Lines can be written in the slope-intercept form: 

\(\displaystyle y=mx+b\)

In this equation, \(\displaystyle m\) is the slope and \(\displaystyle b\) is the y-intercept.

Lines that are perpendicular to each other have slopes that are negative reciprocals of each other. This means that you need to flip the numerator and denominator of the given slope and then change the sign.

First, find the reciprocal of \(\displaystyle \frac{4}{5}\).

\(\displaystyle \frac{4}{5}\rightarrow \frac{5}{4}\)

Next, change the sign.

\(\displaystyle \frac{5}{4}\rightarrow -\frac{5}{4}\)

Lines can be written in the slope-intercept form: 

\(\displaystyle y=mx+b\)

In this equation, \(\displaystyle m\) is the slope and \(\displaystyle b\) is the y-intercept.

Lines that are perpendicular to each other have slopes that are negative reciprocals of each other. This means that you need to flip the numerator and denominator of the given slope and then change the sign.

First, find the reciprocal of \(\displaystyle -6\).

\(\displaystyle -6=-\frac{6}{1}\)

Flip the numerator and the denominator.

\(\displaystyle -\frac{6}{1}=-\frac{1}{6}\)

Next, change the sign.

\(\displaystyle -\frac{1}{6}\rightarrow \frac{1}{6}\)

This problem can be easily solved for by remembering what defines a perpendicular line. A line is perpendicular to another when the product of the two lines' slope is \(\displaystyle -1.\) For instance, if a line has a slope of \(\displaystyle \small 2\), the line perpendicular to it will have a slope of \(\displaystyle \small -\frac{1}2\) because \(\displaystyle \small 2 \cdot -\frac{1}{2}=-1\). Using this example, if the reference line has a slope of \(\displaystyle -3\), that means the line of interest must have a slope of \(\displaystyle +\frac{1}{3}\). The product of these two numbers is \(\displaystyle -1\).

This problem can be easily solved for by remembering what defines a perpendicular line. A line is perpendicular to another when the product of the two lines' slope is \(\displaystyle -1.\) 

For instance, if a line has a slope of \(\displaystyle \small 2\), the line perpendicular to it will have a slope of \(\displaystyle \small -\frac{1}2\) because \(\displaystyle \small 2 \cdot -\frac{1}{2}=-1\).

Using this example, if the reference line has a slope of \(\displaystyle -\frac{1}{7}\), that means the line of interest must have a slope of \(\displaystyle +7\).

The product of these two numbers is \(\displaystyle -1\).

To determine if two lines are perpendicular, we must compare the slopes.  To do that, we must write the equations in slope-intercept form

\(\displaystyle y = mx + b\)

where m is the slope.  Perpendicular lines have slopes that are opposite reciprocals of each other.  In other words, different signs and switch the numerator and the denominator.  

In the original equation

\(\displaystyle -5y = 25x + 30\)

we must write it in slope-intercept form.  To do that, we will divide each term by -5.

\(\displaystyle \frac{-5y}{-5} = \frac{25x}{-5} + \frac{30}{-5}\)

\(\displaystyle y = -5x - 6\)

We see that the slope of this line is -5.  A line that is perpendicular to this line will have a slope that is the opposite reciprocal of -5.  So a perpendicular line will have a slope of \(\displaystyle \frac{1}{5}\).

A line perpendicular to another line has a slope that is the negative reciprocal of the other. In our case, the line given has a slope of \(\displaystyle 3\) (\(\displaystyle m\) in the form \(\displaystyle y=mx+b\)), so the line perpendicular to it must have a slope equal to \(\displaystyle -\frac{1}{3}\).

We will need to rewrite this equation given in standard form to slope intercept form.

Subtract \(\displaystyle 2x\) on both sides.

\(\displaystyle 2x+3y-2x=1-2x\)

Simplify.

\(\displaystyle 3y= -2x+1\)

Divide by three on both sides.

\(\displaystyle \frac{3y}{3}= \frac{-2x+1}{3}\)

\(\displaystyle y=-\frac{2}{3}x+\frac{1}{3}\)

The slope of this line is:  \(\displaystyle -\frac{2}{3}\)

The perpendicular slope is the negative reciprocal of this slope.

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

\(\displaystyle -\frac{1}{-\frac{2}{3}} = 1 \div \frac{2}{3} = 1 \times \frac{3}{2} = \frac{3}{2}\)

The answer is:  \(\displaystyle \frac{3}{2}\)

When finding the slope of a perpendicular line, we need to ensure we have \(\displaystyle y=mx+b\) form. 

\(\displaystyle m\) stands for slope.

Our \(\displaystyle m\) is \(\displaystyle 2\).

To find the perpendicular slope, we need to take the negative reciprocal of that value which is \(\displaystyle -\frac{1}{2}\).

When finding the slope of a perpendicular line, we need to ensure we have \(\displaystyle y=mx+b\) form. 

\(\displaystyle 5x+4y=7\) 

We need to solve for \(\displaystyle y\).

By subtracting \(\displaystyle 5x\) both sides and dividing \(\displaystyle 4\) on both sides, we get 

\(\displaystyle y=\frac{7-5x}{4}\)  

Recall that \(\displaystyle m\) stands for slope.

Our \(\displaystyle m\) is \(\displaystyle -\frac{5}{4}\).

To find the perpendicular slope, we need to take the negative reciprocal of that value which is \(\displaystyle \frac{4}{5}\).

When finding the slope of a perpendicular line, we need to ensure we have \(\displaystyle y=mx+b\) form. 

\(\displaystyle 2y+12=3x\) 

We need to solve for \(\displaystyle y\).

By subtracting \(\displaystyle 12\) both sides and dividing \(\displaystyle 2\) on both sides, we get 

\(\displaystyle y=\frac{3x-12}{2}\)  

Recall that \(\displaystyle m\) stands for slope.

Our \(\displaystyle m\) is \(\displaystyle \frac{3}{2}\).

To find the perpendicular slope, we need to take the negative reciprocal of that value which is \(\displaystyle -\frac{2}{3}\).