Find the Perpendicular Line What is an equation of the line that passes through the point (4,2) and is perpendicular to the line 4x+3y=21 ?

What is an equation of the line that passes through the point $(4, 2)$ and is perpendicular to the line $4 x + 3 y = 21$ ?

Step 1

Solve $4 x + 3 y = 21$ .

Subtract $4 x$ from both sides of the equation.

$3 y = 21 - 4 x$

Divide each term in $3 y = 21 - 4 x$ by $3$ and simplify.

Divide each term in $3 y = 21 - 4 x$ by $3$ .

$\frac{3 y}{3} = \frac{21}{3} + \frac{- 4 x}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} = \frac{21}{3} + \frac{- 4 x}{3}$

Divide $y$ by $1$ .

$y = \frac{21}{3} + \frac{- 4 x}{3}$

Simplify the right side.

Simplify each term.

Divide $21$ by $3$ .

$y = 7 + \frac{- 4 x}{3}$

Move the negative in front of the fraction.

$y = 7 - \frac{4 x}{3}$

Step 2

Find the slope when $y = 7 - \frac{4 x}{3}$ .

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.

$y = m x + b$

Reorder $7$ and $- \frac{4 x}{3}$ .

$y = - \frac{4 x}{3} + 7$

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

Reorder terms.

$y = - (\frac{4}{3} x) + 7$

Remove parentheses.

$y = - \frac{4}{3} x + 7$

Using the slope-intercept form, the slope is $- \frac{4}{3}$ .

$m = - \frac{4}{3}$

Step 3

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

$m_{perpendicular} = - \frac{1}{- \frac{4}{3}}$

Step 4

Simplify $- \frac{1}{- \frac{4}{3}}$ to find the slope of the perpendicular line.

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{4}{3}}$

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{4}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{3}{4})$

Multiply $\frac{3}{4}$ by $1$ .

$m_{perpendicular} = \frac{3}{4}$

Multiply $- - \frac{3}{4}$ .

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{3}{4})$

Multiply $\frac{3}{4}$ by $1$ .

$m_{perpendicular} = \frac{3}{4}$

Step 5

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

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

$y - (2) = \frac{3}{4} \cdot (x - (4))$

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

$y - 2 = \frac{3}{4} \cdot (x - 4)$

Step 6

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

Solve for $y$ .

Simplify $\frac{3}{4} \cdot (x - 4)$ .

Rewrite.

$y - 2 = 0 + 0 + \frac{3}{4} \cdot (x - 4)$

Simplify by adding zeros.

$y - 2 = \frac{3}{4} \cdot (x - 4)$

Apply the distributive property.

$y - 2 = \frac{3}{4} x + \frac{3}{4} \cdot - 4$

Combine $\frac{3}{4}$ and $x$ .

$y - 2 = \frac{3 x}{4} + \frac{3}{4} \cdot - 4$

Cancel the common factor of $4$ .

Factor $4$ out of $- 4$ .

$y - 2 = \frac{3 x}{4} + \frac{3}{4} \cdot (4 (- 1))$

Cancel the common factor.

$y - 2 = \frac{3 x}{4} + \frac{3}{4} \cdot (4 \cdot - 1)$

Rewrite the expression.

$y - 2 = \frac{3 x}{4} + 3 \cdot - 1$

Multiply $3$ by $- 1$ .

$y - 2 = \frac{3 x}{4} - 3$

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

Add $2$ to both sides of the equation.

$y = \frac{3 x}{4} - 3 + 2$

Add $- 3$ and $2$ .

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

Reorder terms.

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

Step 7

Find the Perpendicular Line What is an equation of the line that passes through the point (4,2) and is perpendicular to the line 4x+3y=21 ?

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