How do you find the slope that is perpendicular to the line 4x+5y= -5?

Jun 7, 2017

The slope of the line perpendicular to the line $4 x + 5 y = - 5$ is $\frac{5}{4}$.

Explanation:

$4 x + 5 y = - 5$

From here, we can manipulate the equation into the slope-intercept form. We first move the $4 x$ over to the right side by subtracting $4 x$ from both sides:

$5 y = - 5 - 4 x$

Next, we divide both sides by $5$ to isolate the $y$ variable:

$y = \frac{- 5 - 4 x}{5}$

We then simplify the right portion of the equation:

$y = - \frac{5}{5} - \frac{4 x}{5}$

And further simplification follows:

$y = - 1 - \frac{4 x}{5}$

We then rearrange the entire equation to clearly show the equation in slope-intercept form:

$y = - \frac{4 x}{5} - 1$

Now that we have the equation in slope-intercept form, we can clearly see that $- \frac{4}{5}$ is the slope here.

From here, it is easy. The product of a slope and its perpendicular slope is always $- 1$. (Proof: perpendicular lines have negative reciprocal slope)

If we set $a$ to be the perpendicular slope of $- \frac{4}{5}$, then

$- \frac{4}{5} a = - 1$

Then, we can isolate $a$ by dividing by $- \frac{4}{5}$ on both sides and then simplifying the result:

$a = \frac{- 1}{- \frac{4}{5}} = - 1 \cdot - \frac{5}{4} = \frac{5}{4}$

Thus, the slope of the line perpendicular to the line $4 x + 5 y = - 5$ is $\frac{5}{4}$.

Jun 7, 2017

The slope that is perpendicular to the line $4 x + 5 y = - 5$ is $\frac{5}{4}$.

Explanation:

The slope perpendicular to a line is the opposite reciprocal of the slope of the given line.

For example, if the slope of a line is 2, the slope of a line perpendicular to the line with slope 2 must be $- \frac{1}{2}$. "Opposite" refers to the opposite sign (e.x. the opposite of $- 5$ is $5$), and "reciprocal" just means 1 over whatever you're given (e.x. the reciprocal of 20 is just $\frac{1}{20}$).

Because in your case the line is in standard form (i.e. $a x + b y = c$, such that $a$ is a positive integer, and $b$ and $c$ are integers), the slope of the line is given by the equation: $m = - \frac{a}{b}$. Thus, the slope of the line is $- \frac{4}{5}$.

You can also find the slope, $m$, by putting the equation into slope-intercept form ($y = m x + b$):
1. $4 x + 5 y = - 5$
2. $5 y = - 4 x - 5$
3. $y = - \frac{4}{5} x - 1$
4. $m = - \frac{4}{5}$

The slope of a line perpendicular to this line must be the opposite reciprocal, so $- \left(\frac{1}{- \frac{4}{5}}\right) = \frac{5}{4}$. The slope that is perpendicular to the line $4 x + 5 y = - 5$ is $\frac{5}{4}$.