Find the Perpendicular Line What is an equation of the line that passes through the point (-1,-3) and is perpendicular to the line x-2y=14

What is an equation of the line that passes through the point $(- 1, - 3)$ and is perpendicular to the line $x - 2 y = 14$

Step 1

Write the problem as a mathematical expression.

$(- 1, - 3)$ , $x - 2 y = 14$

Step 2

Solve $x - 2 y = 14$ .

Subtract $x$ from both sides of the equation.

$- 2 y = 14 - x$

Divide each term in $- 2 y = 14 - x$ by $- 2$ and simplify.

Divide each term in $- 2 y = 14 - x$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{14}{- 2} + \frac{- x}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{14}{- 2} + \frac{- x}{- 2}$

Divide $y$ by $1$ .

$y = \frac{14}{- 2} + \frac{- x}{- 2}$

Simplify the right side.

Simplify each term.

Divide $14$ by $- 2$ .

$y = - 7 + \frac{- x}{- 2}$

Dividing two negative values results in a positive value.

$y = - 7 + \frac{x}{2}$

Step 3

Find the slope when $y = - 7 + \frac{x}{2}$ .

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 $- 7$ and $\frac{x}{2}$ .

$y = \frac{x}{2} - 7$

Reorder terms.

$y = \frac{1}{2} x - 7$

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

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

Step 4

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

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

Step 5

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 $- (1 \cdot 2)$ .

Multiply $2$ by $1$ .

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

Multiply $- 1$ by $2$ .

$m_{perpendicular} = - 2$

Step 6

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

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

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

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

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

Step 7

Solve for $y$ .

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

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Multiply $- 2$ by $1$ .

$y + 3 = - 2 x - 2$

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

Subtract $3$ from both sides of the equation.

$y = - 2 x - 2 - 3$

Subtract $3$ from $- 2$ .

$y = - 2 x - 5$

Step 8

Find the Perpendicular Line What is an equation of the line that passes through the point (-1,-3) and is perpendicular to the line x-2y=14

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