Equations of parallel & perpendicular lines
We have three lines and we have to figure out which of the three are parallel. So line A-- and it can't be parallel on its own, it has to be parallel to another of the three lines. So the equation for line A is y is equal to 3/4 x minus four. Line B is 4y minus 20 is equal to negative 3x. And then line C is negative 3x plus 4y is equal to 40. So to figure out if any of these lines are parallel to any of the other lines, we just have to compare their slopes. If any two of these lines have the same slope and they're different lines, they have different y-intercepts, then they're going to be parallel. Now line A, it's very easy to figure out its slope. It's already in slope-intercept form. This is mx plus b, the slope is 3/4 and the y-intercept, which isn't as relevant when you're figuring out parallel lines, is negative 4. So let's see what the other character's slopes are. This isn't in any kind of standard form. It's not a standard form, slope-intercept, or point-slope form, but let's see what the slope of this line is. So to get it into slope-intercept form, which is really the easiest one to pick out the slope from, let's add 20 to both sides of this equation. The left-hand side, those cancel out, that was the whole point, you get 4y is equal to negative 3x plus 20. And now we can divide everything by 4. We are left with y is equal to negative 3/4 x plus 5. So in this case, y-intercept is 5, but most importantly, the slope is negative 3/4, so it's different than this guy. This is negative 3/4, this is positive 3/4, so these two guys definitely aren't parallel. Let's move on of this guy in standard form. So let's get the x term on the other side. So let's add 3x to both sides of this equation. Left-hand side, these cancel out. We're just left with 4y is equal to 3x plus 40, or 40 plus 3x, either way. Now we can divide both sides by 4 you have to divide every term by 4. The left-hand side, you're left with y. The right-hand side, you have 3/4 x plus 10. So here, our slope is 3/4 and our y-intercept, if we care about it, is 10. So this line and this line have the exact same slope, 3/4, and they're different lines because their y-intercept is different. So we know that A and C are parallel lines and B is not parallel to either one of the other two.