The equations are already in slope-intercept form:  \(\displaystyle y=mx+b\).  In order to determine whether the slopes are perpendicular to each other, write the formula for the slope of the perpendicular line.

\(\displaystyle m_{\textup{perpendicular}} = -\frac{1}{m_{\textup{original}}}\)

The perpendicular slope is the negative reciprocal of the original slope.  Take an original equation, perhaps \(\displaystyle y=-3x-3\).

The slope of the original equation is \(\displaystyle - 3\).  Substitute this in the equation.

\(\displaystyle m_{\textup{perpendicular}} = -\frac{1}{-3} = \frac{1}{3}\)

The perpendicular line must have a slope of \(\displaystyle \frac{1}{3}\).  Since this does not match the slope of \(\displaystyle y=-\frac{1}{3}x+\frac{1}{3}\), the lines are NOT perpendicular to each other.

The answer is:  

\(\displaystyle \textup{No, because the slopes are not the negative reciprocal of each other.}\)

Which of the following equations is perpendicular to the given function:

\(\displaystyle y=4x+15\)

To find a line perpendicular to a given linear function, simply find the opposit reciprocal of the slope of the given function.

So, we begin with 4, the opposite reciprocal will have the opposite sign and will be the flipped fraction of 4, so it will look like this:

\(\displaystyle \frac{-1}{4}\)

So we need to choose the answer with the correct slope, choose the only option with slope: 

\(\displaystyle \frac{-1}{4}\)

To find a line that is perpendicular to another line, we must look at the slope.  Two lines that are perpendicular have slopes that are opposite reciprocals of one another.  

Opposite reciprocals mean we change the sign and switch the numerator and denominator. Here is an example:

The reciprocal of \(\displaystyle 4\) is \(\displaystyle - \frac{1}{4}\)

The reciprocal of \(\displaystyle \frac{1}{2}\) is \(\displaystyle -2\)

So, in the given equation of a line, 

\(\displaystyle y = 4x + 7\)

We see that the slope is \(\displaystyle 4\).  To find a line perpendicular to this one, we need to find a line with a slope of \(\displaystyle - \frac{1}{4}\). 

In the following equation,

\(\displaystyle -4y = x - 5\)

We must first write it in slope intercept form.  To do that, we divide each term by \(\displaystyle -4\).

\(\displaystyle \frac{-4y}{-4} = \frac{x}{-4} - \frac{5}{-4}\)

\(\displaystyle y = \frac{x}{-4} + \frac{5}{4}\)

\(\displaystyle y = -\frac{1}{4}x + \frac{5}{4}\)

So, the slope of this line is \(\displaystyle -\frac{1}{4}\) which makes it perpendicular to the original line.

Perpendicular lines have slopes that are negative reciprocals of each other.

For example, the negative reciprocal of 2 is \(\displaystyle -\frac{1}{2}\)  and the negative reciprocal of \(\displaystyle -\frac{1}{5}\)  would be 5.

Thus,  

\(\displaystyle y=5x+2\) 

is perpendicular to   

\(\displaystyle y=-\frac{1}{5}x+4\).

Two lines are perpendicular if their slopes are opposite reciprocals of each other (opposite:  different signs,  reciprocal:  numerator and denominator are switched).

To find the slopes, we will write the original equation in slope-intercept form

\(\displaystyle y = mx+b\)

where m is the slope.  Given the original equation

\(\displaystyle -9y = 54x + 27\)

we must solve for y.  To do that, we will divide each term by -9.  We get

\(\displaystyle \frac{-9y}{-9} = \frac{54x}{-9} + \frac{27}{-9}\)

\(\displaystyle y = -6x -3\)

Therefore, the slope of this line is -6.  We must find a line that has a slope that is the opposite reciprocal of this line.  The opposite reciprocal slope of -6 is \(\displaystyle \frac{1}{6}\).  

