $42$  to  $55$ (which happens to be $42$ itself) so we transition from $42x$ to $(42-55)x=-13x $ because $|-13|<|42|.$ 
In the 3rd line we take nearest multiple of $-13$ to  $-55,$ which is $(-13)4,$ so in the 4th line we transition from $-13x$ to $((-13)4+55)x=3x$ because $|3|<|-13|.$
In the 5th line we take the nearest  multiple of $3$ to $55,$ which is $3(18),$ so we transition in the 6th line from $3x $ to $(3(18)-55)x=-1x$ because $|-1|<|3|.$  And that is as far as these steps can go.
