Find the Perpendicular Line Passing through (3,4) and perpendicular to the line whose equation is y=6x+7

Passing through $(3, 4)$ and perpendicular to the line whose equation is $y = 6 x + 7$

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 $6$ .

$m = 6$

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

$m_{perpendicular} = - \frac{1}{6}$

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

Use the slope $- \frac{1}{6}$ and a given point $(3, 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (4) = - \frac{1}{6} \cdot (x - (3))$

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

$y - 4 = - \frac{1}{6} \cdot (x - 3)$

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

Solve for $y$ .

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

Rewrite.

$y - 4 = 0 + 0 - \frac{1}{6} \cdot (x - 3)$

Simplify by adding zeros.

$y - 4 = - \frac{1}{6} \cdot (x - 3)$

Apply the distributive property.

$y - 4 = - \frac{1}{6} x - \frac{1}{6} \cdot - 3$

Combine $x$ and $\frac{1}{6}$ .

$y - 4 = - \frac{x}{6} - \frac{1}{6} \cdot - 3$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$y - 4 = - \frac{x}{6} + \frac{- 1}{6} \cdot - 3$

Factor $3$ out of $6$ .

$y - 4 = - \frac{x}{6} + \frac{- 1}{3 (2)} \cdot - 3$

Factor $3$ out of $- 3$ .

$y - 4 = - \frac{x}{6} + \frac{- 1}{3 \cdot 2} \cdot (3 \cdot - 1)$

Cancel the common factor.

$y - 4 = - \frac{x}{6} + \frac{- 1}{3 \cdot 2} \cdot (3 \cdot - 1)$

Rewrite the expression.

$y - 4 = - \frac{x}{6} + \frac{- 1}{2} \cdot - 1$

Combine $\frac{- 1}{2}$ and $- 1$ .

$y - 4 = - \frac{x}{6} + \frac{- - 1}{2}$

Multiply $- 1$ by $- 1$ .

$y - 4 = - \frac{x}{6} + \frac{1}{2}$

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

Add $4$ to both sides of the equation.

$y = - \frac{x}{6} + \frac{1}{2} + 4$

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{x}{6} + \frac{1}{2} + 4 \cdot \frac{2}{2}$

Combine $4$ and $\frac{2}{2}$ .

$y = - \frac{x}{6} + \frac{1}{2} + \frac{4 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = - \frac{x}{6} + \frac{1 + 4 \cdot 2}{2}$

Simplify the numerator.

Multiply $4$ by $2$ .

$y = - \frac{x}{6} + \frac{1 + 8}{2}$

Add $1$ and $8$ .

$y = - \frac{x}{6} + \frac{9}{2}$

Reorder terms.

$y = - (\frac{1}{6} x) + \frac{9}{2}$

Remove parentheses.

$y = - \frac{1}{6} x + \frac{9}{2}$

Find the Perpendicular Line Passing through (3,4) and perpendicular to the line whose equation is y=6x+7

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