Since line  follows the equation , we can surmise its slope is . Thus, it follows that any line perpendicular to  will have a slope of .

Since we also know at least one point on line , , we can use point slope form to find an initial equation for our line.

----> .

Next, we can simplify to reach point-slope form.

---->.

Thus, line  in slope-intercept form is .

In order to solve this problem, we need first to transform the equation from standard form to slope-intercept form:

Transform the original equation to find its slope.

First, subtract  from both sides of the equation.

Simplify and rearrange.

Next, divide both sides of the equation by 6.

The slope of our first line is equal to . Perpendicular lines have slopes that are opposite reciprocals of each other; therefore, if the slope of one is x, then the slope of the other is equal to the following: 

Let's calculate the opposite reciprocal of our slope:

The slope of our line is equal to 2. We now have the following partial equation:

We are missing the y-intercept, . Substitute the x- and y-values in the given point  to solve for the missing y-intercept. 

Add 4 to both sides of the equation.

Substitute this value into our partial equation to construct the equation of our line:

First, we must solve the equation for y to determine the slope:  y = 2x + ³/₂

By looking at the coefficient in front of x, we know that the slope of this line has a value of 2. To fine the slope of any line perpendicular to this one, we take the negative reciprocal of it:

slope = m , perpendicular slope = – ¹/_m

slope = 2 , perpendicular slope = – ¹/₂

Find the slope of the given line.  The perpendicular slope will be the opposite reciprocal of the original slope.  Use the slope-intercept form (y = mx + b) and substitute in the given point and the new slope to find the intercept, b.  Convert back to standard form of an equation:  ax + by =  c.

First rearrange the equation so that it is in slope-intercept form, resulting in y=2/3 x + 14/9. The slope of this line is 2/3, so the slope of the line perpendicular will have the opposite reciprocal as a slope, which is -3/2.

Perpendicular lines have slopes that are the opposite of the reciprocal of each other. In this case, the slope of the first line is -2. The reciprocal of -2 is -1/2, so the opposite of the reciprocal is therefore 1/2.

First we must find the slope of the given line. The slope of y = –3x – 4 is –3. The slope of the perpendicular line is the negative reciprocal. This means you change the sign of the slope to its opposite: in this case to 3. Then find the reciprocal by switching the denominator and numerator to get 1/3; therefore the slope of the perpendicular line is 1/3.

The question puts the line in point-slope form y – y₁ = m(x – x₁), where m is the slope. Therefore, the slope of the original line is 1/2.  A line perpendicular to another has a slope that is the negative reciprocal of the slope of the other line. The negative reciprocal of the original line is –2, and is thus the slope of its perpendicular line. 

The equation of a line is  where  is the slope.

Rearrange the equation to match this:

For the perpendicular line, the slope is the negative reciprocal;

therefore 

First, you must find the slope of the line given to you. Remember that the slope is calculated: 

Thus, for our data, this is:

Now, the perpendicular slope to this is opposite and reciprocal. Hence, it must be . This only holds for the equation 

To know this, solve the equation for the format . This will let you find the slope very quickly, for it is . First, add  to both sides:

Next, divide everything by :

You really just need to pay attention to the  term. This reduces to , which is just what you need!