Algebra Examples

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

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

(9, 12)

$(9, 12)$

Step 1

Find the slope when

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

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

Tap for more steps...

Step 1.1

Rewrite in slope-intercept form.

Tap for more steps...

Step 1.1.1

The slope-intercept form is

y = m x + b

$y = m x + b$ , where

m

$m$ is the slope and

b

$b$ is the y-intercept.

y = m x + b

$y = m x + b$

Step 1.1.2

Simplify the right side.

Tap for more steps...

Step 1.1.2.1

Simplify each term.

Tap for more steps...

Step 1.1.2.1.1

Combine

x

$x$ and

\frac{3}{4}

$\frac{3}{4}$ .

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

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

Step 1.1.2.1.2

Move

3

$3$ to the left of

x

$x$ .

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

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

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

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

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

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

Step 1.1.3

Write in

y = m x + b

$y = m x + b$ form.

Tap for more steps...

Step 1.1.3.1

Reorder terms.

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

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

Step 1.1.3.2

Remove parentheses.

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

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

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

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

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

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

Step 1.2

Using the slope-intercept form, the slope is

- \frac{3}{4}

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

m = - \frac{3}{4}

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

m = - \frac{3}{4}

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

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}{- \frac{3}{4}}

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

Step 3

Simplify

- \frac{1}{- \frac{3}{4}}

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

Tap for more steps...

Step 3.1

Cancel the common factor of

1

$1$ and

- 1

$- 1$ .

Tap for more steps...

Step 3.1.1

Rewrite

1

$1$ as

- 1 (- 1)

$- 1 (- 1)$ .

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

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

Step 3.1.2

Move the negative in front of the fraction.

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

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

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

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

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

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

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

Step 3.3

Multiply

\frac{4}{3}

$\frac{4}{3}$ by

1

$1$ .

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

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

Step 3.4

Multiply

- - \frac{4}{3}

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

Tap for more steps...

Step 3.4.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 3.4.2

Multiply

\frac{4}{3}

$\frac{4}{3}$ by

1

$1$ .

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

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

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

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

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

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

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{4}{3}

$\frac{4}{3}$ and a given point

(9, 12)

$(9, 12)$ to substitute for

x_{1}

$x_{1}$ and

y_{1}

$y_{1}$ in the point-slope form

y - y_{1} = m (x - x_{1})

$y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation

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

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

y - (12) = \frac{4}{3} \cdot (x - (9))

$y - (12) = \frac{4}{3} \cdot (x - (9))$

Step 4.2

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

y - 12 = \frac{4}{3} \cdot (x - 9)

$y - 12 = \frac{4}{3} \cdot (x - 9)$

y - 12 = \frac{4}{3} \cdot (x - 9)

$y - 12 = \frac{4}{3} \cdot (x - 9)$

Step 5

Write in

y = m x + b

$y = m x + b$ form.

Tap for more steps...

Step 5.1

Solve for

y

$y$ .

Tap for more steps...

Step 5.1.1

Simplify

\frac{4}{3} \cdot (x - 9)

$\frac{4}{3} \cdot (x - 9)$ .

Tap for more steps...

Step 5.1.1.1

Rewrite.

y - 12 = 0 + 0 + \frac{4}{3} \cdot (x - 9)

$y - 12 = 0 + 0 + \frac{4}{3} \cdot (x - 9)$

Step 5.1.1.2

Simplify by adding zeros.

y - 12 = \frac{4}{3} \cdot (x - 9)

$y - 12 = \frac{4}{3} \cdot (x - 9)$

Step 5.1.1.3

Apply the distributive property.

y - 12 = \frac{4}{3} x + \frac{4}{3} \cdot - 9

$y - 12 = \frac{4}{3} x + \frac{4}{3} \cdot - 9$

Step 5.1.1.4

Combine

\frac{4}{3}

$\frac{4}{3}$ and

x

$x$ .

y - 12 = \frac{4 x}{3} + \frac{4}{3} \cdot - 9

$y - 12 = \frac{4 x}{3} + \frac{4}{3} \cdot - 9$

Step 5.1.1.5

Cancel the common factor of

3

$3$ .

Tap for more steps...

Step 5.1.1.5.1

Factor

3

$3$ out of

- 9

$- 9$ .

y - 12 = \frac{4 x}{3} + \frac{4}{3} \cdot (3 (- 3))

$y - 12 = \frac{4 x}{3} + \frac{4}{3} \cdot (3 (- 3))$

Step 5.1.1.5.2

Cancel the common factor.

$y - 12 = \frac{4 x}{3} + \frac{4}{3} \cdot (3 \cdot - 3)$

Step 5.1.1.5.3

Rewrite the expression.

$y - 12 = \frac{4 x}{3} + 4 \cdot - 3$

Step 5.1.1.6

Multiply

$4$ by

$- 3$ .

$y - 12 = \frac{4 x}{3} - 12$

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

$12$ to both sides of the equation.

$y = \frac{4 x}{3} - 12 + 12$

Step 5.1.2.2

Combine the opposite terms in

$\frac{4 x}{3} - 12 + 12$ .

Tap for more steps...

Step 5.1.2.2.1

Add

$- 12$ and

$12$ .

$y = \frac{4 x}{3} + 0$

Step 5.1.2.2.2

Add

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

$0$ .

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

Step 5.2

Reorder terms.

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

Step 6