# How do you determine the slope of the line passing through (-1,0), (1,2)?

Jun 25, 2015

The slope of a line between two points is the change in $y$ values (positive OR negative change) divided by the change in the $x$ values (which may also be positive or negative>

#### Explanation:

To find the changes in the values, we must pick one ordered pair as the first, and the other is the second. In the end it does not matter which way we do it, as long as we decide which pair is first and which is second,

With the ordered pairs (the points) labelled: $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$, the slope of the line through the points is

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

(Note that, if ${x}_{1} = {x}_{2}$, then the definition does not work, because division by $0$ is not defined.)

The change in a value is the ending value minus the starting value.

One way
With $\left({x}_{1} , {y}_{1}\right) = \left(- 1 , 0\right)$ and $\left({x}_{2} , {y}_{2}\right) = \left(1 , 2\right)$ , we get:

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{2 - 0}{1 - \left(- 1\right)} = \frac{2}{2} = 1$

Another way
If we choose our first and second points the other way around, it looks like this:

$\left({x}_{1} , {y}_{1}\right) = \left(1 , 2\right)$ and $\left({x}_{2} , {y}_{2}\right) = \left(- 1 , 0\right)$ , we get:

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{0 - 2}{\left(- 1\right) - 1} = \left(-\right) \frac{2}{- 2} = 1$

Because both signs of the numerator and the denominator changed, the sign of the fraction did not.