Find the Perpendicular Line (-2/3,7/8) , 3x+4y=7

$(- \frac{2}{3}, \frac{7}{8})$ , $3 x + 4 y = 7$

Solve $3 x + 4 y = 7$ .

Subtract $3 x$ from both sides of the equation.

$4 y = 7 - 3 x$

Divide each term in $4 y = 7 - 3 x$ by $4$ and simplify.

Divide each term in $4 y = 7 - 3 x$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Find the slope when $y = \frac{7}{4} - \frac{3 x}{4}$ .

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

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

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

Reorder terms.

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

Remove parentheses.

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

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

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

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

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

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

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

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

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Multiply $- 1$ by $- 1$ .

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

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

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

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

Use the slope $\frac{4}{3}$ and a given point $(- \frac{2}{3}, \frac{7}{8})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{7}{8}) = \frac{4}{3} \cdot (x - (- \frac{2}{3}))$

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

$y - \frac{7}{8} = \frac{4}{3} \cdot (x + \frac{2}{3})$

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

Solve for $y$ .

Simplify $\frac{4}{3} \cdot (x + \frac{2}{3})$ .

Rewrite.

$y - \frac{7}{8} = 0 + 0 + \frac{4}{3} \cdot (x + \frac{2}{3})$

Simplify by adding zeros.

$y - \frac{7}{8} = \frac{4}{3} \cdot (x + \frac{2}{3})$

Apply the distributive property.

$y - \frac{7}{8} = \frac{4}{3} x + \frac{4}{3} \cdot \frac{2}{3}$

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

$y - \frac{7}{8} = \frac{4 x}{3} + \frac{4}{3} \cdot \frac{2}{3}$

Multiply $\frac{4}{3} \cdot \frac{2}{3}$ .

Multiply $\frac{4}{3}$ by $\frac{2}{3}$ .

$y - \frac{7}{8} = \frac{4 x}{3} + \frac{4 \cdot 2}{3 \cdot 3}$

Multiply $4$ by $2$ .

$y - \frac{7}{8} = \frac{4 x}{3} + \frac{8}{3 \cdot 3}$

Multiply $3$ by $3$ .

$y - \frac{7}{8} = \frac{4 x}{3} + \frac{8}{9}$

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

Add $\frac{7}{8}$ to both sides of the equation.

$y = \frac{4 x}{3} + \frac{8}{9} + \frac{7}{8}$

To write $\frac{8}{9}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = \frac{4 x}{3} + \frac{8}{9} \cdot \frac{8}{8} + \frac{7}{8}$

To write $\frac{7}{8}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y = \frac{4 x}{3} + \frac{8}{9} \cdot \frac{8}{8} + \frac{7}{8} \cdot \frac{9}{9}$

Write each expression with a common denominator of $72$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{8}{9}$ by $\frac{8}{8}$ .

$y = \frac{4 x}{3} + \frac{8 \cdot 8}{9 \cdot 8} + \frac{7}{8} \cdot \frac{9}{9}$

Multiply $9$ by $8$ .

$y = \frac{4 x}{3} + \frac{8 \cdot 8}{72} + \frac{7}{8} \cdot \frac{9}{9}$

Multiply $\frac{7}{8}$ by $\frac{9}{9}$ .

$y = \frac{4 x}{3} + \frac{8 \cdot 8}{72} + \frac{7 \cdot 9}{8 \cdot 9}$

Multiply $8$ by $9$ .

$y = \frac{4 x}{3} + \frac{8 \cdot 8}{72} + \frac{7 \cdot 9}{72}$

Combine the numerators over the common denominator.

$y = \frac{4 x}{3} + \frac{8 \cdot 8 + 7 \cdot 9}{72}$

Simplify the numerator.

Multiply $8$ by $8$ .

$y = \frac{4 x}{3} + \frac{64 + 7 \cdot 9}{72}$

Multiply $7$ by $9$ .

$y = \frac{4 x}{3} + \frac{64 + 63}{72}$

Add $64$ and $63$ .

$y = \frac{4 x}{3} + \frac{127}{72}$

Reorder terms.

$y = \frac{4}{3} x + \frac{127}{72}$

Find the Perpendicular Line (-2/3,7/8) , 3x+4y=7

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