# How do you find the slope of a line that is a) parallel and b) perpendicular to the given line: -2x-5y=-9?

Mar 29, 2018

$- \frac{2}{5} \text{ and } \frac{5}{2}$

#### Explanation:

$\text{the equation of a line in "color(blue)"slope-intercept form}$ is.

•color(white)(x)y=mx+b

$\text{where m is the slope and b the y-intercept}$

$\text{rearrange "-2x-5y=-9" into this form}$

$\Rightarrow - 5 y = 2 x - 9$

$\Rightarrow y = - \frac{2}{5} x + \frac{9}{5} \leftarrow \textcolor{b l u e}{\text{in slope intercept form}}$

$\text{with slope m } = - \frac{2}{5}$

• " Parallel lines have equal slopes"

$\Rightarrow \text{slope of parallel line is } m = - \frac{2}{5}$

$\text{Given a line with slope m then the slope of a line}$
$\text{perpendicular to it is}$

•color(white)(x)m_(color(red)"perpendicular")=-1/m

$\Rightarrow {m}_{\text{perpendicular}} = - \frac{1}{- \frac{2}{5}} = \frac{5}{2}$

Mar 29, 2018

$\frac{2}{5} \setminus \setminus \setminus$ parallel.

$- \frac{5}{2} \setminus \setminus \setminus$ perpendicular.

#### Explanation:

First rearrange the equation to get the form:

$y = m x + b$

It is only in this form that we can see what the gradient is.

$- 2 x - 5 y = - 9$

$y = \frac{2}{5} x + \frac{9}{5}$

If two lines have the same gradient, then they will be parallel. So any line of the form:

$y = \frac{2}{5} x + b$ , will be parallel to $\setminus \setminus y = \frac{2}{5} x + \frac{9}{5}$

If two lines are perpendicular, then the product of their gradients is $- 1$

For two lines with gradients ${m}_{1}$ and ${m}_{2}$:

${m}_{1} \cdot {m}_{2} = - 1$

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

Let ${m}_{1} = \frac{2}{5}$

$\therefore$

${m}_{2} = \frac{1}{m} _ 1 = - \frac{1}{\frac{2}{5}} = - \frac{5}{2}$

So, any line of the form $\boldsymbol{y = - \frac{5}{2} x + b}$ will be perpendicular to $\boldsymbol{y = \frac{2}{5} x + \frac{9}{5}}$