Find the Perpendicular Line What is the equation of the line that is perpendicular to the line y=3x+7 that goes through the point (3,4) ?

What is the equation of the line that is perpendicular to the line $y = 3 x + 7$ that goes through the point $(3, 4)$ ?

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 $3$ .

$m = 3$

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

$m_{perpendicular} = - \frac{1}{3}$

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

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

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

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

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

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

Solve for $y$ .

Simplify $- \frac{1}{3} \cdot (x - 3)$ .

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

$y - 4 = - \frac{1}{3} x - \frac{1}{3} \cdot - 3$

Combine $x$ and $\frac{1}{3}$ .

$y - 4 = - \frac{x}{3} - \frac{1}{3} \cdot - 3$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$y - 4 = - \frac{x}{3} + \frac{- 1}{3} \cdot - 3$

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

$y - 4 = - \frac{x}{3} - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$y - 4 = - \frac{x}{3} + 1$

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

Add $4$ to both sides of the equation.

$y = - \frac{x}{3} + 1 + 4$

Add $1$ and $4$ .

$y = - \frac{x}{3} + 5$

Reorder terms.

$y = - (\frac{1}{3} x) + 5$

Remove parentheses.

$y = - \frac{1}{3} x + 5$

Find the Perpendicular Line What is the equation of the line that is perpendicular to the line y=3x+7 that goes through the point (3,4) ?

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