Parallel lines from equation (example 3)

we're asked which of these lines are parallel so they give us three equations of three different lines and if they're parallel then they have to have the same slope so all we have to do over here is figure out the slopes of each of these lines and defending they're equal they're parallel so let's do line a line a it's 2y is equal to 12x plus 10 we're almost in slope intercept form we can just divide both sides of this equation by 2 let's divide all the terms by 2 we get Y is equal to 6x all right 12 divided by 2 6x plus 5 so our slope in this case we have it in slope intercept form our slope in this case is equal to 6 let's try line B line B is y is equal to 6 now this might be you might say hey Jesus is a bizarre character how do I get this into slope-intercept form where's the X and my answer to you is that it already is in slope-intercept form I could just rewrite it as Y is equal to 0x plus 6 the X term is being multiplied by 0 because the slope here is 0 Y is going to be equal to 6 no matter how much you change X change in Y is always going to be 0 it's always going to be 6 so here our slope is 0 our slope is equal to 0 so these two lines are definitely not parallel they have different slopes so let's try a line C line C I'll do it down here line C so that's Y minus 2 is equal to 6 times X plus 2 and this is actually in point-slope form where the point what the point X is equal to negative 2y is equal to 2 so it's the point negative 2 2 is being represented here because you're subtracting the points and the slope is 6 so we already know that the slope is equal to 6 and sometimes people are more comfortable with slope intercept form so let's put it in slope intercept form just to confirm that if we put it in this form form the slope will still be equal to 6 so if we distribute this 6 we get Y minus 2 is equal to 6 times X 6x 6 times 2 is 12 and then if you add this to if you add 2 to both sides of the equation you get why because these guys cancel out is equal to 6x plus 14 so you see once again the slope is 6 so line a and line C have the same slope so line a and line C are parallel and they're different lines if they had the same y-intercept then they would just be the same line