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
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.