How do you find the slope that is perpendicular to the line  y= -1?

Jun 22, 2016

It is undefined.

Explanation:

The slope $m$ of a line passing through points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ is given by the formula:

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

The line $y = - 1$ passes through the points $\left(0 , - 1\right)$ and $\left(1 , - 1\right)$. So it has slope:

$\frac{\left(- 1\right) - \left(- 1\right)}{1 - 0} = \frac{0}{1} = 0$

Alternatively, simply note that $y = - 1$ can be rewritten:

$y = 0 x + \left(- 1\right)$

which is in standard slope intercept form:

$y = m x + b$

with slope $m = 0$ and intercept $b = - 1$.

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If a line has non-zero slope $m$, then any line perpendicular to it will have slope $- \frac{1}{m}$.

The line $y = - 1$ has slope $0$ so any line perpendicular to it will have undefined slope. If you try to evaluate $- \frac{1}{m}$, it involves division by $0$, which has undefined result.