Find the Perpendicular Line y=-10x+1 (5,7)

$y = - 10 x + 1$ $(5, 7)$

Step 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 $- 10$ .

$m = - 10$

Step 2

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

$m_{perpendicular} = - \frac{1}{- 10}$

Step 3

Simplify $- \frac{1}{- 10}$ to find the slope of the perpendicular line.

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{10}$

Multiply $- - \frac{1}{10}$ .

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{1}{10})$

Multiply $\frac{1}{10}$ by $1$ .

$m_{perpendicular} = \frac{1}{10}$

Step 4

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

Use the slope $\frac{1}{10}$ and a given point $(5, 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (7) = \frac{1}{10} \cdot (x - (5))$

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

$y - 7 = \frac{1}{10} \cdot (x - 5)$

Step 5

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

Solve for $y$ .

Simplify $\frac{1}{10} \cdot (x - 5)$ .

Rewrite.

$y - 7 = 0 + 0 + \frac{1}{10} \cdot (x - 5)$

Simplify by adding zeros.

$y - 7 = \frac{1}{10} \cdot (x - 5)$

Apply the distributive property.

$y - 7 = \frac{1}{10} x + \frac{1}{10} \cdot - 5$

Combine $\frac{1}{10}$ and $x$ .

$y - 7 = \frac{x}{10} + \frac{1}{10} \cdot - 5$

Cancel the common factor of $5$ .

Factor $5$ out of $10$ .

$y - 7 = \frac{x}{10} + \frac{1}{5 (2)} \cdot - 5$

Factor $5$ out of $- 5$ .

$y - 7 = \frac{x}{10} + \frac{1}{5 \cdot 2} \cdot (5 \cdot - 1)$

Cancel the common factor.

$y - 7 = \frac{x}{10} + \frac{1}{5 \cdot 2} \cdot (5 \cdot - 1)$

Rewrite the expression.

$y - 7 = \frac{x}{10} + \frac{1}{2} \cdot - 1$

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

$y - 7 = \frac{x}{10} + \frac{- 1}{2}$

Move the negative in front of the fraction.

$y - 7 = \frac{x}{10} - \frac{1}{2}$

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

Add $7$ to both sides of the equation.

$y = \frac{x}{10} - \frac{1}{2} + 7$

To write $7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{x}{10} - \frac{1}{2} + 7 \cdot \frac{2}{2}$

Combine $7$ and $\frac{2}{2}$ .

$y = \frac{x}{10} - \frac{1}{2} + \frac{7 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{x}{10} + \frac{- 1 + 7 \cdot 2}{2}$

Simplify the numerator.

Multiply $7$ by $2$ .

$y = \frac{x}{10} + \frac{- 1 + 14}{2}$

Add $- 1$ and $14$ .

$y = \frac{x}{10} + \frac{13}{2}$

Reorder terms.

$y = \frac{1}{10} x + \frac{13}{2}$

Step 6

Find the Perpendicular Line y=-10x+1 (5,7)

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