Find the Perpendicular Line 5x=-1-y , (4.5,-2.6)

$5 x = - 1 - y$ , $(4.5, - 2.6)$

Step 1

Solve $5 x = - 1 - y$ .

Rewrite the equation as $- 1 - y = 5 x$ .

$- 1 - y = 5 x$

Add $1$ to both sides of the equation.

$- y = 5 x + 1$

Divide each term in $- y = 5 x + 1$ by $- 1$ and simplify.

Divide each term in $- y = 5 x + 1$ by $- 1$ .

$\frac{- y}{- 1} = \frac{5 x}{- 1} + \frac{1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{5 x}{- 1} + \frac{1}{- 1}$

Divide $y$ by $1$ .

$y = \frac{5 x}{- 1} + \frac{1}{- 1}$

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{5 x}{- 1}$ .

$y = - 1 \cdot (5 x) + \frac{1}{- 1}$

Rewrite $- 1 \cdot (5 x)$ as $- (5 x)$ .

$y = - (5 x) + \frac{1}{- 1}$

Multiply $5$ by $- 1$ .

$y = - 5 x + \frac{1}{- 1}$

Divide $1$ by $- 1$ .

$y = - 5 x - 1$

Step 2

Use the slope-intercept form to find the slope.

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

$y = m x + b$

Using the slope-intercept form, the slope is $- 5$ .

$m = - 5$

Step 3

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

$m_{perpendicular} = - \frac{1}{- 5}$

Step 4

Simplify $- \frac{1}{- 5}$ to find the slope of the perpendicular line.

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{5}$

Multiply $- - \frac{1}{5}$ .

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{1}{5})$

Multiply $\frac{1}{5}$ by $1$ .

$m_{perpendicular} = \frac{1}{5}$

Step 5

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

Use the slope $\frac{1}{5}$ and a given point $(4.5, - 2.6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 2.6) = \frac{1}{5} \cdot (x - (4.5))$

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

$y + 2.6 = \frac{1}{5} \cdot (x - 4.5)$

Step 6

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

Solve for $y$ .

Simplify $\frac{1}{5} \cdot (x - 4.5)$ .

Rewrite.

$y + 2.6 = 0 + 0 + \frac{1}{5} \cdot (x - 4.5)$

Simplify by adding zeros.

$y + 2.6 = \frac{1}{5} \cdot (x - 4.5)$

Apply the distributive property.

$y + 2.6 = \frac{1}{5} x + \frac{1}{5} \cdot - 4.5$

Combine $\frac{1}{5}$ and $x$ .

$y + 2.6 = \frac{x}{5} + \frac{1}{5} \cdot - 4.5$

Combine $\frac{1}{5}$ and $- 4.5$ .

$y + 2.6 = \frac{x}{5} + \frac{- 4.5}{5}$

Divide $- 4.5$ by $5$ .

$y + 2.6 = \frac{x}{5} - 0.9$

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

Subtract $2.6$ from both sides of the equation.

$y = \frac{x}{5} - 0.9 - 2.6$

Subtract $2.6$ from $- 0.9$ .

$y = \frac{x}{5} - 3.5$

Reorder terms.

$y = \frac{1}{5} x - 3.5$

Step 7

Find the Perpendicular Line 5x=-1-y , (4.5,-2.6)

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