Algebra Examples

y = \frac{10}{9} x + 1

$y = \frac{10}{9} x + 1$ and

(- 3, 3)

$(- 3, 3)$

Step 1

Find the slope when

y = \frac{10}{9} x + 1

$y = \frac{10}{9} x + 1$ .

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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

Combine

\frac{10}{9}

$\frac{10}{9}$ and

x

$x$ .

y = \frac{10 x}{9} + 1

$y = \frac{10 x}{9} + 1$

y = \frac{10 x}{9} + 1

$y = \frac{10 x}{9} + 1$

Step 1.1.3

Reorder terms.

y = \frac{10}{9} x + 1

$y = \frac{10}{9} x + 1$

y = \frac{10}{9} x + 1

$y = \frac{10}{9} x + 1$

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

Multiply the numerator by the reciprocal of the denominator.

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

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

Step 3.2

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}

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

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

(- 3, 3)

$(- 3, 3)$ 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 - (3) = - \frac{9}{10} \cdot (x - (- 3))

$y - (3) = - \frac{9}{10} \cdot (x - (- 3))$

Step 4.2

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

y - 3 = - \frac{9}{10} \cdot (x + 3)

$y - 3 = - \frac{9}{10} \cdot (x + 3)$

y - 3 = - \frac{9}{10} \cdot (x + 3)

$y - 3 = - \frac{9}{10} \cdot (x + 3)$

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 + 3)

$- \frac{9}{10} \cdot (x + 3)$ .

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Step 5.1.1.1

Rewrite.

y - 3 = 0 + 0 - \frac{9}{10} \cdot (x + 3)

$y - 3 = 0 + 0 - \frac{9}{10} \cdot (x + 3)$

Step 5.1.1.2

Simplify by adding zeros.

y - 3 = - \frac{9}{10} \cdot (x + 3)

$y - 3 = - \frac{9}{10} \cdot (x + 3)$

Step 5.1.1.3

Apply the distributive property.

y - 3 = - \frac{9}{10} x - \frac{9}{10} \cdot 3

$y - 3 = - \frac{9}{10} x - \frac{9}{10} \cdot 3$

Step 5.1.1.4

Combine

x

$x$ and

\frac{9}{10}

$\frac{9}{10}$ .

y - 3 = - \frac{x \cdot 9}{10} - \frac{9}{10} \cdot 3

$y - 3 = - \frac{x \cdot 9}{10} - \frac{9}{10} \cdot 3$

Step 5.1.1.5

Multiply

- \frac{9}{10} \cdot 3

$- \frac{9}{10} \cdot 3$ .

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Step 5.1.1.5.1

Multiply

3

$3$ by

- 1

$- 1$ .

y - 3 = - \frac{x \cdot 9}{10} - 3 (\frac{9}{10})

$y - 3 = - \frac{x \cdot 9}{10} - 3 (\frac{9}{10})$

Step 5.1.1.5.2

Combine

- 3

$- 3$ and

\frac{9}{10}

$\frac{9}{10}$ .

y - 3 = - \frac{x \cdot 9}{10} + \frac{- 3 \cdot 9}{10}

$y - 3 = - \frac{x \cdot 9}{10} + \frac{- 3 \cdot 9}{10}$

Step 5.1.1.5.3

Multiply

- 3

$- 3$ by

9

$9$ .

y - 3 = - \frac{x \cdot 9}{10} + \frac{- 27}{10}

$y - 3 = - \frac{x \cdot 9}{10} + \frac{- 27}{10}$

y - 3 = - \frac{x \cdot 9}{10} + \frac{- 27}{10}

$y - 3 = - \frac{x \cdot 9}{10} + \frac{- 27}{10}$

Step 5.1.1.6

Simplify each term.

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Step 5.1.1.6.1

Move

9

$9$ to the left of

x

$x$ .

y - 3 = - \frac{9 \cdot x}{10} + \frac{- 27}{10}

$y - 3 = - \frac{9 \cdot x}{10} + \frac{- 27}{10}$

Step 5.1.1.6.2

Move the negative in front of the fraction.

y - 3 = - \frac{9 x}{10} - \frac{27}{10}

$y - 3 = - \frac{9 x}{10} - \frac{27}{10}$

y - 3 = - \frac{9 x}{10} - \frac{27}{10}

$y - 3 = - \frac{9 x}{10} - \frac{27}{10}$

y - 3 = - \frac{9 x}{10} - \frac{27}{10}

$y - 3 = - \frac{9 x}{10} - \frac{27}{10}$

Step 5.1.2

Move all terms not containing

y

$y$ to the right side of the equation.

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Step 5.1.2.1

Add

3

$3$ to both sides of the equation.

y = - \frac{9 x}{10} - \frac{27}{10} + 3

$y = - \frac{9 x}{10} - \frac{27}{10} + 3$

Step 5.1.2.2

To write

3

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

\frac{10}{10}

$\frac{10}{10}$ .

y = - \frac{9 x}{10} - \frac{27}{10} + 3 \cdot \frac{10}{10}

$y = - \frac{9 x}{10} - \frac{27}{10} + 3 \cdot \frac{10}{10}$

Step 5.1.2.3

Combine

3

$3$ and

\frac{10}{10}

$\frac{10}{10}$ .

y = - \frac{9 x}{10} - \frac{27}{10} + \frac{3 \cdot 10}{10}

$y = - \frac{9 x}{10} - \frac{27}{10} + \frac{3 \cdot 10}{10}$

Step 5.1.2.4

Combine the numerators over the common denominator.

y = - \frac{9 x}{10} + \frac{- 27 + 3 \cdot 10}{10}

$y = - \frac{9 x}{10} + \frac{- 27 + 3 \cdot 10}{10}$

Step 5.1.2.5

Simplify the numerator.

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Step 5.1.2.5.1

Multiply

3

$3$ by

10

$10$ .

y = - \frac{9 x}{10} + \frac{- 27 + 30}{10}

$y = - \frac{9 x}{10} + \frac{- 27 + 30}{10}$

Step 5.1.2.5.2

Add

- 27

$- 27$ and

30

$30$ .

y = - \frac{9 x}{10} + \frac{3}{10}

$y = - \frac{9 x}{10} + \frac{3}{10}$

y = - \frac{9 x}{10} + \frac{3}{10}

$y = - \frac{9 x}{10} + \frac{3}{10}$

y = - \frac{9 x}{10} + \frac{3}{10}

$y = - \frac{9 x}{10} + \frac{3}{10}$

y = - \frac{9 x}{10} + \frac{3}{10}

$y = - \frac{9 x}{10} + \frac{3}{10}$

Step 5.2

Reorder terms.

y = - (\frac{9}{10} x) + \frac{3}{10}

$y = - (\frac{9}{10} x) + \frac{3}{10}$

Step 5.3

Remove parentheses.

y = - \frac{9}{10} x + \frac{3}{10}

$y = - \frac{9}{10} x + \frac{3}{10}$

y = - \frac{9}{10} x + \frac{3}{10}

$y = - \frac{9}{10} x + \frac{3}{10}$

Step 6