Algebra Examples

A line is perpendicular to

$y = 3 x - 8$ and intersects the point

$(6, 1)$

Step 1

Write the problem as a mathematical expression.

$y = 3 x - 8$ ,

$(6, 1)$

Step 2

Use the slope-intercept form to find the slope.

Tap for more steps...

Step 2.1

The slope-intercept form is

$y = m x + b$ , where

$m$ is the slope and

$b$ is the y-intercept.

$y = m x + b$

Step 2.2

Using the slope-intercept form, the slope is

$3$ .

$m = 3$

Step 3

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

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

Step 4

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

Tap for more steps...

Step 4.1

Use the slope

$- \frac{1}{3}$ and a given point

$(6, 1)$ 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

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = - \frac{1}{3} \cdot (x - (6))$

Step 4.2

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

$y - 1 = - \frac{1}{3} \cdot (x - 6)$

Step 5

Write in

$y = m x + b$ form.

Tap for more steps...

Step 5.1

Solve for

$y$ .

Tap for more steps...

Step 5.1.1

Simplify

$- \frac{1}{3} \cdot (x - 6)$ .

Tap for more steps...

Step 5.1.1.1

Rewrite.

$y - 1 = 0 + 0 - \frac{1}{3} \cdot (x - 6)$

Step 5.1.1.2

Simplify by adding zeros.

$y - 1 = - \frac{1}{3} \cdot (x - 6)$

Step 5.1.1.3

Apply the distributive property.

$y - 1 = - \frac{1}{3} x - \frac{1}{3} \cdot - 6$

Step 5.1.1.4

Combine

$x$ and

$\frac{1}{3}$ .

$y - 1 = - \frac{x}{3} - \frac{1}{3} \cdot - 6$

Step 5.1.1.5

Cancel the common factor of

$3$ .

Tap for more steps...

Step 5.1.1.5.1

Move the leading negative in

$- \frac{1}{3}$ into the numerator.

$y - 1 = - \frac{x}{3} + \frac{- 1}{3} \cdot - 6$

Step 5.1.1.5.2

Factor

$3$ out of

$- 6$ .

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

Step 5.1.1.5.3

Cancel the common factor.

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

Step 5.1.1.5.4

Rewrite the expression.

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

Step 5.1.1.6

Multiply

$- 1$ by

$- 2$ .

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

Step 5.1.2

Move all terms not containing

$y$ to the right side of the equation.

Tap for more steps...

Step 5.1.2.1

Add

$1$ to both sides of the equation.

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

Step 5.1.2.2

Add

$2$ and

$1$ .

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

Step 5.2

Reorder terms.

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

Step 5.3

Remove parentheses.

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

Step 6