# Gradient (or slope) of a Line, and Inclination

A 15% road gradient is equivalent to m = 0.15.

The gradient (also known as slope) of a line is defined as

"gradient"= text(vertical rise)/text(horizontal run

In the following diagram, the gradient of the line AB is given by: a/b

In general, for the line joining the points (x1, y1) and (x2, y2), we have:

(x_1,y_1)
(x_2,y_1)
(x_2,y_2)
y_2-y_1
x_2-x_1
"slope" = (y_2-y_1)/(x_2-x_1)

Slope of the line joining the points (x1, y1) and (x2, y2).

We can now write the formula for the slope of a line.

## Gradient of a Line Formula

We see from the diagram above, that the gradient (usually written m) is given by:

m=(y_2-y_1)/(x_2-x_1

Continues below

## Interactive graph - slope of a line

You can explore the concept of slope of a line in the following interactive graph (it's not a fixed image).

Drag either point A (x1, y1) or point B (x2, y2) to investigate how the gradient formula works. The numbers will update as you interact with the graph.

Notice what happens to the sign (plus or minus) of the slope when point B is above or below A.

Slope = (y_2 - y_1)/(x_2 - x_1)

=(BC)/(AC)

You can move the graph up-down, left-right if you hold down the "Shift" key and then drag the graph.

If you get lost, you can always refresh the page.

### Example

Find the slope of the line joining the points (−4, −1) and (2, −5).

These are the points involved:

So the slope is:

m=(y_2-y_1)/(x_2-x_1

=(-5-(-1))/(2-(-4)

=(-4)/6

=-2/3

Note the slope is negative. The line is going "down hill" as we move left to right.

### Positive and Negative Slopes

In general, a positive slope indicates the value of the dependent variable (usually y) increases as we go left to right:

"slope " m " is positive"

The line has positive slope.

The dependent variable in the above graph is the y-value, while the independent variable is x.

A negative slope means that the value of the dependent variable (usually y) is decreasing as we go left to right:

"slope " m " is negative"

The line has negative slope.

## Inclination

We have a line with slope m and the angle that the line makes with the x-axis is α.

From trigonometry, we recall that the tan of angle α is given by:

tan\ alpha=text(opposite)/text(adjacent)

Now, since slope is also defined as opposite/adjacent, we have:

alpha
tan alpha = "opposite"/"adjacent" = m

Diagram illustrating tan α = m.

This gives us the result:

tan α = m

Then we can find angle α using

α = arctan m

(That is, α = tan-1 m)

This angle α is called the inclination of the line.

### Exercise 1

Find the inclination of the line with slope 2.

Here, tan α = 2, so

alpha=arctan(2)

=63.43^@

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NOTE: The size of angle α is (by definition) only between 0° and 180°`.

### Exercise 2

Find the slope of the line with inclination α = 137°.

The situation is as follows:

So the slope is:

m = tan α

= tan 137°

= −0.933

Note that the slope is negative.