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

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

Find the slope when $y = \frac{7}{5} x - \frac{3}{5}$ .

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 $\frac{7}{5}$ and $x$ .

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

Reorder terms.

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

Using the slope-intercept form, the slope is $\frac{7}{5}$ .

$m = \frac{7}{5}$

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

$m_{perpendicular} = - \frac{1}{\frac{7}{5}}$

Simplify $- \frac{1}{\frac{7}{5}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{5}{7}))$

Multiply $\frac{5}{7}$ by $1$ .

$m_{perpendicular} = - \frac{5}{7}$

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

Use the slope $- \frac{5}{7}$ and a given point $(- 4, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = - \frac{5}{7} \cdot (x - (- 4))$

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

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

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

Solve for $y$ .

Simplify $- \frac{5}{7} \cdot (x + 4)$ .

Rewrite.

$y - 3 = 0 + 0 - \frac{5}{7} \cdot (x + 4)$

Simplify by adding zeros.

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

Apply the distributive property.

$y - 3 = - \frac{5}{7} x - \frac{5}{7} \cdot 4$

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

$y - 3 = - \frac{x \cdot 5}{7} - \frac{5}{7} \cdot 4$

Multiply $- \frac{5}{7} \cdot 4$ .

Multiply $4$ by $- 1$ .

$y - 3 = - \frac{x \cdot 5}{7} - 4 (\frac{5}{7})$

Combine $- 4$ and $\frac{5}{7}$ .

$y - 3 = - \frac{x \cdot 5}{7} + \frac{- 4 \cdot 5}{7}$

Multiply $- 4$ by $5$ .

$y - 3 = - \frac{x \cdot 5}{7} + \frac{- 20}{7}$

Simplify each term.

Move $5$ to the left of $x$ .

$y - 3 = - \frac{5 \cdot x}{7} + \frac{- 20}{7}$

Move the negative in front of the fraction.

$y - 3 = - \frac{5 x}{7} - \frac{20}{7}$

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

Add $3$ to both sides of the equation.

$y = - \frac{5 x}{7} - \frac{20}{7} + 3$

To write $3$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = - \frac{5 x}{7} - \frac{20}{7} + 3 \cdot \frac{7}{7}$

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

$y = - \frac{5 x}{7} - \frac{20}{7} + \frac{3 \cdot 7}{7}$

Combine the numerators over the common denominator.

$y = - \frac{5 x}{7} + \frac{- 20 + 3 \cdot 7}{7}$

Simplify the numerator.

Multiply $3$ by $7$ .

$y = - \frac{5 x}{7} + \frac{- 20 + 21}{7}$

Add $- 20$ and $21$ .

$y = - \frac{5 x}{7} + \frac{1}{7}$

Reorder terms.

$y = - (\frac{5}{7} x) + \frac{1}{7}$

Remove parentheses.

$y = - \frac{5}{7} x + \frac{1}{7}$

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

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