Find the Perpendicular Line y-4=x (4,-3)

$y - 4 = x$ $(4, - 3)$

Step 1

Add $4$ to both sides of the equation.

$y = x + 4$

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 $1$ .

$m = 1$

Step 3

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

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

Step 4

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

Cancel the common factor of $1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $1$ .

$m_{perpendicular} = - 1$

Step 5

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

Use the slope $- 1$ and a given point $(4, - 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 3) = - 1 \cdot (x - (4))$

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

$y + 3 = - 1 \cdot (x - 4)$

Step 6

Solve for $y$ .

Simplify $- 1 \cdot (x - 4)$ .

Rewrite.

$y + 3 = 0 + 0 - 1 \cdot (x - 4)$

Simplify by adding zeros.

$y + 3 = - 1 \cdot (x - 4)$

Apply the distributive property.

$y + 3 = - 1 x - 1 \cdot - 4$

Simplify the expression.

Rewrite $- 1 x$ as $- x$ .

$y + 3 = - x - 1 \cdot - 4$

Multiply $- 1$ by $- 4$ .

$y + 3 = - x + 4$

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

Subtract $3$ from both sides of the equation.

$y = - x + 4 - 3$

Subtract $3$ from $4$ .

$y = - x + 1$

Step 7

Find the Perpendicular Line y-4=x (4,-3)

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