How do you write an equation of a line passing through (2, -8), perpendicular to 4x+5y=7?

Mar 14, 2017

Answer:

$y = \frac{5}{4} x - \frac{21}{2}$

Explanation:

First, you would want to get only $y$ on one side of the given equation (for converting the equation to point slope form), first by subtracting $4 x$ from both sides:

$5 y = 7 - 4 x$ or $5 y = - 4 x + 7$

Then divide by 5:

$y = \frac{7}{5} - \frac{4}{5} x$ or $y = - \frac{4}{5} x + \frac{7}{5}$

Product of slopes of two perpendicular lines is $- 1$, so you know that the slope of the line perpendicular to given line would be

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

Now, make your new equation in point-slope format:

$y = m x + b$

$y = \frac{5}{4} x + b$

Now, to figure out the $y$-intercept (b), you can substitute $2$ and $- 8$ into x and y, respectively i.e.

$\left(- 8\right) = \frac{5}{4} \times 2 + b$

Multiply $\frac{5}{4}$ and $2$:

$\left(- 8\right) = \frac{10}{4} + b$ or $\left(- 8\right) = \frac{5}{2} + b$

Adding $- \frac{5}{2}$ to both sides:

$\left(- 8\right) - \frac{5}{2} = b$

Make -8 and 5/2 have a common denominator:

$- \frac{16}{2} - \frac{5}{2} = b$

$- \frac{21}{2} = b$

Then you can put that into the point-slope formula and you're done!

$y = \frac{5}{4} x - \frac{21}{2}$
graph{(y-5/4x+21/2)(5y-7+4x)=0 [-8.62, 11.38, -8.44, 1.56]}