Find the Perpendicular Line through: (1,-4) , perp. to y=1/9x+3

through: $(1, - 4)$ , perp. to $y = \frac{1}{9} x + 3$

Step 1

Find the slope when $y = \frac{1}{9} x + 3$ .

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 $\frac{1}{9}$ and $x$ .

$y = \frac{x}{9} + 3$

Reorder terms.

$y = \frac{1}{9} x + 3$

Using the slope-intercept form, the slope is $\frac{1}{9}$ .

$m = \frac{1}{9}$

Step 2

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

$m_{perpendicular} = - \frac{1}{\frac{1}{9}}$

Step 3

Simplify $- \frac{1}{\frac{1}{9}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 \cdot 9)$

Multiply $- (1 \cdot 9)$ .

Multiply $9$ by $1$ .

$m_{perpendicular} = - 1 \cdot 9$

Multiply $- 1$ by $9$ .

$m_{perpendicular} = - 9$

Step 4

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

Use the slope $- 9$ and a given point $(1, - 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 4) = - 9 \cdot (x - (1))$

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

$y + 4 = - 9 \cdot (x - 1)$

Step 5

Solve for $y$ .

Simplify $- 9 \cdot (x - 1)$ .

Rewrite.

$y + 4 = 0 + 0 - 9 \cdot (x - 1)$

Simplify by adding zeros.

$y + 4 = - 9 \cdot (x - 1)$

Apply the distributive property.

$y + 4 = - 9 x - 9 \cdot - 1$

Multiply $- 9$ by $- 1$ .

$y + 4 = - 9 x + 9$

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

Subtract $4$ from both sides of the equation.

$y = - 9 x + 9 - 4$

Subtract $4$ from $9$ .

$y = - 9 x + 5$

Step 6

Find the Perpendicular Line through: (1,-4) , perp. to y=1/9x+3

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