Find the Perpendicular Line (1,8) and perpendicular to 2x+5y=1

$(1, 8)$ and perpendicular to $2 x + 5 y = 1$

Solve $2 x + 5 y = 1$ .

Subtract $2 x$ from both sides of the equation.

$5 y = 1 - 2 x$

Divide each term in $5 y = 1 - 2 x$ by $5$ and simplify.

Divide each term in $5 y = 1 - 2 x$ by $5$ .

$\frac{5 y}{5} = \frac{1}{5} + \frac{- 2 x}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 y}{5} = \frac{1}{5} + \frac{- 2 x}{5}$

Divide $y$ by $1$ .

$y = \frac{1}{5} + \frac{- 2 x}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{1}{5} - \frac{2 x}{5}$

Find the slope when $y = \frac{1}{5} - \frac{2 x}{5}$ .

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.

$y = m x + b$

Reorder $\frac{1}{5}$ and $- \frac{2 x}{5}$ .

$y = - \frac{2 x}{5} + \frac{1}{5}$

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

Reorder terms.

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

Remove parentheses.

$y = - \frac{2}{5} x + \frac{1}{5}$

Using the slope-intercept form, the slope is $- \frac{2}{5}$ .

$m = - \frac{2}{5}$

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

$m_{perpendicular} = - \frac{1}{- \frac{2}{5}}$

Simplify $- \frac{1}{- \frac{2}{5}}$ to find the slope of the perpendicular line.

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{2}{5}}$

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{2}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{5}{2})$

Multiply $\frac{5}{2}$ by $1$ .

$m_{perpendicular} = \frac{5}{2}$

Multiply $- - \frac{5}{2}$ .

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{5}{2})$

Multiply $\frac{5}{2}$ by $1$ .

$m_{perpendicular} = \frac{5}{2}$

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

Use the slope $\frac{5}{2}$ and a given point $(1, 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (8) = \frac{5}{2} \cdot (x - (1))$

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

$y - 8 = \frac{5}{2} \cdot (x - 1)$

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

Solve for $y$ .

Simplify $\frac{5}{2} \cdot (x - 1)$ .

Rewrite.

$y - 8 = 0 + 0 + \frac{5}{2} \cdot (x - 1)$

Simplify by adding zeros.

$y - 8 = \frac{5}{2} \cdot (x - 1)$

Apply the distributive property.

$y - 8 = \frac{5}{2} x + \frac{5}{2} \cdot - 1$

Combine $\frac{5}{2}$ and $x$ .

$y - 8 = \frac{5 x}{2} + \frac{5}{2} \cdot - 1$

Multiply $\frac{5}{2} \cdot - 1$ .

Combine $\frac{5}{2}$ and $- 1$ .

$y - 8 = \frac{5 x}{2} + \frac{5 \cdot - 1}{2}$

Multiply $5$ by $- 1$ .

$y - 8 = \frac{5 x}{2} + \frac{- 5}{2}$

Move the negative in front of the fraction.

$y - 8 = \frac{5 x}{2} - \frac{5}{2}$

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

Add $8$ to both sides of the equation.

$y = \frac{5 x}{2} - \frac{5}{2} + 8$

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{5 x}{2} - \frac{5}{2} + 8 \cdot \frac{2}{2}$

Combine $8$ and $\frac{2}{2}$ .

$y = \frac{5 x}{2} - \frac{5}{2} + \frac{8 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{5 x}{2} + \frac{- 5 + 8 \cdot 2}{2}$

Simplify the numerator.

Multiply $8$ by $2$ .

$y = \frac{5 x}{2} + \frac{- 5 + 16}{2}$

Add $- 5$ and $16$ .

$y = \frac{5 x}{2} + \frac{11}{2}$

Reorder terms.

$y = \frac{5}{2} x + \frac{11}{2}$

Find the Perpendicular Line (1,8) and perpendicular to 2x+5y=1

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