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Rule

Slope Formula

The slope of a line can be found algebraically using the following rule.

$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$

Here, $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ are two points on the line.

Proof

To prove the slope formula, use the following definition of slope.

m = \frac{change in y - value}{change in x - value}

Consider a line passing through the points

(x_{1}, y_{1})

and

(x_{2}, y_{2}) .

Notice that the change in $y -$ value is the difference between the $y -$ coordinates of the points as well as the change in $x -$ value is the difference between the $x -$ coordinates.

From here, the slope formula can be found by substituting

y_{2} - y_{1}

and

x_{2} - x_{1}

for the change in

y -

value and the change in

x -

value, respectively.

m = \frac{change in y - value}{change in x - value} ⇓ m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Exercises

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