Find the Perpendicular Line passing through (5,-7) and perpendicular to the line whose equation is y=1/2x+5

passing through $(5, - 7)$ and perpendicular to the line whose equation is $y = \frac{1}{2} x + 5$

Step 1

Find the slope when $y = \frac{1}{2} x + 5$ .

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.

$y = m x + b$

Simplify the right side.

Combine $\frac{1}{2}$ and $x$ .

$y = \frac{x}{2} + 5$

Reorder terms.

$y = \frac{1}{2} x + 5$

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

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

Step 2

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

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

Step 3

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 4

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

Use the slope $- 2$ and a given point $(5, - 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 7) = - 2 \cdot (x - (5))$

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

$y + 7 = - 2 \cdot (x - 5)$

Step 5

Solve for $y$ .

Simplify $- 2 \cdot (x - 5)$ .

Rewrite.

$y + 7 = 0 + 0 - 2 \cdot (x - 5)$

Simplify by adding zeros.

$y + 7 = - 2 \cdot (x - 5)$

Apply the distributive property.

$y + 7 = - 2 x - 2 \cdot - 5$

Multiply $- 2$ by $- 5$ .

$y + 7 = - 2 x + 10$

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

Subtract $7$ from both sides of the equation.

$y = - 2 x + 10 - 7$

Subtract $7$ from $10$ .

$y = - 2 x + 3$

Step 6

Find the Perpendicular Line passing through (5,-7) and perpendicular to the line whose equation is y=1/2x+5

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