# Slope intercept form

The slope intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. The graph of a linear equation is a line and y = mx + b is one of the most common forms to write the equation of a line.

## How to derive the slope intercept form of a linear equation

Consider an arbitrary point (x,y) on the line and a point (0,b) on the y-axis. Then, you can use the slope formula to derive the slope-intercept form.

Slope = Rise / Run = (y1 - y2) / (x1 - x2)

Using (x1,y1) = (x,y) and (x2,y2) = (0,b), compute the slope m.

m = (y - b) / (x - 0)

m = (y - b) / x

Multiply both sides of the equation by x

(x)m = y - b

mx = y - b

Add b to both sides of the equation

mx + b = y - b + b

y = mx + b

## Slope intercept formula

The slope intercept formula is the equation y = mx + b

m is the slope of the line

b is the y-intercept or the point on the y-axis

x and y are the x and y coordinates

## Examples showing how to find the slope-intercept form of a straight line

The goal of these exercises is to write the equation of a straight line in slope-intercept form (y = mx  + b) by considering the following cases:

• The slope and the y-intercept are given (example #1)
• The slope and a point on the line are given (example #2)
• Two points on the line are given (example #3)

Example #1

If the slope of a line is m = 2 and the y-intercept is b = 5, write the slope intercept form of the line.

The equation is y = 2x + 5.

Example #2

If the slope of a line is m = 5 and (1, 6) is a point on the line, find the slope intercept form of the line.

This time we have m, but b is missing, so we have to find b.

Since m = 5, y = mx + b becomes y = 5x + b.

Now, use (1, 6) to get b.

Since x = 1 and y = 6, you can replace them into the equation.

Substituting 1 for x and 6 for y gives 6 = 5×1 + b.

6 = 5×1 + b is just a linear equation that you can solve to get b.

6 = 5×1 + b

6 = 5 + b

Subtract 5 from both sides.

6 − 5 = 5 − 5 + b

1 = 0 + b

1 = b

Now since we have b, y = 5x + 1

Example #3

Suppose (2, 3) and (4, 9) are two points on a line. Find the slope intercept form of the line.

This time both m and b are missing, so the first thing to do is to get m and then use m and a point, either (2, 3) or (4, 9) to get b.

Let x1 = 4, y1 = 9 and x2 = 2, y2 = 3

m = (y1 − y2) / (x1 − x2) = (9 − 3) / (4 − 2 ) = 6 / 2 = 3

Now we can use the value for m and one point to get b as already done in example #2.

Although you have two points, It does not matter which point you choose. Since both points are on the line, they will yield similar results.

Choosing (2, 3), x = 2 and y = 3

Substituting 2 for x, 3 for y, and 3 for m into the equation y = mx + b we get:

3 = 3 × 2 + b

3 = 6 + b

Subtract 6 from both sides

3 − 6 = 6 − 6 + b

-3 = 0 + b

-3 = b

Now we have b = -3 and m = 3, y = 3x + -3

## Slope intercept form special cases

The y-intercept b is equal to zero

In this case, y = mx and the line always goes through the origin of the coordinate system.

The slope of the line is equal to zero

In this case, y = (0)x + b = b and when you graph the line, it will be a horizontal line as it crosses the y-intercept.

The slope of the line is undefined

In this case, x is equal to the x-coordinate of any point on the line or any other number that is given to you in an exercise. When you graph the line, it will be a vertical line as it crosses the x-intercept.

Notice that you cannot find the equation in slope intercept form when the slope is undefined since you cannot find a value for the slope or m.

For example, if the line has an undefined slope and passes through the points (-2, 1) and (-2, 5), the equation of the line is just x = -2

## Take the slope intercept form quiz below to test your knowledge of this lesson.

100 Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

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