Find the Perpendicular Line Through (1,2) ; perpendicular to x-4y=4

Through $(1, 2)$ ; perpendicular to $x - 4 y = 4$

Solve $x - 4 y = 4$ .

Subtract $x$ from both sides of the equation.

$- 4 y = 4 - x$

Divide each term in $- 4 y = 4 - x$ by $- 4$ and simplify.

Divide each term in $- 4 y = 4 - x$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{4}{- 4} + \frac{- x}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{4}{- 4} + \frac{- x}{- 4}$

Divide $y$ by $1$ .

$y = \frac{4}{- 4} + \frac{- x}{- 4}$

Simplify the right side.

Simplify each term.

Divide $4$ by $- 4$ .

$y = - 1 + \frac{- x}{- 4}$

Dividing two negative values results in a positive value.

$y = - 1 + \frac{x}{4}$

Find the slope when $y = - 1 + \frac{x}{4}$ .

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 $- 1$ and $\frac{x}{4}$ .

$y = \frac{x}{4} - 1$

Reorder terms.

$y = \frac{1}{4} x - 1$

Using the slope-intercept form, the slope is $\frac{1}{4}$ .

$m = \frac{1}{4}$

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

$m_{perpendicular} = - \frac{1}{\frac{1}{4}}$

Simplify $- \frac{1}{\frac{1}{4}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 \cdot 4)$

Multiply $4$ by $1$ .

$m_{perpendicular} = - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$m_{perpendicular} = - 4$

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

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

$y - (2) = - 4 \cdot (x - (1))$

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

$y - 2 = - 4 \cdot (x - 1)$

Solve for $y$ .

Simplify $- 4 \cdot (x - 1)$ .

Rewrite.

$y - 2 = 0 + 0 - 4 \cdot (x - 1)$

Simplify by adding zeros.

$y - 2 = - 4 \cdot (x - 1)$

Apply the distributive property.

$y - 2 = - 4 x - 4 \cdot - 1$

Multiply $- 4$ by $- 1$ .

$y - 2 = - 4 x + 4$

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

Add $2$ to both sides of the equation.

$y = - 4 x + 4 + 2$

Add $4$ and $2$ .

$y = - 4 x + 6$

Find the Perpendicular Line Through (1,2) ; perpendicular to x-4y=4

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