Find the Perpendicular Line A line is perpendicular to y=3x-8 and intersects the point (6,1)

A line is perpendicular to $y = 3 x - 8$ and intersects the point $(6, 1)$

Write the problem as a mathematical expression.

$y = 3 x - 8$ , $(6, 1)$

Use the slope-intercept form to find the slope.

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Using the slope-intercept form, the slope is $3$ .

$m = 3$

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

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

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

Use the slope $- \frac{1}{3}$ and a given point $(6, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = - \frac{1}{3} \cdot (x - (6))$

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

$y - 1 = - \frac{1}{3} \cdot (x - 6)$

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

Solve for $y$ .

Simplify $- \frac{1}{3} \cdot (x - 6)$ .

Rewrite.

$y - 1 = 0 + 0 - \frac{1}{3} \cdot (x - 6)$

Simplify by adding zeros.

$y - 1 = - \frac{1}{3} \cdot (x - 6)$

Apply the distributive property.

$y - 1 = - \frac{1}{3} x - \frac{1}{3} \cdot - 6$

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

$y - 1 = - \frac{x}{3} - \frac{1}{3} \cdot - 6$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$y - 1 = - \frac{x}{3} + \frac{- 1}{3} \cdot - 6$

Factor $3$ out of $- 6$ .

$y - 1 = - \frac{x}{3} + \frac{- 1}{3} \cdot (3 (- 2))$

Cancel the common factor.

$y - 1 = - \frac{x}{3} + \frac{- 1}{3} \cdot (3 \cdot - 2)$

Rewrite the expression.

$y - 1 = - \frac{x}{3} - 1 \cdot - 2$

Multiply $- 1$ by $- 2$ .

$y - 1 = - \frac{x}{3} + 2$

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

Add $1$ to both sides of the equation.

$y = - \frac{x}{3} + 2 + 1$

Add $2$ and $1$ .

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

Reorder terms.

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

Remove parentheses.

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

Find the Perpendicular Line A line is perpendicular to y=3x-8 and intersects the point (6,1)

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