# 5.1: Determining the Equation of a Line

**At Grade**Created by: CK-12

**Practice**Determining the Equation of a Line

Suppose you were given the slope of a line and either its

### Watch This

CK-12 Foundation: 0501S Linear Equations (H264)

### Try This

An applet at http://www.cut-the-knot.org/Curriculum/Calculus/StraightLine.shtml lets you create multiple lines and see how they intersect. Each line is defined by two points; you can change the slope of a line by moving either of the points, or just drag the whole line around without changing its slope. To create another line, just click *duplicate* and then drag one of the lines that are already there.

### Guidance

You are probably aware that many real-world situations can be described with linear graphs and equations. In this lesson, we’ll see how to find equations in a variety of situations.

**Write an Equation Given Slope and y−Intercept**

Recall that you may write an equation in slope–intercept form with a few simple steps: start with the general equation for the slope-intercept form of a line,

If you are given just the graph of a line, you can identify the slope and

**Write an Equation Given the Slope and a Point**

Often, we don’t know the value of the **point-slope form.** An equation in point-slope form is written as

#### Example A

*A line has a slope of 35, and the point (2, 6) is on the line. Write the equation of the line in point-slope form.*

**Solution**

Start with the formula

Plug in

**The equation in point-slope form is**

Notice that the equation in point-slope form is not solved for

**Write an Equation Given Two Points**

Point-slope form also comes in useful when we need to find an equation given just two points on a line.

For example, suppose we are told that the line passes through the points (-2, 3) and (5, 2). To find the equation of the line, we can start by finding the slope.

Starting with the slope formula,

Now we just need to pick one of the two points to plug into the formula. Let’s use (5, 2); that gives us

What if we’d picked the other point instead? Then we’d have ended up with the equation

Starting with

On the other hand, if we start with

So whichever point we choose to get an equation in point-slope form, the equation is still mathematically the same, and we can see this when we convert it to

#### Example B

*A line contains the points (3, 2) and (-2, 4). Write an equation for the line in point-slope form; then write an equation in y−intercept form.*

**Solution**

Find the slope of the line:

Plug in the value of the slope:

Plug point (3, 2) into the equation:

**The equation in point-slope form is**

To convert to

**The equation in y−intercept form is y=−25x+315.**

**Graph an Equation in Point-Slope Form**

Another useful thing about point-slope form is that you can use it to graph an equation without having to convert it to slope-intercept form. From the equation

#### Example C

*Make a graph of the line given by the equation y+2=23(x−2).*

**Solution**

To read off the right values, we need to rewrite the equation slightly:

First plot point (2, -2) on the graph:

A slope of

Now draw a line through the two points and extend it in both directions:

Watch this video for help with the Examples above.

CK-12 Foundation: Linear Equations

### Guided Practice

*A line contains the points (1, -2) and (0, 0). Write an equation for the line in point-slope form; then write an equation in y−intercept form.*

**Solution**

Find the slope of the line:

Plug in the value of the slope:

Plug point (1, -2) into the equation:

**The equation in point-slope form is**

To convert to

**The equation in y−intercept form is y=−2x.**

### Explore More

Find the equation of each line in slope–intercept form.

- The line has a slope of 7 and a
y− intercept of -2. - The line has a slope of -5 and a
y− intercept of 6. - The line has a slope of
−14 and contains the point (4, -1). - The line contains points (3, 5) and (-3, 0).
- The line contains points (10, 15) and (12, 20).

Write the equation of each line in slope-intercept form.

Find the equation of each linear function in slope–intercept form.

m=5,f(0)=−3 m=−7,f(2)=−1 m=13,f(−1)=23 m=4.2,f(−3)=7.1 f(14)=34,f(0)=54 f(1.5)=−3,f(−1)=2

Write the equation of each line in point-slope form.

- The line has slope
−110 and goes through the point (10, 2). - The line has slope -75 and goes through the point (0, 125).
- The line has slope 10 and goes through the point (8, -2).
- The line goes through the points (-2, 3) and (-1, -2).
- The line contains the points (10, 12) and (5, 25).
- The line goes through the points (2, 3) and (0, 3).
- The line has a slope of
35 and ay− intercept of -3. - The line has a slope of -6 and a
y− intercept of 0.5.

Write the equation of each linear function in point-slope form.

m=−15 andf(0)=7 m=−12 andf(−2)=5 f(−7)=5 andf(3)=−4 f(6)=0 andf(0)=6 m=3 andf(2)=−9 m=−95 andf(0)=32

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 5.1.

### Image Attributions

## Description

## Learning Objectives

Here you'll learn how to write the equations of lines given their slope and

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## Date Created:

Aug 13, 2012## Last Modified:

Jan 31, 2016## Vocabulary

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