i'm trying to find the equation of a straight line in y-mx+c from that is parallel to the line with the equation $\displaystyle 3x+6y=8, passing through(2,2)$
Write the parallel line you are given in $\displaystyle y = mx + c$ form.
$\displaystyle 3x + 6y = 8$
$\displaystyle 6y = -3x + 8$
$\displaystyle y = -\frac{1}{2}x + \frac{4}{3}$.
Parallel lines have the same gradient. The gradient of the parallel line is $\displaystyle -\frac{1}{2}$, so the line you are trying to find has gradient $\displaystyle -\frac{1}{2}$.
The line you are trying to find has gradient $\displaystyle m = -\frac{1}{2}$, and you know the point $\displaystyle (x, y) = (2, 2)$ lies on the line.
So $\displaystyle y = mx + c$
$\displaystyle 2 = -\frac{1}{2}(2) + c$
$\displaystyle 2 = -1 + c$
$\displaystyle 3 = c$.
So the equation of the line is $\displaystyle y = -\frac{1}{2}x + 3$.