Algebra Examples

Through

$(\frac{1}{2}, - \frac{2}{3})$ ; perpendicular to the line

$5 x - 10 y = 1$

Step 1

Solve

$5 x - 10 y = 1$ .

Tap for more steps...

Step 1.1

Subtract

$5 x$ from both sides of the equation.

$- 10 y = 1 - 5 x$

Step 1.2

Divide each term in

$- 10 y = 1 - 5 x$ by

$- 10$ and simplify.

Tap for more steps...

Step 1.2.1

Divide each term in

$- 10 y = 1 - 5 x$ by

$- 10$ .

$\frac{- 10 y}{- 10} = \frac{1}{- 10} + \frac{- 5 x}{- 10}$

Step 1.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.1

Cancel the common factor of

$- 10$ .

Tap for more steps...

Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 10 y}{- 10} = \frac{1}{- 10} + \frac{- 5 x}{- 10}$

Step 1.2.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{1}{- 10} + \frac{- 5 x}{- 10}$

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.

$y = - \frac{1}{10} + \frac{- 5 x}{- 10}$

Step 1.2.3.1.2

Cancel the common factor of

$- 5$ and

$- 10$ .

Tap for more steps...

Step 1.2.3.1.2.1

Factor

$- 5$ out of

$- 5 x$ .

$y = - \frac{1}{10} + \frac{- 5 (x)}{- 10}$

Step 1.2.3.1.2.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.1.2.2.1

Factor

$- 5$ out of

$- 10$ .

$y = - \frac{1}{10} + \frac{- 5 (x)}{- 5 (2)}$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$y = - \frac{1}{10} + \frac{- 5 x}{- 5 \cdot 2}$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$y = - \frac{1}{10} + \frac{x}{2}$

Step 2

Find the slope when

$y = - \frac{1}{10} + \frac{x}{2}$ .

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

Reorder

$- \frac{1}{10}$ and

$\frac{x}{2}$ .

$y = \frac{x}{2} - \frac{1}{10}$

Step 2.1.3

Reorder terms.

$y = \frac{1}{2} x - \frac{1}{10}$

Step 2.2

Using the slope-intercept form, the slope is

$\frac{1}{2}$ .

$m = \frac{1}{2}$

Step 3

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

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

Step 4

Simplify

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

Tap for more steps...

Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 \cdot 2)$

Step 4.2

Multiply

$- (1 \cdot 2)$ .

Tap for more steps...

Step 4.2.1

Multiply

$2$ by

$1$ .

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

Step 4.2.2

Multiply

$- 1$ by

$2$ .

$m_{perpendicular} = - 2$

Step 5

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

Tap for more steps...

Step 5.1

Use the slope

$- 2$ and a given point

$(\frac{1}{2}, - \frac{2}{3})$ 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

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

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

Step 5.2

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

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

Step 6

Solve for

$y$ .

Tap for more steps...

Step 6.1

Simplify

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

Tap for more steps...

Step 6.1.1

Rewrite.

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

Step 6.1.2

Simplify by adding zeros.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.1.4

Cancel the common factor of

$2$ .

Tap for more steps...

Step 6.1.4.1

Move the leading negative in

$- \frac{1}{2}$ into the numerator.

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

Step 6.1.4.2

Factor

$2$ out of

$- 2$ .

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

Step 6.1.4.3

Cancel the common factor.

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

Step 6.1.4.4

Rewrite the expression.

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

Step 6.1.5

Multiply

$- 1$ by

$- 1$ .

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

Step 6.2

Move all terms not containing

$y$ to the right side of the equation.

Tap for more steps...

Step 6.2.1

Subtract

$\frac{2}{3}$ from both sides of the equation.

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

Step 6.2.2

Write

$1$ as a fraction with a common denominator.

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

Step 6.2.3

Combine the numerators over the common denominator.

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

Step 6.2.4

Subtract

$2$ from

$3$ .

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

Step 7