Equations of parallel & perpendicular lines
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 if any of them are 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. We get y is equal to 6x-- 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 six. You might say this hey, this 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 six 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, so these two lines are definitely not parallel, they have different slopes. So let's try line C. Line C-- I'll do it down here. Line C, so it's y minus 2 is equal to 6 times x plus 2. And this is actually in point-slope form, where the point x is equal to negative 2, y is equal to 2. So 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, the slope will still be equal to 6. So if we distribute the 6, we get y minus 2 is equal to 6 times x, 6x, plus 6 times 2 is 12. And if you add this 2 -- if you add 2 to both sides of the equation, you get y-- 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 the 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.