Video: Recognizing Perpendicular Lines

Video Transcript

Which of the following lines is perpendicular to the line 19𝑥 minus three 𝑦 equals five?

(a) three 𝑥 minus 19𝑦 equals five, (b) two minus 19𝑦 equals three 𝑥, (c) three 𝑦 equals one minus 19𝑥, (d) three 𝑦 equals 19𝑥 plus four, (e) three plus 19𝑦 equals two 𝑥.

Before we choose one of these functions, let’s remember what perpendicular lines are, more specifically what perpendicular lines have. Perpendicular lines have negative reciprocal slopes. So first we’ll take 19𝑥 minus three 𝑦 equals five and find its slope. To do that, we’ll take the function given in standard form and convert it into slope-intercept form by isolating 𝑦.

First subtract 19𝑥 from both sides of the equation. 19𝑥 minus 19𝑥 cancels out, leaving us with negative three 𝑦 equals negative 19𝑥 plus five. Remember the goal: isolate 𝑦. We divide 𝑦 by negative three and that means we’ll have to divide both terms on the right side by negative three. On the left, negative three divided by negative three equals one, and one times 𝑦 equals 𝑦, equals — our 𝑥 term has a negative in the numerator and the denominator. We can simplify that by saying 19 over three 𝑥 minus five over three. This is the slope-intercept form of the same equation we started with.

Slope-intercept form is 𝑦 equals 𝑚𝑥 plus 𝑏. The 𝑚 value is the slope. The slope of the line we were given is 19 over three. We want the negative reciprocal of 19 over three. The reciprocal of 19 over three is three over 19, and we need the negative value. Any function with the slope of negative three over 19 will be perpendicular to our line. We’ll examine all five of these functions to see which of them has a slope of negative three over 19.

Instead of trying to find the slope of all five of these lines, let’s see if we can eliminate any of the options. To do that, I want you to notice the relationship between 𝑥 and 𝑦 in the original function. We need the opposite to be true. We need a constant value of three associated with the 𝑥 and 19 associated with the 𝑦. It’s also important to note that the 𝑦-intercept doesn’t matter. In our new equation, the constant value, the 𝑦- intercept, can be anything.

So we’ll walk through these five functions and see which of them do not have a relationship of three 𝑥 to 19𝑦. (a) follows this pattern. (b) follows this pattern. (c) does not, neither does (d) or (e). Now we’re down to two functions we have to check. We’ll find the slope of both (a) and (b). In function (a), we subtract three 𝑥 from both sides. Three 𝑥 minus three 𝑥 cancels out. Negative 19𝑦 equals negative three 𝑥 plus five. To find the slope-intercept form, we need to isolate 𝑦. We’ll divide every term by negative 19. 𝑦 equals three over 19𝑥 minus five over 19. The slope of function (a) is three over 19.

Three over 19 is the reciprocal of our first function, but it is not the negative reciprocal of our first function. So now we move to function (b). Our first step: subtract two from both sides of the equation. Two minus two equals zero. And now we have negative 19𝑦 equals three 𝑥 minus two. We divide every term by negative 19, which simplifies to 𝑦 equals negative three over 19𝑥 plus two over 19. The slope here is negative three over 19, which is the negative reciprocal slope we’re looking for. The function two minus 19𝑦 equals three 𝑥 is perpendicular to the function 19𝑥 minus three 𝑦 equals five.