Precalculus Examples

$(- 2, 2)$ , $2 x - y - 4 = 0$

Step 1

Solve $2 x - y - 4 = 0$ .

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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

$- y - 4 = - 2 x$

Step 1.1.2

Add $4$ to both sides of the equation.

$- y = - 2 x + 4$

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.2.2

Divide $y$ by $1$ .

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

Step 1.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.1

Simplify each term.

Tap for more steps...

Step 1.2.3.1.1

Move the negative one from the denominator of $\frac{- 2 x}{- 1}$ .

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

Step 1.2.3.1.2

Rewrite $- 1 \cdot (- 2 x)$ as $- (- 2 x)$ .

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

Step 1.2.3.1.3

Multiply $- 2$ by $- 1$ .

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

Step 1.2.3.1.4

Divide $4$ by $- 1$ .

$y = 2 x - 4$

Step 2

Use the slope-intercept form to find the slope.

Tap for more steps...

Step 2.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 2.2

Using the slope-intercept form, the slope is $2$ .

$m = 2$

Step 3

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

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

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 5

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

Tap for more steps...

Step 5.1

Solve for $y$ .

Tap for more steps...

Step 5.1.1

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

Tap for more steps...

Step 5.1.1.1

Rewrite.

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

Step 5.1.1.2

Simplify by adding zeros.

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

Step 5.1.1.3

Apply the distributive property.

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

Step 5.1.1.4

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

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

Step 5.1.1.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.1.1.5.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 5.1.1.5.2

Cancel the common factor.

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

Step 5.1.1.5.3

Rewrite the expression.

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

Step 5.1.2

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

Tap for more steps...

Step 5.1.2.1

Add $2$ to both sides of the equation.

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

Step 5.1.2.2

Add $- 1$ and $2$ .

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

Step 5.2

Reorder terms.

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

Step 5.3

Remove parentheses.

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

Step 6