# Intermediate Geometry : How to find the equation of a line

## Example Questions

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### Example Question #1 : How To Find The Equation Of A Line

Given two points  and , find the equation for the line connecting those two points in slope-intercept form.

Explanation:

If we have two points, we can find the slope of the line between them by using the definition of the slope:

where the triangle is the greek letter 'Delta', and is used as a symbol for 'difference' or 'change in'

Now that we have our slope ( , simplified to ), we can write the equation for slope-intercept form:

where  is the slope and  is the y-intercept

In order to find the y-intercept, we simply plug in one of the points on our line

So our equation looks like

### Example Question #2 : How To Find The Equation Of A Line

Which of the following is an equation for a line with a slope of  and a y-intercept of ?

Explanation:

Because we have the desired slope and the y-intercept, we can easily write this as an equation in slope-intercept form (y=mx+b).

This gives us . Because this does not match either of the answers in this form (y=mx+b), we must solve the equation for x. Adding 5 to each side gives us . We can then multiple both sides by 3 and divide both sides by 4, giving us .

### Example Question #3 : How To Find The Equation Of A Line

If the -intercept of a line is , and the -intercept is , what is the equation of this line?

Explanation:

If the y-intercept of a line is , then the -value is  when  is zero. Write the point:

If the -intercept of a line is , then the -value is  when  is zero. Write the point:

Use the following formula for slope and the two points to determine the slope:

Use the slope intercept form and one of the points, suppose , to find the equation of the line by substituting in the values of the point and solving for , the -intercept.

Therefore, the equation of this line is .

### Example Question #4 : How To Find The Equation Of A Line

What is the equation of a line that has a slope of  and a -intercept of ?

Explanation:

The slope intercept form can be written as:

where  is the slope and  is the y-intercept. Plug in the values of the slope and -intercept into the equation.

### Example Question #5 : How To Find The Equation Of A Line

What is the equation of a line with a slope of  and an -intercept of ?

Explanation:

The -intercept is the value of  when the  value is equal to zero. The actual point located on the graph for an -intercept of  is . The slope, , is 2.

Write the slope-intercept equation and substitute the point and slope to solve for the -intercept:

Plug the slope and -intercept back in the slope-intercept formula:

### Example Question #6 : How To Find The Equation Of A Line

A line goes through the following points  and .

Find the equation of the line.

Explanation:

First, find the slope of the line using the slope formula:

.

Next we plug one of the points, and the slope, into the point-intercept line forumula:

where m is our slope.

Then  and when we plug in point (2,3) the formula reads  then solve for b.

.

To find the equation of the line, we plug in our m and b into the slope-intercept equation.

So,  or simplified, .

### Example Question #7 : How To Find The Equation Of A Line

Write the equation for the line passing through the points  and

Explanation:

To determine the equation, first find the slope:

We want this equation in slope-intercept form, . We know  and  because we have two coordinate pairs to choose from representing an  and a . We know  because that represents the slope. We just need to solve for , and then we can write the equation.

We can choose either point and get the correct answer. Let's choose

multiply ""

This means that the form is

### Example Question #8 : How To Find The Equation Of A Line

Write the equation for a line that passes through the points and .

Explanation:

To determine the equation, first find the slope:

We want this equation in slope-intercept form, . We know  and  because we have two coordinate pairs to choose from representing an  and a . We know  because that represents the slope. We just need to solve for , and then we can write the equation.

We can choose either point and get the correct answer. Let's choose

multiply ""

subtract  from both sides

This means that the form is

### Example Question #9 : How To Find The Equation Of A Line

Find the equation for a line passing through the points and .

Explanation:

To determine the equation, first find the slope:

We want this equation in slope-intercept form, . We know  and  because we have two coordinate pairs to choose from representing an  and a  . We know  because that represents the slope. We just need to solve for , and then we can write the equation.

We can choose either point and get the correct answer. Let's choose

multiply ""

subtract  from both sides

This means that the form is

### Example Question #10 : How To Find The Equation Of A Line

Find the equation for the line passing through the points and .

Explanation:

To determine the equation, first find the slope:

We want this equation in slope-intercept form, . We know  and  because we have two coordinate pairs to choose from representing an  and a . We know  because that represents the slope. We just need to solve for , and then we can write the equation.

We can choose either point and get the correct answer. Let's choose

multiply ""

subtract  from both sides

This means that the form is

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