Find the Perpendicular Line y-2=7/3(x+5) , (-4,9)

$y - 2 = \frac{7}{3} (x + 5)$ , $(- 4, 9)$

Step 1

Solve $y - 2 = \frac{7}{3} \cdot (x + 5)$ .

Simplify $\frac{7}{3} (x + 5)$ .

Rewrite.

$y - 2 = 0 + 0 + \frac{7}{3} (x + 5)$

Simplify by adding zeros.

$y - 2 = \frac{7}{3} (x + 5)$

Apply the distributive property.

$y - 2 = \frac{7}{3} x + \frac{7}{3} \cdot 5$

Combine $\frac{7}{3}$ and $x$ .

$y - 2 = \frac{7 x}{3} + \frac{7}{3} \cdot 5$

Multiply $\frac{7}{3} \cdot 5$ .

Combine $\frac{7}{3}$ and $5$ .

$y - 2 = \frac{7 x}{3} + \frac{7 \cdot 5}{3}$

Multiply $7$ by $5$ .

$y - 2 = \frac{7 x}{3} + \frac{35}{3}$

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

Add $2$ to both sides of the equation.

$y = \frac{7 x}{3} + \frac{35}{3} + 2$

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{7 x}{3} + \frac{35}{3} + 2 \cdot \frac{3}{3}$

Combine $2$ and $\frac{3}{3}$ .

$y = \frac{7 x}{3} + \frac{35}{3} + \frac{2 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = \frac{7 x}{3} + \frac{35 + 2 \cdot 3}{3}$

Simplify the numerator.

Multiply $2$ by $3$ .

$y = \frac{7 x}{3} + \frac{35 + 6}{3}$

Add $35$ and $6$ .

$y = \frac{7 x}{3} + \frac{41}{3}$

Step 2

Find the slope when $y = \frac{7 x}{3} + \frac{41}{3}$ .

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.

$y = m x + b$

Reorder terms.

$y = \frac{7}{3} x + \frac{41}{3}$

Using the slope-intercept form, the slope is $\frac{7}{3}$ .

$m = \frac{7}{3}$

Step 3

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

$m_{perpendicular} = - \frac{1}{\frac{7}{3}}$

Step 4

Simplify $- \frac{1}{\frac{7}{3}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{3}{7}))$

Multiply $\frac{3}{7}$ by $1$ .

$m_{perpendicular} = - \frac{3}{7}$

Step 5

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

Use the slope $- \frac{3}{7}$ and a given point $(- 4, 9)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (9) = - \frac{3}{7} \cdot (x - (- 4))$

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

$y - 9 = - \frac{3}{7} \cdot (x + 4)$

Step 6

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

Solve for $y$ .

Simplify $- \frac{3}{7} \cdot (x + 4)$ .

Rewrite.

$y - 9 = 0 + 0 - \frac{3}{7} \cdot (x + 4)$

Simplify by adding zeros.

$y - 9 = - \frac{3}{7} \cdot (x + 4)$

Apply the distributive property.

$y - 9 = - \frac{3}{7} x - \frac{3}{7} \cdot 4$

Combine $x$ and $\frac{3}{7}$ .

$y - 9 = - \frac{x \cdot 3}{7} - \frac{3}{7} \cdot 4$

Multiply $- \frac{3}{7} \cdot 4$ .

Multiply $4$ by $- 1$ .

$y - 9 = - \frac{x \cdot 3}{7} - 4 (\frac{3}{7})$

Combine $- 4$ and $\frac{3}{7}$ .

$y - 9 = - \frac{x \cdot 3}{7} + \frac{- 4 \cdot 3}{7}$

Multiply $- 4$ by $3$ .

$y - 9 = - \frac{x \cdot 3}{7} + \frac{- 12}{7}$

Simplify each term.

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

$y - 9 = - \frac{3 \cdot x}{7} + \frac{- 12}{7}$

Move the negative in front of the fraction.

$y - 9 = - \frac{3 x}{7} - \frac{12}{7}$

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

Add $9$ to both sides of the equation.

$y = - \frac{3 x}{7} - \frac{12}{7} + 9$

To write $9$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = - \frac{3 x}{7} - \frac{12}{7} + 9 \cdot \frac{7}{7}$

Combine $9$ and $\frac{7}{7}$ .

$y = - \frac{3 x}{7} - \frac{12}{7} + \frac{9 \cdot 7}{7}$

Combine the numerators over the common denominator.

$y = - \frac{3 x}{7} + \frac{- 12 + 9 \cdot 7}{7}$

Simplify the numerator.

Multiply $9$ by $7$ .

$y = - \frac{3 x}{7} + \frac{- 12 + 63}{7}$

Add $- 12$ and $63$ .

$y = - \frac{3 x}{7} + \frac{51}{7}$

Reorder terms.

$y = - (\frac{3}{7} x) + \frac{51}{7}$

Remove parentheses.

$y = - \frac{3}{7} x + \frac{51}{7}$

Step 7

Find the Perpendicular Line y-2=7/3(x+5) , (-4,9)

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