Algebra Examples

y = - \frac{10}{9} x - \frac{22}{9}

$y = - \frac{10}{9} x - \frac{22}{9}$ ;

(4, 2)

$(4, 2)$

Step 1

Find the slope when

y = - \frac{10}{9} x - \frac{22}{9}

$y = - \frac{10}{9} x - \frac{22}{9}$ .

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Step 1.1

Rewrite in slope-intercept form.

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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.

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

Combine

x

$x$ and

\frac{10}{9}

$\frac{10}{9}$ .

y = - \frac{x \cdot 10}{9} - \frac{22}{9}

$y = - \frac{x \cdot 10}{9} - \frac{22}{9}$

Step 1.1.2.1.2

Move

10

$10$ to the left of

x

$x$ .

y = - \frac{10 x}{9} - \frac{22}{9}

$y = - \frac{10 x}{9} - \frac{22}{9}$

y = - \frac{10 x}{9} - \frac{22}{9}

$y = - \frac{10 x}{9} - \frac{22}{9}$

y = - \frac{10 x}{9} - \frac{22}{9}

$y = - \frac{10 x}{9} - \frac{22}{9}$

Step 1.1.3

Write in

y = m x + b

$y = m x + b$ form.

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Step 1.1.3.1

Reorder terms.

y = - (\frac{10}{9} x) - \frac{22}{9}

$y = - (\frac{10}{9} x) - \frac{22}{9}$

Step 1.1.3.2

Remove parentheses.

y = - \frac{10}{9} x - \frac{22}{9}

$y = - \frac{10}{9} x - \frac{22}{9}$

y = - \frac{10}{9} x - \frac{22}{9}

$y = - \frac{10}{9} x - \frac{22}{9}$

y = - \frac{10}{9} x - \frac{22}{9}

$y = - \frac{10}{9} x - \frac{22}{9}$

Step 1.2

Using the slope-intercept form, the slope is

- \frac{10}{9}

$- \frac{10}{9}$ .

m = - \frac{10}{9}

$m = - \frac{10}{9}$

m = - \frac{10}{9}

$m = - \frac{10}{9}$

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{10}{9}}

$m_{perpendicular} = - \frac{1}{- \frac{10}{9}}$

Step 3

Simplify

- \frac{1}{- \frac{10}{9}}

$- \frac{1}{- \frac{10}{9}}$ to find the slope of the perpendicular line.

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Step 3.1

Cancel the common factor of

1

$1$ and

- 1

$- 1$ .

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Step 3.1.1

Rewrite

1

$1$ as

- 1 (- 1)

$- 1 (- 1)$ .

m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{10}{9}}

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{10}{9}}$

Step 3.1.2

Move the negative in front of the fraction.

m_{perpendicular} = \frac{1}{\frac{10}{9}}

$m_{perpendicular} = \frac{1}{\frac{10}{9}}$

m_{perpendicular} = \frac{1}{\frac{10}{9}}

$m_{perpendicular} = \frac{1}{\frac{10}{9}}$

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

m_{perpendicular} = 1 (\frac{9}{10})

$m_{perpendicular} = 1 (\frac{9}{10})$

Step 3.3

Multiply

\frac{9}{10}

$\frac{9}{10}$ by

1

$1$ .

m_{perpendicular} = \frac{9}{10}

$m_{perpendicular} = \frac{9}{10}$

Step 3.4

Multiply

- - \frac{9}{10}

$- - \frac{9}{10}$ .

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Step 3.4.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

m_{perpendicular} = 1 (\frac{9}{10})

$m_{perpendicular} = 1 (\frac{9}{10})$

Step 3.4.2

Multiply

\frac{9}{10}

$\frac{9}{10}$ by

1

$1$ .

m_{perpendicular} = \frac{9}{10}

$m_{perpendicular} = \frac{9}{10}$

m_{perpendicular} = \frac{9}{10}

$m_{perpendicular} = \frac{9}{10}$

m_{perpendicular} = \frac{9}{10}

$m_{perpendicular} = \frac{9}{10}$

Step 4

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

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Step 4.1

Use the slope

\frac{9}{10}

$\frac{9}{10}$ and a given point

(4, 2)

$(4, 2)$ 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 - (2) = \frac{9}{10} \cdot (x - (4))

$y - (2) = \frac{9}{10} \cdot (x - (4))$

Step 4.2

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

y - 2 = \frac{9}{10} \cdot (x - 4)

$y - 2 = \frac{9}{10} \cdot (x - 4)$

y - 2 = \frac{9}{10} \cdot (x - 4)

$y - 2 = \frac{9}{10} \cdot (x - 4)$

Step 5

Write in

y = m x + b

$y = m x + b$ form.

