In this post, I’ll demonstrate how we can use *y* = *mx* +* b* to find equations of lines. No need to memorize other equations of lines…it is easier to focus on the data given to us an use it to find *m* and *b*.

Let’s look at the most basic example that illustrates this process.

a. Find the equation of a line through the point (4, -6) with slope – ^{3}/_{4}.

Start by substituting *m* = -3/4 in the slope-intercept form to yield

*y* = – ^{3}/_{4 }*x* + *b*

Now substitute the point into the line by setting* x* = 4 and *y* = -6. This leads to

Using this value for *b* in the slope-intercept form above give the line *y* = – ^{3}/_{4} *x* -3. The answer to the problem is the equation of the line with the appropriate values for *m* and *b*.

Now let’s complicate matters a bit by finding the equation of a line passing through two points.

b. Find a line through the points (-1, 3) and (2, 6).

The slope through the points is

Substitute the slope into the slope-intercept form to give

*y* = *x* + *b*

Now take one of the points and substitute it into this equation. Using the ordered pair (2, 6), we can solve for *b*:

This gives us the equation *y* = *x* + 4.

In both of these examples we applied the same strategy of putting in the slope* m* and then solving for the intercept *b*.

Now let’s look at a problem whose wording might throw you off.

c. Find a line with *x*-intercept of -5 and a *y*-intercept of 4.

Don’t let the fact that they talk about intercepts throw you off the strategy. These intercepts can be written as ordered pairs (-5, 0) and (0, 4). The slope between these points is

This leads to the line

*y* = ^{4}/_{5} *x* + *b*

Since the *y*-intercept is 4, we can substitute it into this line for b to give *y* = ^{4}/_{5} *x* + 4 .

This problem is even easier since the y intercept was given to us. In each case we can start from* y* = *mx* +* b* and then find the value of *m* and *b*.