Let's look at the line

\(\displaystyle 12y = 2x+1\)

We must write it in slope-intercept form.  To do that, we will divide each term by 12.  We get

\(\displaystyle \frac{12y}{12} = \frac{2x}{12} + \frac{1}{12}\)

\(\displaystyle y = \frac{2}{12}x + \frac{1}{12}\)

We can simplify to

\(\displaystyle y = \frac{1}{6}x + \frac{1}{12}\)

The slope of this line is \(\displaystyle \frac{1}{6}\).  Therefore, it is perpendicular to the original line.

Two lines are perpendicular if they have slopes that are opposite reciprocals of each other (opposite:  different signs, reciprocal:  switch the numerator and denominator). 

To find the slope of a line, we write it in slope-intercept form

\(\displaystyle y = mx + b\)

where m is the slope.

Given the equation

\(\displaystyle -3y = -9x + 12\)

we will solve for y by dividing each term by -3.  

\(\displaystyle \frac{-3y}{-3} = \frac{-9x}{-3} + \frac{12}{-3}\)

\(\displaystyle y = 3x - 4\)

We can see that the slope of this line is 3.  The slope of a line perpendicular to this one will have a slope of \(\displaystyle -\frac{1}{3}\).  

Therefore, the line

\(\displaystyle y = -\frac{1}{3}x - 1\)

is perpendicular to the original line.

In order to compare these lines, start by transforming the second equation into slope-intercept form:

\(\displaystyle y=mx+b\)

In this equation, the variable \(\displaystyle m\) represents the slope. Identify the slope of the second line's equation by transforming it until y-variable is isolated on the left side.

\(\displaystyle \small x=-8-2y\)

First, we will add 8 to both sides of the equation.

\(\displaystyle \small x+8=-8+8-2y\)

\(\displaystyle \small x + 8 = -2y\) 

Divide both sides of the equation by -2.

\(\displaystyle \small \frac{x+8}{-2}=\frac{-2y}{-2}\)

Rearrange and simplify.

\(\displaystyle \small y=-\frac{1}{2}x-4\) 

This means that the second line possesses the following slope:

\(\displaystyle \small -\frac{1}{2}\).

We know that the slope of the first line is 2; therefore, the slope of the second line is the negative reciprocal of the first. Perpendicular lines have slopes that are negative reciprocals of one another. 

To find a line that is perpendicular to another line, the slopes must be opposite reciprocals of each other. "Opposite" means there is a sign change, and "reciprocal" means that the numerator and denominator are inverted.  

For example, the following are opposite reciprocals of each other:

\(\displaystyle -4 \ \text{and} \ \frac{1}{4}\)

\(\displaystyle -\frac{1}{2} \ \text{and} \ 2\)

\(\displaystyle 3 \ \text{and} \ -\frac{1}{3}\)

First, convert the equation to slope-intercept form:

\(\displaystyle 3x+y=-9\)

\(\displaystyle y=-3x-9\)

The slope is \(\displaystyle -3\)  To find a line that is perpendicular, find a line with a slope that is the opposite reciprocal of \(\displaystyle -3\), or \(\displaystyle \frac{1}{3}\).

Looking at the equation

\(\displaystyle x-3y=-9\)

convert the equation to slope-intercept form.

\(\displaystyle x-3y-x=-9-x\)

\(\displaystyle -3y=-x-9\)

\(\displaystyle y = \frac{1}{3}x + 3\)

The slope is \(\displaystyle \frac{1}{3}\), making it perpendicular to the original line.

To find out if two lines are perpendicular, we just need to see whether their slopes are opposite reciprocals of each other.

The reciprocal of 6 is \(\displaystyle \frac{1}{6}\), so the opposite reciprocal is \(\displaystyle -\frac{1}{6}\).

The only answer choice with a slope of \(\displaystyle -\frac{1}{6}\) is

 \(\displaystyle y=-\frac{1}{6}x+4\)