Slope of a Line

Slope of a Line

How to find the slope of a line and the linear equation y = mx: slope formula, the linear equation y = mx, proof, examples, and their solutions.

Formula: Slope

The slope of a line that passes through (x1, y1) and (x2, y2) is m = (y2 - y1)/(x2 - x1).

The slope means
the ratio of [change of y]/[change of x].

Ratio

If a line passes through (x1, y1) and (x2, y2),
then [change of x] = x2 - x1
and [change of y] = y2 - y1.

So the slope of the line, m, is
m = (y2 - y1)/(x2 - x1).

Example 1: Positive Slope

Find the slope of the line that passes through the given points. (1, 1), (3, 5).

The change of x is
3 - 1 = 2.

And the change of y is
5 - 1 = 4.

Then m = 4/2.

As you can see,
if the graph goes upward as x increases,
then the slope is (+).

Example 2: Negative Slope

Find the slope of the line that passes through the given points. (-4, 1), (1, -4)

The change of x is
1 - (-4) = 5.

And the change of y is
-2 - 1 = -3.

Then m = -3/5.

As you can see,
if the graph goes downward as x increases,
then the slope is (-).

Example 3: Zero Slope

Find the slope of the line that passes through the given points. (-1, 2), (5, 2)

The change of x is
5 - (-1) = 6.

And the change of y is
2 - 2 = 0.

Then m = 0/6.

As you can see,
the slope of a horizontal line is 0.

Example 4: No Slope

Find the slope of the line that passes through the given points. (1, 2), (1, 3)

The change of x is
1 - 1 = 0.

And the change of y is
2 - (-3) = 5.

Then m = 5/0.

The denominator is 0,
which cannot be defined.

So the given line has [no slope].

As you can see,
a vertical line has no slope.

Linear Equation: y = mx

If a line passes through the origin and has a slope m, then the linear equation is y = mx.

If a line passes through the [origin]
and has a [slope m],

then the linear equation is y = mx.

In other words,
y = mx is a function
that passes through the origin
and that shows the slope m.

Function

This is true
unless the line is a vertical line.
(There's no slope: m cannot be defined.)

Proof: y = mx

Linear Equation: y = mx (Proof)

Think of a slope
between the origin (0, 0) and (x, y).
(x, y) is the point on the line.

The change of x is
x - 0 = x.

And the change of y is
y - 0 = y.

Then the slope is y/x.

Then set y/x = m.

Multiply x on both sides.

Then y = mx.

Example 5

Write the linear equation of the given line. (slope) = 3

The given line passes through the origin.
And its slope is 3/1.

So the linear equation is y = (3/1)x
which is y = 3x.

Example 6

Write the linear equation of the given line. (slope) = -1/2

The given line passes through the origin.
And its slope is -1/2.

So the linear equation is y = -1/2x.

Example 7

Write the linear equation of the given line. (slope) = 0

The given line passes through the origin.

And it's a horizontal line.
So the slope of the line is 0.

So the linear equation is y = 0x,
which is y = 0.

If you see the vertical number line (y-axis),
the given horizontal line shows y = 0.

You can check the answer like this
when you're wondering
if the answer is x = 0 or y = 0.

Example 8

Write the linear equation of the given line: y-axis.

The given line passes through the origin.

And it's a vertical line.
So it has no slope.

Then see the horizontal number line (x-axis).
The given vertical line shows x = 0.

So x = 0 is the answer.