Learn about Parallel, Perpendicular, Intersecting, and Coinciding Lines In this lesson, let’s learn how we identify lines that are parallel, perpendicular, intersecting or coinciding and we’ll do it by just looking at the equations. So we need to determine of these two lines are parallel, perpendicular, intersecting or coinciding, how do we do that? We know the formula for lines, slope formula is y=mx+b. the slope intercept we’re writing the equation of a line. So let’s look at both these lines I’m going to draw a line here and actually do the calculation on both ends. Here, what do we know? What is M? The slope equals to. Here, what do we know is the slope mx slope equals to. What is the Y intercept? Here is three y intercept here is negative one. So we can see two lines have the same slope but different Y intercepts. If they have the same slope they are parallel lines. And they are parallel and not coinciding because their Y intercepts are different. So another of doing it would be, this would be one line, Y intercept and this would be another line with the different y intercept. If the slopes are the same and Y intercepts are different then these two lines are parallel. So in this case, the lines are parallel and we did that by looking at the slopes and the Y intercepts. Let’s do another problem. In this case, we will look at these two lines. Now, this is Y=3x-5, slope equals mx, and b equals negative five. Let’s find the slope and the Y intercept here. I’m going to solve this equation. So we will leave the 2y here and we will subtract 6x from both sides so let’s subtract that 6x from here and 6x from here to get -2y=10-6x. Now, let’s divide by negative two. What do I get? y equals 10 over negative two is negative five and 6x over negative two is plus 3x. Now, if I look at the slope of this particular line slope m equals three and the Y intercept B equals negative five. So we can see slopes are the same, Y intercepts are the same. If slope and Y intercepts are exactly the same then the lines are coinciding lines. They’re exactly the same lines. As you can see the equation is the same so these two are exact same lines. Let’s try and solve the third one. What do I see? Y=5x+8 which means slope equals five, y intercept equals eight. In this scenario, Y=-1/5x-3 which means the slope is -1/5 and the Y intercept is -3 that’s what we’re given. As we can see, the slopes of the two lines are reciprocal which means it is opposite. One is five the other one is negative one over five. So since they are reciprocal and 5x-1/5=-1 so the multiplication of the two slopes gives us negative that means the lines are perpendicular. For any two lines be perpendicular the slopes when multiplied have to result in negative one. So I multiplied the two slopes 5 and -1/5 the answer I get is negative one. If the multiplication of both slopes is negative one, then the lines are perpendicular.