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Step 5.1

Solve for

y

$y$ .

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Step 5.1.1

Simplify

\frac{9}{10} \cdot (x - 4)

$\frac{9}{10} \cdot (x - 4)$ .

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Step 5.1.1.1

Rewrite.

y - 2 = 0 + 0 + \frac{9}{10} \cdot (x - 4)

$y - 2 = 0 + 0 + \frac{9}{10} \cdot (x - 4)$

Step 5.1.1.2

Simplify by adding zeros.

y - 2 = \frac{9}{10} \cdot (x - 4)

$y - 2 = \frac{9}{10} \cdot (x - 4)$

Step 5.1.1.3

Apply the distributive property.

y - 2 = \frac{9}{10} x + \frac{9}{10} \cdot - 4

$y - 2 = \frac{9}{10} x + \frac{9}{10} \cdot - 4$

Step 5.1.1.4

Combine

\frac{9}{10}

$\frac{9}{10}$ and

x

$x$ .

y - 2 = \frac{9 x}{10} + \frac{9}{10} \cdot - 4

$y - 2 = \frac{9 x}{10} + \frac{9}{10} \cdot - 4$

Step 5.1.1.5

Cancel the common factor of

2

$2$ .

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Step 5.1.1.5.1

Factor

2

$2$ out of

10

$10$ .

y - 2 = \frac{9 x}{10} + \frac{9}{2 (5)} \cdot - 4

$y - 2 = \frac{9 x}{10} + \frac{9}{2 (5)} \cdot - 4$

Step 5.1.1.5.2

Factor

2

$2$ out of

- 4

$- 4$ .

y - 2 = \frac{9 x}{10} + \frac{9}{2 \cdot 5} \cdot (2 \cdot - 2)

$y - 2 = \frac{9 x}{10} + \frac{9}{2 \cdot 5} \cdot (2 \cdot - 2)$

Step 5.1.1.5.3

Cancel the common factor.

$y - 2 = \frac{9 x}{10} + \frac{9}{2 \cdot 5} \cdot (2 \cdot - 2)$

Step 5.1.1.5.4

Rewrite the expression.

$y - 2 = \frac{9 x}{10} + \frac{9}{5} \cdot - 2$

Step 5.1.1.6

Combine

$\frac{9}{5}$ and

$- 2$ .

$y - 2 = \frac{9 x}{10} + \frac{9 \cdot - 2}{5}$

Step 5.1.1.7

Simplify the expression.

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Step 5.1.1.7.1

Multiply

$9$ by

$- 2$ .

$y - 2 = \frac{9 x}{10} + \frac{- 18}{5}$

Step 5.1.1.7.2

Move the negative in front of the fraction.

$y - 2 = \frac{9 x}{10} - \frac{18}{5}$

Step 5.1.2

Move all terms not containing

$y$ to the right side of the equation.

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Step 5.1.2.1

Add

$2$ to both sides of the equation.

$y = \frac{9 x}{10} - \frac{18}{5} + 2$

Step 5.1.2.2

To write

$2$ as a fraction with a common denominator, multiply by

$\frac{5}{5}$ .

$y = \frac{9 x}{10} - \frac{18}{5} + 2 \cdot \frac{5}{5}$

Step 5.1.2.3

Combine

$2$ and

$\frac{5}{5}$ .

$y = \frac{9 x}{10} - \frac{18}{5} + \frac{2 \cdot 5}{5}$

Step 5.1.2.4

Combine the numerators over the common denominator.

$y = \frac{9 x}{10} + \frac{- 18 + 2 \cdot 5}{5}$

Step 5.1.2.5

Simplify the numerator.

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Step 5.1.2.5.1

Multiply

$2$ by

$5$ .

$y = \frac{9 x}{10} + \frac{- 18 + 10}{5}$

Step 5.1.2.5.2

Add

$- 18$ and

$10$ .

$y = \frac{9 x}{10} + \frac{- 8}{5}$

Step 5.1.2.6

Move the negative in front of the fraction.

$y = \frac{9 x}{10} - \frac{8}{5}$

Step 5.2

Reorder terms.

$y = \frac{9}{10} x - \frac{8}{5}$

Step 6