I have the following 2 points and need to find other points along this line and the y- intercept.
$$\left(-\frac13, \frac54\right), \left(\frac12, \frac34\right)$$
I have determined the slope $m = -\frac56.$
If these were whole numbers I could just find points using the slope, but these are fractions of differing denominators. How should I go about calculating accurate points. Thanks.
The fact that the points are not given as whole numbers has no effect on the calculation.
You have the equation
$$y=mx+n$$
and you have two pairs of $x, y$ that you know you can plug into the equation.
You can calculate $m$ as the difference in the $y$ values, divided by the difference in $x$-values. You should re-do your calculation, because the result is not $-\frac56$. I don't know where you made the mistake, because you didn't provide the method by which you calculated $-\frac56$, but the result is wrong. If you want help with where your result is wrong, please first show how you got to it.
Once you calculated $m$, just plug in one of the numbers and calculate $n$.