Characterization of Parallel Lines 05:49 minutes

Video Transcript

Transcript Characterization of Parallel Lines

Welcome to Aerial City, built on two interconnected islands in the sky. Citizens travel between Upper Aerial and Lower Aerial using the world-famous Parallel Cable Cars. Two strangers, Cassandra and Spark, pass each other on the cable cars every day. Cassandra is always on her way to Upper Aerial and Spark is always headed down to Lower Aerial. We can see their paths will never cross. But do you know why? To find out, let’s take a look at the characteristics of parallel lines. The Parallel Cable Car routes in Aerial City are represented by these linear equations. Are these lines truly parallel? Lines are parallel if they have the same slope but different y-intercepts. We can easily identify the slope and y-intercepts in these equations, because they are already in slope-intercept form, 'y equals mx plus b'. The slope 'm' in both of these equations is 3, so the slopes are the same, but the y-intercepts, or b, are different: 5 and 6, respectively. These linear equations have the same slope and different y-intercepts, so they must be parallel. This is what the graphs of the two lines look like when we plot them on a coordinate plane. Both lines have the same shape and steepness, but they pass through an entirely separate set of points. Which means these lines will never intersect! Remember how we solved systems of equations to find a point of intersection? What do you think happens when we try to solve this system, when we know the lines never intersect? Let's assume the 'y's in both equations are equal. Then, we can set '3x plus 5' equal to '3x plus 6'. Subtracting 3x from both sides to isolate the variable gives us 5 is equal to 6. Wait, that’s not right! When solving a system of equations leaves you with a false mathematical statement like '5 equals 6,' it means there's no ordered pair that works as a solution to this system of equations. That also means there are no points of intersection so the lines don’t cross: they’re parallel. Let’s look at another system of equations. It's hard to tell if these lines are parallel, so let's rewrite their equations using slope-intercept form, or 'y' equals mx plus 'b'. This will allow us to easily identify their slopes and y-intercepts. We'll start with the first equation, '2x plus 8y equals 10'. To isloate the 'y', subtract 2x from both sides and divide both sides by 8. Then, simplify the fractions. Now we have the equation "y equals negative one-fourth 'x' plus five-fourths. From this, we can determine the slope 'm' is negative one fourth and the y-intercept 'b' is five-fourths. Now, let's get the second equation in slope-intercept form. To isolate the 'y', use opposite operations and add 2 to both sides. Then divide both sides by 4 to cancel out the coefficient of 'y'. Then, simplify any remaining fractions and we are left with the equation "negative one-fourth x plus one-half equals 'y'". To put this in the more familiar form of "y equals mx plus b", let's rearrange things so the 'y' variable is on the left side. Now we can easily see that the slope 'm' is negative one fourth, and the y-intercept 'b' is one-half. Comparing our two equations, we can see that the slopes are the same but the y-intercepts are different. Hey, that's the definition of parallel lines! If we want to check that they are in fact parallel, we can set the two equations equal to each other. Sure enough, when we try to solve for 'x' by adding one-fourth 'x' to both sides, we end up with a false statement of five-fourths equals one-half. Looks like this system of equations has no solution, so these two lines will never meet. Let's quickly review. When determining if two lines are parallel, put their equations in slope-intercept form, 'y equals mx plus b'. Lines are parallel if they have the same slope, but different y-intercepts. You can also set them equal to each other and try to solve for 'x'. If you end up with a false statement, like 5 equals 6, then you know these lines will never intersect, so they are parallel. But if these parallel lines never intersect, does that mean Cassandra and Spark will never meet?