# 2.3. Equation of a Line

# Determining the Equation of a Line

In this section, you will learn to:

- Find an equation of a line if a point and the slope are given.
- Find an equation of a line if two points are given.

So far, we were given an equation of a line and were asked to give information about it. For example, we were asked to find points on it, find its slope, and find intercepts. Now we are going to reverse the process. That is, we will be given either two points, or a point and the slope of a line, and we will be asked to find its equation.

An equation of a line can be written in two forms, the **slope-intercept form** or the **standard form**.

**The Slope-Intercept Form of a Line**: *y* = *mx* + *b *

A line is completely determined by two points, or a point and slope. So it makes sense to ask to find the equation of a line if one of these two situations is given.

Example 2.3.1

**Solution**

*b*is

*y*=

*mx*+

*b*.

*m*= 5, and

*b*= 3, the equation is

*y*= 5

*x*+ 3.

Example 2.3.2

**Solution**

*m*= 3, the partial equation is

*y*= 3

*x*+ b.

*b*can be determined by substituting the point (2, 7) in the equation

*y*= 3

*x*+

*b*.

*y*= 3

*x*+ 1.

Example 2.3.3

**Solution**

*y*= 3

*x*+

*b*

*b*. Substituting (-1, 2) gives

*y*= 3

*x*+ 5.

Example 2.3.4

**Solution**

*y*= – 4/3

*x*+ 4.

**The Standard form of a Line: A**

*x*+ B*y*= C*L*be a line with slope m, and containing a point (

*x*

_{1},

*y*

_{1}). If (

*x*,

*y*) is any other point on the line

*L*, then by the definition of a slope, we get

**point-slope form**or point-slope formula. If we simplify this formula, we get the equation of the line in the standard form, Ax + By = C.

Example 2.3.5

**Solution**

*m*= – 3/5 in the point-slope formula, we get

Example 2.3.6

Find the standard form of the line that passes through the points (1, -2), and (4, 0).

**Solution**

The point-slope form is:

Multiplying both sides by 3 gives us:

We should always be able to convert from one form of an equation to another. That is, if we are given a line in the slope-intercept form, we should be able to express it in the standard form, and vice versa.

Example 2.3.7

Write the equation *y* = – 2/3*x* +3 in the standard form.

**Solution **

Multiplying both sides of the equation by 3, we get

Example 2.3.8

*x*− 4

*y*= 10 in the slope-intercept form.

**Solution**

*y*, we get:

Finally, we learn a very quick and easy way to write an equation of a line in the standard form. But first we must learn to find the slope of a line in the standard form by inspection.

By solving for *y*, it can easily be shown that the slope of the line A*x* + B*y* = C is −A/B. The reader should verify.

Example 2.3.9

**a.**3

*x*− 5

*y*= 10

**b.**2

*x*+ 7

*y*= 20

**c.**4

*x*− 3

*y*= 8

**Solution**

**a.**A = 3, B = −5, therefore,

**b.**A = 2, B = 7, therefore,

**c.**

Now that we know how to find the slope of a line in the standard form by inspection, our job in finding the equation of a line is going to be very easy.

Example 2.3.10

**Solution**

*x*+5

*y*, and the partial equation is going to be:

*c*can easily be found by substituting for

*x*and

*y*.

# Practice questions

**1.** Write an equation of the line satisfying the following conditions. Write the equation in the form *y* = *mx* + *b*.

**a.** Passes through (3, 5) and (2, -1).

**b. **Passes through (5, -2) and *m* = .

**c.** Passes through (2, -5) and its *x*-intercept is 4.

**d.** Passes through (-3, -4), and (-5, 2).

**e. **Is a horizontal line passing through (2, -1).

**f.** Has an *x*-intercept = 3 and *y*-intercept = 4.

**2.** Write an equation of the line that satisfies the following conditions. Write the equation in the form A*x *+ B*y* = C.

**a. **Passes through (-4, -2) and *m* =

**b. **Passes through (2, -3) and (5, 1).