which equation would you use to find out if the two lines in the graph are parallel

There is no equation that corresponds directly to parallel lines. However, when given two points for each line, the slope equation (m=y2-y1/x2-x1) can help. If after doing the equation and reducing (if necessary) you come up with the same answers, then the lines are parallel. Other ways to tell if you have parallel lines are:

A) If you have slope intercept form and the slopes are the same but the y-intercept is different (m=m; "b" does not equal "b") then the lines are parallel

B) If you have point slope form and you know the lines are different, but the slope (m) is the same, then the lines are parallel

B1) If you are given point slope form and don't know that the lines are not the same line, place the equation into slope intercept and then the above applies.

C) Horizontal lines, providing they are not same lines, are always parallel

D) Vertical lines, providing they are not same lines, are always parallel.

E) If given standard form (Ax+By=C), and the lines are not same lines,take the opposite of "A"
and divide by "B" to obtain slope.

Ex: 2x+3y=5 m=-2/3