Find the Perpendicular Line f(x)=2x+3 , (7,3)

$f (x) = 2 x + 3$ , $(7, 3)$

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 $2$ .

$m = 2$

Step 2

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

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

Step 3

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

Use the slope $- \frac{1}{2}$ and a given point $(7, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = - \frac{1}{2} \cdot (x - (7))$

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

$y - 3 = - \frac{1}{2} \cdot (x - 7)$

Step 4

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

Solve for $y$ .

Simplify $- \frac{1}{2} \cdot (x - 7)$ .

Rewrite.

$y - 3 = 0 + 0 - \frac{1}{2} \cdot (x - 7)$

Simplify by adding zeros.

$y - 3 = - \frac{1}{2} \cdot (x - 7)$

Apply the distributive property.

$y - 3 = - \frac{1}{2} x - \frac{1}{2} \cdot - 7$

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

$y - 3 = - \frac{x}{2} - \frac{1}{2} \cdot - 7$

Multiply $- \frac{1}{2} \cdot - 7$ .

Multiply $- 7$ by $- 1$ .

$y - 3 = - \frac{x}{2} + 7 (\frac{1}{2})$

Combine $7$ and $\frac{1}{2}$ .

$y - 3 = - \frac{x}{2} + \frac{7}{2}$

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

Add $3$ to both sides of the equation.

$y = - \frac{x}{2} + \frac{7}{2} + 3$

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{x}{2} + \frac{7}{2} + 3 \cdot \frac{2}{2}$

Combine $3$ and $\frac{2}{2}$ .

$y = - \frac{x}{2} + \frac{7}{2} + \frac{3 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = - \frac{x}{2} + \frac{7 + 3 \cdot 2}{2}$

Simplify the numerator.

Multiply $3$ by $2$ .

$y = - \frac{x}{2} + \frac{7 + 6}{2}$

Add $7$ and $6$ .

$y = - \frac{x}{2} + \frac{13}{2}$

Reorder terms.

$y = - (\frac{1}{2} x) + \frac{13}{2}$

Remove parentheses.

$y = - \frac{1}{2} x + \frac{13}{2}$

Step 5

Find the Perpendicular Line f(x)=2x+3 , (7,3)

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