Coordinates

My math is very rusty, admittedly, so perhaps I shouldn't pick back up the keyboard to help, but I still will regardless. If I recall, this is going to be a point-slope formula problem, followed by a slope-intercept formula problem.

Point slope formula: $y − y_{1} = m(x − x_{1})$

Substitute the values accordingly: Treat $(3, 17)$ as $y$ and $(6, 32)$ as $y_{1}$.

Post-substitution: $17 - 32 = m(3 - 6)$

From there, simplify as needed.

Step 1: $17 - 32 = m(3 - 6)$

Step 2: $-15 = m(-3)$

Step 3: $\frac{-15}{-3} = \frac{m(-3)}{-3}$

Step 4: $5 = m$

From there, take either point and, knowing the value of $y$, $m$, and $x$, plug in those numbers to the slope intercept formula to find $b$ and make sure they match.

Slope intercept formula: $y=mx+b$

Substitutions: $17=5(3)+b$ and $32=5(6)+b$

Calculations 1: $17=15+b$ and $32=30+b$

Calculations 2: $(17-15)=(15-15)+b$ and $(32-30)=(30-30)+b$

Solution(s): $2=b$ for both equations.

The likely solution is that your answer wants to be done up in slope-intercept form, in which case you would plug in your values for $m$ and $b$, which would yield a final answer of $y=5x+2$. Hope this helped with understanding how you reached such an answer!

WELCOME BACK GOLDENLEAF!    

Your method looks fine. (I have not checked all your working)

I do this in a way that is not quite mathematically correct but it works fine.

any equation of a line question I start with 

gradient = gradient

$\frac{rise}{run}=\frac{rise}{run}\\ \frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}\\ \frac{y-17}{x-3}=\frac{32-17}{6-3}\\ $

and then just simply

Technically this presentation is not quite right because it would make a hole at x=3, but just ignore that.