Answer:
The correct answers are answers 4, 7 and 8.
Step-by-step explanation:
The slope of a line perpendicular to a given line will be the negative reciprocal of the given line's slope. A negative reciprocal means that the signs are switched (+ to -, or - to +) and the numerator and denominator switch places. For example, the reciprocal of 2/3 is -3/2.
The negative reciprocal of 3/4 is -4/3. All answers with the slope -4/3 are correct. Any other slope is incorrect.
Option 1 is in point slope form, which gives us the slope without needing to manipulate the equation. It's slope is 3/4, which is not -4/3. It is the same slope as the given line so it is parallel to it, not perpendicular. Option 1 is incorrect.
Option 2 is in slope intercept form, which also gives us the slope just by looking at it. It's slope is -3/4, which is the negative version of the given slope, but not the reciprocal and therefore is incorrect.
Option 3 also does not have the slope -4/3, so it is incorrect.
Option 4 has the slope -4/3, so we know this is correct regardless of the y-intercept. It does not matter where the line is, as long as it has the negative reciprocal slope it is perpendicular. Option 4 is correct.
To find the slope of option 5's equation, we must isolate y and put it into slope-intercept form. WE are starting out with the equation:
3x-4y = 2
First, subtract 3x on both sides.
-4y = -3x +2
Divide by -4 on both sides
[tex]y = \frac{3}{4}x -\frac{1}{2}[/tex]
The slope is not -4/3, so option 5 is not perpendicular to the given line.
Choice 6 is in slope-intercept form, so the slpe is given as 4/3. While it is the reciprocal of 3/4, it is not the negative reciprocal so it is also incorrect.
Option 7 must also be manipulated into slope-intercept form to find the slope. We are starting with the equation:
4x+ 3y = 15
Subtract 4x on both sides
3y = -4x + 15
Divide by 3 on both sides.
[tex]y = \frac{-4}{3}x + 5[/tex]
The slope of this line is the negative reciprocal of 3/4, so option 7 is correct.
Option 8 has the slope -4/3, so it is also correct.
Options 4, 7, and 8 are correct.
[tex]y=\frac{-4}{3}x-1[/tex]
[tex]4x+3y=15[/tex]
[tex]y+15=-\frac{4}{3}(x-9)[/tex]
They are all correct because they all have the slope -4/3, which is the negative reciprocal of 3/4 and are therefore perpendicular to the given line.
