Find Any Equation Perpendicular to the Line y-3=5(x-9)

$y - 3 = 5 (x - 9)$

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

Solve $y - 3 = 5 (x - 9)$ .

Simplify $5 (x - 9)$ .

Rewrite.

$y - 3 = 0 + 0 + 5 (x - 9)$

Simplify by adding zeros.

$y - 3 = 5 (x - 9)$

Apply the distributive property.

$y - 3 = 5 x + 5 \cdot - 9$

Multiply $5$ by $- 9$ .

$y - 3 = 5 x - 45$

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

Add $3$ to both sides of the equation.

$y = 5 x - 45 + 3$

Add $- 45$ and $3$ .

$y = 5 x - 42$

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 $5$ .

$m = 5$

Step 4

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

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

Step 5

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

Use the slope $- \frac{1}{5}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

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

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

Step 6

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

Solve for $y$ .

Add $y$ and $0$ .

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

Simplify $- \frac{1}{5} \cdot (x + 0)$ .

Add $x$ and $0$ .

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

Combine $x$ and $\frac{1}{5}$ .

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

Reorder terms.

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

Remove parentheses.

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

Step 7

Find Any Equation Perpendicular to the Line y-3=5(x-9)

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