Find the Perpendicular Line What is an equation of the line that passes through the point (1,7) and is perpendicular to the line x+6y=42 ?

What is an equation of the line that passes through the point $(1, 7)$ and is perpendicular to the line $x + 6 y = 42$ ?

Step 1

Solve $x + 6 y = 42$ .

Subtract $x$ from both sides of the equation.

$6 y = 42 - x$

Divide each term in $6 y = 42 - x$ by $6$ and simplify.

Divide each term in $6 y = 42 - x$ by $6$ .

$\frac{6 y}{6} = \frac{42}{6} + \frac{- x}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} = \frac{42}{6} + \frac{- x}{6}$

Divide $y$ by $1$ .

$y = \frac{42}{6} + \frac{- x}{6}$

Simplify the right side.

Simplify each term.

Divide $42$ by $6$ .

$y = 7 + \frac{- x}{6}$

Move the negative in front of the fraction.

$y = 7 - \frac{x}{6}$

Step 2

Find the slope when $y = 7 - \frac{x}{6}$ .

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 $7$ and $- \frac{x}{6}$ .

$y = - \frac{x}{6} + 7$

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

Reorder terms.

$y = - (\frac{1}{6} x) + 7$

Remove parentheses.

$y = - \frac{1}{6} x + 7$

Using the slope-intercept form, the slope is $- \frac{1}{6}$ .

$m = - \frac{1}{6}$

Step 3

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

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

Step 4

Simplify $- \frac{1}{- \frac{1}{6}}$ to find the slope of the perpendicular line.

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

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

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 \cdot 6$

Multiply $- - (1 \cdot 6)$ .

Multiply $6$ by $1$ .

$m_{perpendicular} = - (- 1 \cdot 6)$

Multiply $- 1$ by $6$ .

$m_{perpendicular} = 6$

Multiply $- 1$ by $- 6$ .

$m_{perpendicular} = 6$

Step 5

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

Use the slope $6$ and a given point $(1, 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (7) = 6 \cdot (x - (1))$

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

$y - 7 = 6 \cdot (x - 1)$

Step 6

Solve for $y$ .

Simplify $6 \cdot (x - 1)$ .

Rewrite.

$y - 7 = 0 + 0 + 6 \cdot (x - 1)$

Simplify by adding zeros.

$y - 7 = 6 \cdot (x - 1)$

Apply the distributive property.

$y - 7 = 6 x + 6 \cdot - 1$

Multiply $6$ by $- 1$ .

$y - 7 = 6 x - 6$

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

Add $7$ to both sides of the equation.

$y = 6 x - 6 + 7$

Add $- 6$ and $7$ .

$y = 6 x + 1$

Step 7

Find the Perpendicular Line What is an equation of the line that passes through the point (1,7) and is perpendicular to the line x+6y=42 ?

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