Find the Perpendicular Line 4x-y=2 that contains the point (4,-2)

$4 x - y = 2$ that contains the point $(4, - 2)$

Step 1

Solve $4 x - y = 2$ .

Subtract $4 x$ from both sides of the equation.

$- y = 2 - 4 x$

Divide each term in $- y = 2 - 4 x$ by $- 1$ and simplify.

Divide each term in $- y = 2 - 4 x$ by $- 1$ .

$\frac{- y}{- 1} = \frac{2}{- 1} + \frac{- 4 x}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{2}{- 1} + \frac{- 4 x}{- 1}$

Divide $y$ by $1$ .

$y = \frac{2}{- 1} + \frac{- 4 x}{- 1}$

Simplify the right side.

Simplify each term.

Divide $2$ by $- 1$ .

$y = - 2 + \frac{- 4 x}{- 1}$

Move the negative one from the denominator of $\frac{- 4 x}{- 1}$ .

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

Rewrite $- 1 \cdot (- 4 x)$ as $- (- 4 x)$ .

$y = - 2 - (- 4 x)$

Multiply $- 4$ by $- 1$ .

$y = - 2 + 4 x$

Step 2

Find the slope when $y = - 2 + 4 x$ .

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 $- 2$ and $4 x$ .

$y = 4 x - 2$

Using the slope-intercept form, the slope is $4$ .

$m = 4$

Step 3

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

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

Step 4

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

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

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

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

$y + 2 = - \frac{1}{4} \cdot (x - 4)$

Step 5

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

Solve for $y$ .

Simplify $- \frac{1}{4} \cdot (x - 4)$ .

Rewrite.

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

Simplify by adding zeros.

$y + 2 = - \frac{1}{4} \cdot (x - 4)$

Apply the distributive property.

$y + 2 = - \frac{1}{4} x - \frac{1}{4} \cdot - 4$

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

$y + 2 = - \frac{x}{4} - \frac{1}{4} \cdot - 4$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$y + 2 = - \frac{x}{4} + \frac{- 1}{4} \cdot - 4$

Factor $4$ out of $- 4$ .

$y + 2 = - \frac{x}{4} + \frac{- 1}{4} \cdot (4 (- 1))$

Cancel the common factor.

$y + 2 = - \frac{x}{4} + \frac{- 1}{4} \cdot (4 \cdot - 1)$

Rewrite the expression.

$y + 2 = - \frac{x}{4} - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$y + 2 = - \frac{x}{4} + 1$

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

Subtract $2$ from both sides of the equation.

$y = - \frac{x}{4} + 1 - 2$

Subtract $2$ from $1$ .

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

Reorder terms.

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

Remove parentheses.

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

Step 6

Find the Perpendicular Line 4x-y=2 that contains the point (4,-2)

Related Questions

Support LCMGCF.com by disabling your adblocker.