Algebra Examples

$y = - \frac{4}{9} x - 2$ and $(3, 5)$

Step 1

Find the slope when $y = - \frac{4}{9} x - 2$ .

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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 $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

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 $x$ and $\frac{4}{9}$ .

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

Step 1.1.2.1.2

Move $4$ to the left of $x$ .

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

Step 1.1.3

Write in $y = m x + b$ form.

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Step 1.1.3.1

Reorder terms.

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

Step 1.1.3.2

Remove parentheses.

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

Step 1.2

Using the slope-intercept form, the slope is $- \frac{4}{9}$ .

$m = - \frac{4}{9}$

Step 2

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

$m_{perpendicular} = - \frac{1}{- \frac{4}{9}}$

Step 3

Simplify $- \frac{1}{- \frac{4}{9}}$ to find the slope of the perpendicular line.

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Step 3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{4}{9}}$

Step 3.1.2

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{4}{9}}$

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{9}{4})$

Step 3.3

Multiply $\frac{9}{4}$ by $1$ .

$m_{perpendicular} = \frac{9}{4}$

Step 3.4

Multiply $- - \frac{9}{4}$ .

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Step 3.4.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{9}{4})$

Step 3.4.2

Multiply $\frac{9}{4}$ by $1$ .

$m_{perpendicular} = \frac{9}{4}$

Step 4

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

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Step 4.1

Use the slope $\frac{9}{4}$ and a given point $(3, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = \frac{9}{4} \cdot (x - (3))$

Step 4.2

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

$y - 5 = \frac{9}{4} \cdot (x - 3)$

Step 5

Write in $y = m x + b$ form.

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Step 5.1

Solve for $y$ .

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Step 5.1.1

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

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Step 5.1.1.1

Rewrite.

$y - 5 = 0 + 0 + \frac{9}{4} \cdot (x - 3)$

Step 5.1.1.2

Simplify by adding zeros.

$y - 5 = \frac{9}{4} \cdot (x - 3)$

Step 5.1.1.3

Apply the distributive property.

$y - 5 = \frac{9}{4} x + \frac{9}{4} \cdot - 3$

Step 5.1.1.4

Combine $\frac{9}{4}$ and $x$ .

$y - 5 = \frac{9 x}{4} + \frac{9}{4} \cdot - 3$

Step 5.1.1.5

Multiply $\frac{9}{4} \cdot - 3$ .

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Step 5.1.1.5.1

Combine $\frac{9}{4}$ and $- 3$ .

$y - 5 = \frac{9 x}{4} + \frac{9 \cdot - 3}{4}$

Step 5.1.1.5.2

Multiply $9$ by $- 3$ .

$y - 5 = \frac{9 x}{4} + \frac{- 27}{4}$

Step 5.1.1.6

Move the negative in front of the fraction.

$y - 5 = \frac{9 x}{4} - \frac{27}{4}$

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 $5$ to both sides of the equation.

$y = \frac{9 x}{4} - \frac{27}{4} + 5$

Step 5.1.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{9 x}{4} - \frac{27}{4} + 5 \cdot \frac{4}{4}$

Step 5.1.2.3

Combine $5$ and $\frac{4}{4}$ .

$y = \frac{9 x}{4} - \frac{27}{4} + \frac{5 \cdot 4}{4}$

Step 5.1.2.4

Combine the numerators over the common denominator.

$y = \frac{9 x}{4} + \frac{- 27 + 5 \cdot 4}{4}$

Step 5.1.2.5

Simplify the numerator.

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Step 5.1.2.5.1

Multiply $5$ by $4$ .

$y = \frac{9 x}{4} + \frac{- 27 + 20}{4}$

Step 5.1.2.5.2

Add $- 27$ and $20$ .

$y = \frac{9 x}{4} + \frac{- 7}{4}$

Step 5.1.2.6

Move the negative in front of the fraction.

$y = \frac{9 x}{4} - \frac{7}{4}$

Step 5.2

Reorder terms.

$y = \frac{9}{4} x - \frac{7}{4}$

Step 6