Find the Perpendicular Line through: (5,3) , perp. to y=-x+3

through: $(5, 3)$ , perp. to $y = - x + 3$

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.

Divide $1$ by $- 1$ .

$m_{perpendicular} = 1$

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 $(5, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = 1 \cdot (x - (5))$

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

$y - 3 = 1 \cdot (x - 5)$

Step 5

Solve for $y$ .

Multiply $x - 5$ by $1$ .

$y - 3 = x - 5$

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

Add $3$ to both sides of the equation.

$y = x - 5 + 3$

Add $- 5$ and $3$ .

$y = x - 2$

Step 6

Find the Perpendicular Line through: (5,3) , perp. to y=-x+3

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