# Slope of a Line

How to find the slope of a line that passes through two points: formula, 4 examples, and their solutions.

## Formula

### Formula

The slope means the ratio of

[change of y]/[change of x].

If a line passes through (x_{1}, y_{1}) and (x_{2}, y_{2}),

then the change of x is x_{2} - x_{1}

and the change of y is y_{2} - y_{1}.

So the slope of the line, *m*, is*m* = (y_{2} - y_{1})/(x_{2} - x_{1}).

## Example 1

### Example

### Solution

Draw a line that passes through (1, 1) and (3, 5).

The change of x is

3 - 1 = 2.

The change of y is

5 - 1 = 4.

So the change of x is 2.

And the change of y is 4.

Then the slope, m, is

m = 4/2.

4/2 = 2

So the slope of the line, m, is 2.

As you can see,

if a line goes upward as you move to the right,

then the slope of the line is plus.

## Example 2

### Example

### Solution

Draw a line that passes through (-4, 1) and (1, -2).

The change of x is

1 - (-4) = 1 + 4 = 5.

The change of y is

-2 - 1 = -3.

So the change of x is 5.

And the change of y is -3.

Then the slope, m, is

m = (-3)/5.

(-3)/5 = -3/5

So the slope of the line, m, is -3/5.

As you can see,

if a line goes downward as you move to the right,

then the slope of the line is minus.

## Example 3

### Example

### Solution

Draw a line that passes through (-1, 2) and (5, 2).

The change of x is

5 - (-1) = 5 + 1 = 6.

The change of y is

2 - 2 = 0.

Then the slope, m, is

m = 0/6.

If the numerator is 0,

then the fraction is 0.

So 0/6 = 0.

So the slope of the line, m, is 0.

As you can see,

if a line is horizontal,

then the slope of the line is 0.

## Example 4

### Example

### Solution

Draw a line that passes through (1, 2) and (1, -3).

The change of x is

1 - 1 = 0.

The change of y is

-3 - 2 = -5.

Then the slope, m, is

m = (-5)/0.

The denominator of a fraction cannot be 0.

So (-5)/0 doesn't make sense.

So the line has no slope.

So the answer is [no slope].

As you can see,

if a line is vertical,

then the line has no slope.