To find the equation of a line, we need to know the slope and a point that passes through the line.  Once we know this, we can use the equation  where m is the slope of the line, and  is a point on the line.  For perpendicular lines, the slopes are negative reciprocals of each other.  The slope of  is 5, so the slope of the perpendicular line will have a slope of .  We know that the perpendicular line needs to contain the point (5,3), so we have all of the information we need.  We can now use the equation  

The equation of a line is written in the following format: 

1) The first step, then, is to find the slope, .

 is equal to the change in  divided by the change in .

So,

2) The perpendicular slope of a line with a slope of 2 is the opposite reciprocal of 2, which is .

3) Next step is to find . We don't need to find the equation of the original line; all we need from the original line is the slope. So all we need for is the perpendicular line. We can find values for  and  from the one point we have from the perpendicular line, plug them in, and solve for .

Our point is (4,7)

So, 

Then we just fill in our value for , and we have  as a function of .

This problem first relies on the knowledge of the slope-intercept form of a line, , where m is the slope and b is the y-intercept.

In order for a line to be perpendicular to another line, its slope has to be the negative reciprocal. In this case, we are seeking a line to be perpendicular to . This line has a slope of 2, a.k.a. . This means that the negative reciprocal slope will be . We are told that the y-intercept of this new line is 4.

We can now put these two new pieces of information into to get the equation

.

To solve this type of problem, we have to be familiar with the slope-intercept form of a line, where m is the slope and b is the y-intercept. The line that our line is perpendicular to has the slope-intercept equation , which means that the slope is .

The slope of a perpendicular line would be the negative reciprocal, so our slope is .

We don't know the y-intercept of our line yet, so we can only write the equation as:

.

We do know that the point is on this line, so to solve for b we can plug in -2 for x and 3 for y:

First we can multiply to get .

This makes our equation now:

either by subtracting 3 from both sides, or just by looking at this critically, we can see that b = 0.

Our original becomes , or simply .

Two lines are perpendicular if and only if their slopes are "negative reciprocals" of each other such as 1/2 and -2. In our problem we are not given line equations that we can readily see the slope, so we must convert each equation to slope intercept form or .

First find out the slope of the given equation by converting it to slope-intercept form:

So we need a line whose slope is the negative reciprocal of -1/8 (which is 8). Even though this number is not "negative" the idea of the negative reicprocal gives a positive number here because two negative signs cancel each other to make a positive. ...negative recicprocal....

Now we must choose an equation that, after being changed to slope intercept form has a slope of 8.

So   (answer) is the equation of a line that is perpendicular to 

.

A line is perpendicular to another line when they meet at a degree angle. That angle is the result of the slopes of the lines being opposite reciprocals.

The "opposite reciprocal" of  is best described as .

First we reorganize the original equation to isolate . To do this we want to get our equation into slope intercept form .

First subtract 12 from each side.

Now divide by 4 and simplify where possible.

opposite reciprocal

The only equation in the answer choices with a slope of is

.

This problem can be quickly solved through using the point-slope formula considering the given information. Before we start substituting in values, however, it's important to remember what determines a line to be perpendicular relative to another. By definition, lines are perpendicular if their slopes have a product of . For instance, if a line has a slope of , the line perpendicular to it will have a slope of  because . Using this concept, we must first determine the slope of the perpendicular line. We are given that the reference line has a slope of . That means the perpendicular line must have a slope of  because . Now that we know  (slope) and have a coordinate, we have fulfilled the requirements of the point-slope formula and can begin to substitue in information and solve for the equation. 

Here we arbitrarily use 

Before beginning this problem, it's important to remember what defines a perpendicular line. A line is perpendicular to another if the product of the two lines' slope is  For instance, if a line has a slope of , the line perpendicular to it will have a slope of  because . Using this concept, we must first determine the slope of the perpendicular line. The line of interest is perpendicular to a line with a slope of , therefore its slope must be 

The problem does not provide us enough information to use the point-slope formula to solve for the equation. However, we are provided with the line's y-intercept. This allows us to use the skeleton  to solve for the equation, where  is slope and  is the y-intercept.

By definition, lines are perpendicular if the product of their slopes is . For example, if a line has a slope of , the perpendicular line would have a slope of . This same concept can be used to solve this problem. Beginning with the issue of slope, we realize that the reference slope is . In order to create a product of , we must multiply  by . Therefore, the slope of the perpendicular line is . 

Because the problem has not given us a point, we cannot use the point-slope formula to solve for the equation. However, we have been provided with the line's y-intercept. With this information, we are ready to construct the line's equation remembering the  skeleton. We have  and now we have . Substituting in the given information will yield our answer. 

Because  is usally not written in front of variables, we may omit it from the final answer.