Find the Perpendicular Line What is an equation of the line that passes through the point (-3,-1) and is perpendicular to the line x-2y=6 ?

What is an equation of the line that passes through the point $(- 3, - 1)$ and is perpendicular to the line $x - 2 y = 6$ ?

Write the problem as a mathematical expression.

$(- 3, - 1)$ , $x - 2 y = 6$

Solve $x - 2 y = 6$ .

Subtract $x$ from both sides of the equation.

$- 2 y = 6 - x$

Divide each term in $- 2 y = 6 - x$ by $- 2$ and simplify.

Divide each term in $- 2 y = 6 - x$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{6}{- 2} + \frac{- x}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{6}{- 2} + \frac{- x}{- 2}$

Divide $y$ by $1$ .

$y = \frac{6}{- 2} + \frac{- x}{- 2}$

Simplify the right side.

Simplify each term.

Divide $6$ by $- 2$ .

$y = - 3 + \frac{- x}{- 2}$

Dividing two negative values results in a positive value.

$y = - 3 + \frac{x}{2}$

Find the slope when $y = - 3 + \frac{x}{2}$ .

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 $- 3$ and $\frac{x}{2}$ .

$y = \frac{x}{2} - 3$

Reorder terms.

$y = \frac{1}{2} x - 3$

Using the slope-intercept form, the slope is $\frac{1}{2}$ .

$m = \frac{1}{2}$

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

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

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

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 \cdot 2)$

Multiply $2$ by $1$ .

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

Multiply $- 1$ by $2$ .

$m_{perpendicular} = - 2$

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

Use the slope $- 2$ and a given point $(- 3, - 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 1) = - 2 \cdot (x - (- 3))$

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

$y + 1 = - 2 \cdot (x + 3)$

Solve for $y$ .

Simplify $- 2 \cdot (x + 3)$ .

Rewrite.

$y + 1 = 0 + 0 - 2 \cdot (x + 3)$

Simplify by adding zeros.

$y + 1 = - 2 \cdot (x + 3)$

Apply the distributive property.

$y + 1 = - 2 x - 2 \cdot 3$

Multiply $- 2$ by $3$ .

$y + 1 = - 2 x - 6$

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

Subtract $1$ from both sides of the equation.

$y = - 2 x - 6 - 1$

Subtract $1$ from $- 6$ .

$y = - 2 x - 7$

Find the Perpendicular Line What is an equation of the line that passes through the point (-3,-1) and is perpendicular to the line x-2y=6 ?

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