Finding all solutions to $3x + 4y \equiv 1 \pmod 7$ Find all solutions $\pmod 7$: $3x + 4y$ is congruent to $1 \pmod 7$.
I have tried writing out the various equations such as $3x + 4y = 1$, $3x + 4y = 8$, $3x + 4y = 15$, etc., but I do not know how to find the finite solution.

Consider the equation
$$ 3x + 4y = 1 (\mod 7).$$
$ (3,4,1) = 1 | 7$, so there are $7\times 1 = 7$ solutions mod $7$. I’ll solve the equation using a reduction trick.
The given equation is equivalent to
$3x + 4y + 7z = 1$ for some $z$.
Set $$ w=\frac {3x+4y}{(3,4)} $$.
Then
$(3,4)w + 7z = 1, w + 7z = 1$.
$w_0 = -6, z_0 = 1$ is a particular solution. The general solution is
$w = −6 + 7s, z = 1 − s$.
Substitute for $w$:
$$ \frac {3x+4y}{(3,4)} =-6+7s$$, $$\Rightarrow 3x + 4y = −6 + 7s$$
$x_0 = -2, y_0 = \frac{7s}{4}$, is a particular solution. The general solution is
$x = -2 + 4t, y = \frac {7s}{4}-3t $.
$t = 0, 1, . . ., 6$ will produce distinct values of y mod 7. Note, however, that $s$ and $s + r$ produce $7s$ and $7s + 7r$, which are congruent mod $7$. That is, adding a multiple of $1$ to a given value of $s$ makes the
$7s$ term in $x$ repeat itself mod $7$. So I can get all possibilities for $x$ mod $7$ by letting $s = 0$.
All together, the distinct solutions mod $7$ are:  

$$ x = -2 + 4t, y = \frac{7s}{4}-3t$$  

where $s = 0$ and $t = 0,1...6$. Hope it helps you. 
