Find the Perpendicular Line through: (-3,1) , perp. to y=3/5x+2

through: $(- 3, 1)$ , perp. to $y = \frac{3}{5} x + 2$

Find the slope when $y = \frac{3}{5} x + 2$ .

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 $\frac{3}{5}$ and $x$ .

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

Reorder terms.

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

Using the slope-intercept form, the slope is $\frac{3}{5}$ .

$m = \frac{3}{5}$

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

$m_{perpendicular} = - \frac{1}{\frac{3}{5}}$

Simplify $- \frac{1}{\frac{3}{5}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{5}{3}))$

Multiply $\frac{5}{3}$ by $1$ .

$m_{perpendicular} = - \frac{5}{3}$

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

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

$y - (1) = - \frac{5}{3} \cdot (x - (- 3))$

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

$y - 1 = - \frac{5}{3} \cdot (x + 3)$

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

Solve for $y$ .

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

Rewrite.

$y - 1 = 0 + 0 - \frac{5}{3} \cdot (x + 3)$

Simplify terms.

Apply the distributive property.

$y - 1 = - \frac{5}{3} x - \frac{5}{3} \cdot 3$

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

$y - 1 = - \frac{x \cdot 5}{3} - \frac{5}{3} \cdot 3$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$y - 1 = - \frac{x \cdot 5}{3} + \frac{- 5}{3} \cdot 3$

Cancel the common factor.

$y - 1 = - \frac{x \cdot 5}{3} + \frac{- 5}{3} \cdot 3$

Rewrite the expression.

$y - 1 = - \frac{x \cdot 5}{3} - 5$

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

$y - 1 = - \frac{5 x}{3} - 5$

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

Add $1$ to both sides of the equation.

$y = - \frac{5 x}{3} - 5 + 1$

Add $- 5$ and $1$ .

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

Reorder terms.

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

Remove parentheses.

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

Find the Perpendicular Line through: (-3,1) , perp. to y=3/5x+2

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