Find the Perpendicular Line (-2,5) ; y-1=-1/3(x-4)

$(- 2, 5)$ ; $y - 1 = - \frac{1}{3} (x - 4)$

Step 1

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

Simplify $- \frac{1}{3} (x - 4)$ .

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Combine $x$ and $\frac{1}{3}$ .

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

Multiply $- \frac{1}{3} \cdot - 4$ .

Multiply $- 4$ by $- 1$ .

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

Combine $4$ and $\frac{1}{3}$ .

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

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

Add $1$ to both sides of the equation.

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $4$ and $3$ .

$y = - \frac{x}{3} + \frac{7}{3}$

Step 2

Find the slope when $y = - \frac{x}{3} + \frac{7}{3}$ .

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.

$y = m x + b$

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

Reorder terms.

$y = - (\frac{1}{3} x) + \frac{7}{3}$

Remove parentheses.

$y = - \frac{1}{3} x + \frac{7}{3}$

Using the slope-intercept form, the slope is $- \frac{1}{3}$ .

$m = - \frac{1}{3}$

Step 3

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

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

Step 4

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

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 \cdot 3$

Multiply $- - (1 \cdot 3)$ .

Multiply $3$ by $1$ .

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

Multiply $- 1$ by $3$ .

$m_{perpendicular} = 3$

Multiply $- 1$ by $- 3$ .

$m_{perpendicular} = 3$

Step 5

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

Use the slope $3$ and a given point $(- 2, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = 3 \cdot (x - (- 2))$

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

$y - 5 = 3 \cdot (x + 2)$

Step 6

Solve for $y$ .

Simplify $3 \cdot (x + 2)$ .

Rewrite.

$y - 5 = 0 + 0 + 3 \cdot (x + 2)$

Simplify by adding zeros.

$y - 5 = 3 \cdot (x + 2)$

Apply the distributive property.

$y - 5 = 3 x + 3 \cdot 2$

Multiply $3$ by $2$ .

$y - 5 = 3 x + 6$

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

Add $5$ to both sides of the equation.

$y = 3 x + 6 + 5$

Add $6$ and $5$ .

$y = 3 x + 11$

Step 7

Find the Perpendicular Line (-2,5) ; y-1=-1/3(x-4)

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