Find Any Equation Perpendicular to the Line y-5=6(x-7)

$y - 5 = 6 (x - 7)$

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

Solve $y - 5 = 6 (x - 7)$ .

Simplify $6 (x - 7)$ .

Rewrite.

$y - 5 = 0 + 0 + 6 (x - 7)$

Simplify by adding zeros.

$y - 5 = 6 (x - 7)$

Apply the distributive property.

$y - 5 = 6 x + 6 \cdot - 7$

Multiply $6$ by $- 7$ .

$y - 5 = 6 x - 42$

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

Add $5$ to both sides of the equation.

$y = 6 x - 42 + 5$

Add $- 42$ and $5$ .

$y = 6 x - 37$

Step 3

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 $6$ .

$m = 6$

Step 4

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

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

Step 5

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

Use the slope $- \frac{1}{6}$ and a given point $(0, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

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

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

Step 6

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

Solve for $y$ .

Add $y$ and $0$ .

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

Simplify $- \frac{1}{6} \cdot (x + 0)$ .

Add $x$ and $0$ .

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

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

$y = - \frac{x}{6}$

Reorder terms.

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

Remove parentheses.

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

Step 7

Find Any Equation Perpendicular to the Line y-5=6(x-7)

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