Which of the following lines is perpendicular to 4x+5y=9

BTGmoderatorLU wrote:Source: Manhattan Prep

Which of the following lines is perpendicular to 4x+5y=9 on the xy plane?
$$A.\ y=\frac{5}{4}x+2$$
$$B.\ y=\frac{-5}{4}x+9$$
$$C.\ y=-4x+\frac{9}{5}$$
$$D.\ y=\frac{4}{5}x+\frac{-4}{5}$$
$$E.\ y=\frac{-4}{5}x$$
The OA is A

Note that two lines are perpendicular to each other if the product of their slopes is 1.

Let's first find out the slope of the given line 4x + 5y = 9. Let's transform it into y = mx + c form, where m is the slope of the line.

Thus, 4x+5y=9 => y = (–4/5)x + 9. Thus, m = –4/5

Say the slope of the line which is perpendicular to 4x+5y=9 is p.

Thus, mp = – 1. Thus, p = –1/m = –1/(–4/5) = 5/4.

The equation of the required line would be y = px + d, where p = slope of the line and d = y-intercept (We are not bothered about it).

The line presented by option A has a slope of 5/4, the correct answer.

The correct answer: A

Hope this helps!

-Jay
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BTGmoderatorLU wrote:Source: Manhattan Prep

Which of the following lines is perpendicular to 4x+5y=9 on the xy plane?
$$A.\ y=\frac{5}{4}x+2$$
$$B.\ y=\frac{-5}{4}x+9$$
$$C.\ y=-4x+\frac{9}{5}$$
$$D.\ y=\frac{4}{5}x+\frac{-4}{5}$$
$$E.\ y=\frac{-4}{5}x$$
The OA is A

We need to first express the given line in slope-intercept form, which is y = mx + b, where m is the slope of the line:

5y = -4x + 9

y = (-4/5)x + 9/5

We see that the slope of the given line is -4/5. Since the slopes of two perpendicular lines are negative reciprocals of each other, then a line perpendicular to the given line will have a slope of 5/4. The line whose equation is given in choice A has that slope.

Answer: A