Find the Perpendicular Line through: (4,1) , perp. to y=x+4

through: $(4, 1)$ , perp. to $y = x + 4$

Step 1

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 2

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 3

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 4

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

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

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

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

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

Step 5

Solve for $y$ .

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

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Simplify the expression.

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

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

Multiply $- 1$ by $- 4$ .

$y - 1 = - x + 4$

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

Add $1$ to both sides of the equation.

$y = - x + 4 + 1$

Add $4$ and $1$ .

$y = - x + 5$

Step 6

Find the Perpendicular Line through: (4,1) , perp. to y=x+4

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