Using Extended Euclidean Algorithm for $85$ and $45$ 
Apply the Extended Euclidean Algorithm of back-substitution to find
  the value of $\gcd(85, 45)$ and to express $\gcd(85, 45)$ in the form $85x + 45y$ for a pair of integers $x$ and $y$.

I have no idea what is the difference between the regular EEA and the back-substitution EEA. The only thing that I have been taught is constructing the EEA table, for some values a, b. Can anyone help me explain this? Thanks a ton!
 A: You’re probably intended to do the substitutions explicitly. You have
$$\begin{align*}
85&=1\cdot45+40\\
45&=1\cdot40+5\\
40&=8\cdot5\;.
\end{align*}$$
Now work backwards:
$$\begin{align*}
5&=1\cdot45-1\cdot40\\
&=1\cdot45-1\cdot(1\cdot85-1\cdot45)\\
&=(-1)\cdot85+2\cdot45\;.
\end{align*}$$
The tabular method is just a shortcut for this explicit back-substitution.
A: The method here is the back-substitution, but the organization highlights the logic. It can be done on paper but is really easy using the wonders of copy-paste-edit-latex technology.
$\tag 1 [40] = [85] - [45] \,1$
$\tag 2 [05] = [45] - [40] \,1$
$\tag ! [00] = [40] - [05] \,8 $
$\text{ >>> Remove [] from lhs of (2) and work the substitution/simplify algebra in reverse:}$
$\tag 2 5 = [45] - [40] \,1 $
$\tag 1 5 = [45] - [[85] - [45] \,1] \,1 = [85](-1) + [45](2)$
For practice, here is another one:
Write $\text{gcd}(80,62) = 80 x + 62 y$ with integers $x$ and $y$.
$\tag 1 [18] = [80] - [62] \,1$
$\tag 2 [08] = [62] - [18] \,3$
$\tag 3 [02] = [18] - [08] \,2 $
$\tag ! [00] = [08] - [02] \,4 $
$\text{ >>> Remove [] from lhs of (3) and work substitution/simplify algebra in reverse:}$
$\tag 3 2 = [18] - [08] \,2 $
$\tag 2 2 = [18] - [[62] - [18] \,3] \,2 = [18]\,7 - [62]\,2$
$\tag 1 2 =  [[80] - [62] \,1]\,7 - [62]\,(2) = [80] \,(7) +[62]\,(-9) $
