Equations of parallel and perpendicular lines
We are asked which of these lines are parallel. So parallel lines are lines that have the same slope, and they're different lines, so they never, ever intersect. So we need to look for different lines that have the exact same slope. And lucky for us, all of these lines are in y equals mx plus b or slope-intercept form, so you can really just look at these lines and figure out their slope. The slope for line A, m is equal to 2. We see it right over there. For line B, our slope is equal to 3, so these two guys are not parallel. I'll graph it in a second and you'll see that. And then finally, for line C-- I'll do it in purple-- the slope is 2. So m is equal to 2. I don't know if that purple is too dark for you. So line C and line A have the same slope, but they're different lines, they have different y-intercepts, so they're going to be parallel. And to see that, let's actually graph all of these characters. So line A, our y-intercept is negative 6. So the point 0, 1, 2, 3, 4, 5, 6. And our slope is 2. So if we move 1 in the positive x direction, we go up 2 in the positive y direction. One in x, up 2 in y, if we go to in x, we're going to go up 4 in y. And I can just do up 2, then we're going to go 2, 4, and you're going to see it's all on the same line, so line A is going to look something like-- do my best to draw it as straight as possible. Line A-- I can do a better version than that-- line A is going to look like-- well, that's about just as good as what I just drew-- that is line A. Now let's do line B. Line B, the y-intercept is negative 6. 0, negative 6. So it has the same y-intercept, but its slope is 3, so if x goes up by 1, y will go up by 3. So x goes up by 1, y goes up by 3. If x goes up by 2, y is going to go up by 6. 2, 4, 6. So this line is going to look something like this. Trying my best to connect the dots. It has a steeper slope, and you see that when x increases, this blue line increases by more in the y direction. So that is line B-- and notice, they do intersect, there's definitely not two parallel lines. And then finally, let's look at line C. The y-intercept is 5. So 0, 1, 2, 3, 4, 5. The point 0, 5, its y-intercept. And its slope is 2. So you increase by 1 in the x direction, you're going to go up by 2 in the y direction. If you decrease by 1, you're going to go down 2 in the y direction. If you increase by, well, you're going to go to that point, you're going to have a bunch of these points. And then if I were to graph the line-- let me do it one more time-- if I were to decrease by two, I'm going to have to go down by 4, right? Negative 4 over negative 2 still a slope of 2, so 1, 2, 3, 4. And I can do that one more time, get right over there. And then you'll see the line. The line will look like that, it will look just like that. And notice that line C and line A never intersect. They have the exact same slope. Different y-intercepts, same slope, so they're increasing at the exact same rate, but they're never going to intersect each other. So line A and line C are parallel.