Find the Perpendicular Line (-4,-1) and y=4/3x+6

$(- 4, - 1)$ and $y = \frac{4}{3} x + 6$

Step 1

Find the slope when $y = \frac{4}{3} x + 6$ .

Rewrite in slope-intercept form.

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Simplify the right side.

Combine $\frac{4}{3}$ and $x$ .

$y = \frac{4 x}{3} + 6$

Reorder terms.

$y = \frac{4}{3} x + 6$

Using the slope-intercept form, the slope is $\frac{4}{3}$ .

$m = \frac{4}{3}$

Step 2

The equation of a perpendicular line must have a slope that is the negative reciprocal of the original slope.

$m_{perpendicular} = - \frac{1}{\frac{4}{3}}$

Step 3

Simplify $- \frac{1}{\frac{4}{3}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{3}{4}))$

Multiply $\frac{3}{4}$ by $1$ .

$m_{perpendicular} = - \frac{3}{4}$

Step 4

Find the equation of the perpendicular line using the point-slope formula.

Use the slope $- \frac{3}{4}$ and a given point $(- 4, - 1)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Simplify the equation and keep it in point-slope form.

$y + 1 = - \frac{3}{4} \cdot (x + 4)$

Step 5

Write in $y = m x + b$ form.

Solve for $y$ .

Simplify $- \frac{3}{4} \cdot (x + 4)$ .

Rewrite.

$y + 1 = 0 + 0 - \frac{3}{4} \cdot (x + 4)$

Simplify terms.

Apply the distributive property.

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

Combine $x$ and $\frac{3}{4}$ .

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

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{3}{4}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Move $3$ to the left of $x$ .

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

Move all terms not containing $y$ to the right side of the equation.

Subtract $1$ from both sides of the equation.

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

Subtract $1$ from $- 3$ .

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

Reorder terms.

$y = - (\frac{3}{4} x) - 4$

Remove parentheses.

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

Step 6

Find the Perpendicular Line (-4,-1) and y=4/3x+6

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