Algebra Examples

Through

(- 1, - 3)

$(- 1, - 3)$ ; perpendicular to the line

2 x + 7 y + 4 = 0

$2 x + 7 y + 4 = 0$

Step 1

Solve

2 x + 7 y + 4 = 0

$2 x + 7 y + 4 = 0$ .

Tap for more steps...

Step 1.1

Move all terms not containing

y

$y$ to the right side of the equation.

Tap for more steps...

Step 1.1.1

Subtract

2 x

$2 x$ from both sides of the equation.

7 y + 4 = - 2 x

$7 y + 4 = - 2 x$

Step 1.1.2

Subtract

4

$4$ from both sides of the equation.

7 y = - 2 x - 4

$7 y = - 2 x - 4$

7 y = - 2 x - 4

$7 y = - 2 x - 4$

Step 1.2

Divide each term in

7 y = - 2 x - 4

$7 y = - 2 x - 4$ by

7

$7$ and simplify.

Tap for more steps...

Step 1.2.1

Divide each term in

7 y = - 2 x - 4

$7 y = - 2 x - 4$ by

7

$7$ .

\frac{7 y}{7} = \frac{- 2 x}{7} + \frac{- 4}{7}

$\frac{7 y}{7} = \frac{- 2 x}{7} + \frac{- 4}{7}$

Step 1.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.1

Cancel the common factor of

7

$7$ .

Tap for more steps...

Step 1.2.2.1.1

Cancel the common factor.

$\frac{7 y}{7} = \frac{- 2 x}{7} + \frac{- 4}{7}$

Step 1.2.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{- 2 x}{7} + \frac{- 4}{7}$

Step 1.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.1

Simplify each term.

Tap for more steps...

Step 1.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{2 x}{7} + \frac{- 4}{7}$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$y = - \frac{2 x}{7} - \frac{4}{7}$

Step 2

Find the slope when

$y = - \frac{2 x}{7} - \frac{4}{7}$ .

Tap for more steps...

Step 2.1

Rewrite in slope-intercept form.

Tap for more steps...

Step 2.1.1

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 2.1.2

Write in

$y = m x + b$ form.

Tap for more steps...

Step 2.1.2.1

Reorder terms.

$y = - (\frac{2}{7} x) - \frac{4}{7}$

Step 2.1.2.2

Remove parentheses.

$y = - \frac{2}{7} x - \frac{4}{7}$

Step 2.2

Using the slope-intercept form, the slope is

$- \frac{2}{7}$ .

$m = - \frac{2}{7}$

Step 3

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

$m_{perpendicular} = - \frac{1}{- \frac{2}{7}}$

Step 4

Simplify

$- \frac{1}{- \frac{2}{7}}$ to find the slope of the perpendicular line.

Tap for more steps...

Step 4.1

Cancel the common factor of

$1$ and

$- 1$ .

Tap for more steps...

Step 4.1.1

Rewrite

$1$ as

$- 1 (- 1)$ .

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{2}{7}}$

Step 4.1.2

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{2}{7}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{7}{2})$

Step 4.3

Multiply

$\frac{7}{2}$ by

$1$ .

$m_{perpendicular} = \frac{7}{2}$

Step 4.4

Multiply

$- - \frac{7}{2}$ .

Tap for more steps...

Step 4.4.1

Multiply

$- 1$ by

$- 1$ .

$m_{perpendicular} = 1 (\frac{7}{2})$

Step 4.4.2

Multiply

$\frac{7}{2}$ by

$1$ .

$m_{perpendicular} = \frac{7}{2}$

Step 5

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

Tap for more steps...

Step 5.1

Use the slope

$\frac{7}{2}$ and a given point

$(- 1, - 3)$ 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

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

$y - (- 3) = \frac{7}{2} \cdot (x - (- 1))$

Step 5.2

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

$y + 3 = \frac{7}{2} \cdot (x + 1)$

Step 6

Write in

$y = m x + b$ form.

Tap for more steps...

Step 6.1

Solve for

$y$ .

Tap for more steps...

Step 6.1.1

Simplify

$\frac{7}{2} \cdot (x + 1)$ .

Tap for more steps...

Step 6.1.1.1

Rewrite.

$y + 3 = 0 + 0 + \frac{7}{2} \cdot (x + 1)$

Step 6.1.1.2

Simplify by adding zeros.

$y + 3 = \frac{7}{2} \cdot (x + 1)$

Step 6.1.1.3

Apply the distributive property.

$y + 3 = \frac{7}{2} x + \frac{7}{2} \cdot 1$

Step 6.1.1.4

Combine

$\frac{7}{2}$ and

$x$ .

$y + 3 = \frac{7 x}{2} + \frac{7}{2} \cdot 1$

Step 6.1.1.5

Multiply

$\frac{7}{2}$ by

$1$ .

$y + 3 = \frac{7 x}{2} + \frac{7}{2}$

Step 6.1.2

Move all terms not containing

$y$ to the right side of the equation.

Tap for more steps...

Step 6.1.2.1

Subtract

$3$ from both sides of the equation.

$y = \frac{7 x}{2} + \frac{7}{2} - 3$

Step 6.1.2.2

To write

$- 3$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$y = \frac{7 x}{2} + \frac{7}{2} - 3 \cdot \frac{2}{2}$

Step 6.1.2.3

Combine

$- 3$ and

$\frac{2}{2}$ .

$y = \frac{7 x}{2} + \frac{7}{2} + \frac{- 3 \cdot 2}{2}$

Step 6.1.2.4

Combine the numerators over the common denominator.

$y = \frac{7 x}{2} + \frac{7 - 3 \cdot 2}{2}$

Step 6.1.2.5

Simplify the numerator.

Tap for more steps...

Step 6.1.2.5.1

Multiply

$- 3$ by

$2$ .

$y = \frac{7 x}{2} + \frac{7 - 6}{2}$

Step 6.1.2.5.2

Subtract

$6$ from

$7$ .

$y = \frac{7 x}{2} + \frac{1}{2}$

Step 6.2

Reorder terms.

$y = \frac{7}{2} x + \frac{1}{2}$

Step 7