# What is the equation of the line perpendicular to y=-7/16x  that passes through  (5,4) ?

Mar 13, 2018

$y = \frac{16}{7} x - \frac{52}{7}$

See details below

#### Explanation:

If a line has the equation $y = m x$, we call slope to $m$ and whatever perpendicular line to it has then the equation $y = - \frac{1}{m} x$

In our case $y = - \frac{7}{16} x$, then, the slope is $m = - \frac{7}{16}$, so the perpendicular has slope m´=-1/(-7/16)=16/7. Our perpendicular line is

$y = \frac{16}{7} x + b$. But this line passes through $\left(5 , 4\right)$. Then

4=16/7·5+b. Transposing terms we have $b = - \frac{52}{7}$

Finally, perpendicular line equation is $y = \frac{16}{7} x - \frac{52}{7}$

Mar 13, 2018

$y = \frac{16}{7} x - \frac{52}{7}$

#### Explanation:

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

•color(white)(x)y=mx+b

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

$y = - \frac{7}{16} x \text{ is in this form}$

$\text{with } m = - \frac{7}{16}$

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

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

$\Rightarrow {m}_{\text{perpendicular}} = - \frac{1}{- \frac{7}{16}} = \frac{16}{7}$

$\Rightarrow y = \frac{16}{7} x + b \leftarrow \textcolor{b l u e}{\text{is the partial equation}}$

$\text{to find b substitute "(5,4)" into the partial equation}$

$4 = \frac{80}{7} + b \Rightarrow b = \frac{28}{7} - \frac{80}{7} = - \frac{52}{7}$

$\Rightarrow y = \frac{16}{7} x - \frac{52}{7} \leftarrow \textcolor{red}{\text{perpendicular equation}}$