A is the point with coordinates (1, 3) B is the point with coordinates (–2, –1) The line L has equation 3y = 4 – 2x Is line L parallel to AB?

No it is not parallel, here is the reasoning.

First we work out the gradient of the line AB. We do this by using the gradient formula (Y2-Y1)/(X2-X1). Our points are A=(1,3) rewritten as (X1,Y1) and B=(-2,-1) rewritten as (X2,Y2). If we sub in these points into the gradient formula we get: (-1-3)/(-2-1) = (-4/-3) = (4/3)

For two lines to be parallel, the gradients of the two lines must be equal. To find the gradient of the line 3y=4-2x we divide through by the 3 to get y=4/3 - (2/3)x We have then rearranged this line into the form y=mx+c, the standard straight line equation where m = the gradient.

we can see that the m in the equation y=4/3-(2/3)x is (-2/3) which is not the same as 4/3 therefore the gradients of the two lines are not the same and the lines are not parallel.