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# How do you write the equation of the family of line that are perpendicular to 4x+3y-1=0?

Mar 24, 2018

$y = - \frac{3}{4} x + b$

#### Explanation:

Rearrange $4 x + 3 y - 1 = 0$ to be an explicit function of x:

$y = - \frac{4}{3} x - \frac{1}{3}$

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

i.e.

If the gradients are ${m}_{1}$ and ${m}_{2}$:

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

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

Let $\setminus \setminus \setminus \setminus {m}_{1} = - \frac{4}{3}$

Then:

${m}_{2} = \frac{1}{- \frac{4}{3}} = - \frac{3}{4}$

Any line of the form:

$y = - \frac{3}{4} x + b$

Will be perpendicular to $\setminus \setminus \setminus \setminus y = - \frac{4}{3} x - \frac{1}{3}$