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

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

Step 1

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

$y = 3 - 4 x$

Step 2

Find the slope when $y = 3 - 4 x$ .

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 $3$ and $- 4 x$ .

$y = - 4 x + 3$

Using the slope-intercept form, the slope is $- 4$ .

$m = - 4$

Step 3

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

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

Step 4

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

Move the negative in front of the fraction.

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

Multiply $- - \frac{1}{4}$ .

Multiply $- 1$ by $- 1$ .

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

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

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

Step 5

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

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

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

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

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

Step 6

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

Solve for $y$ .

Simplify $\frac{1}{4} \cdot (x - 4)$ .

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Combine $\frac{1}{4}$ and $x$ .

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

Cancel the common factor of $4$ .

Factor $4$ out of $- 4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Subtract $3$ from both sides of the equation.

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

Subtract $3$ from $- 1$ .

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

Reorder terms.

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

Step 7

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

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