Algebra Examples

Passing through

(9, - 3)

$(9, - 3)$ and perpendicular to the line whose equation is

x - 9 y - 4 = 0

$x - 9 y - 4 = 0$

Step 1

Solve

x - 9 y - 4 = 0

$x - 9 y - 4 = 0$ .

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Step 1.1

Move all terms not containing

y

$y$ to the right side of the equation.

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Step 1.1.1

Subtract

x

$x$ from both sides of the equation.

- 9 y - 4 = - x

$- 9 y - 4 = - x$

Step 1.1.2

Add

4

$4$ to both sides of the equation.

- 9 y = - x + 4

$- 9 y = - x + 4$

- 9 y = - x + 4

$- 9 y = - x + 4$

Step 1.2

Divide each term in

- 9 y = - x + 4

$- 9 y = - x + 4$ by

- 9

$- 9$ and simplify.

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Step 1.2.1

Divide each term in

- 9 y = - x + 4

$- 9 y = - x + 4$ by

- 9

$- 9$ .

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

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of

- 9

$- 9$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Divide

$y$ by

$1$ .

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

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Dividing two negative values results in a positive value.

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

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

Step 2

Find the slope when

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

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Step 2.1

Rewrite in slope-intercept form.

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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 terms.

$y = \frac{1}{9} x - \frac{4}{9}$

Step 2.2

Using the slope-intercept form, the slope is

$\frac{1}{9}$ .

$m = \frac{1}{9}$

Step 3

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

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

Step 4

Simplify

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

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Multiply

$- (1 \cdot 9)$ .

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Step 4.2.1

Multiply

$9$ by

$1$ .

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

Step 4.2.2

Multiply

$- 1$ by

$9$ .

$m_{perpendicular} = - 9$

Step 5

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

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Step 5.1

Use the slope

$- 9$ and a given point

$(9, - 3)$ 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

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

$y - (- 3) = - 9 \cdot (x - (9))$

Step 5.2

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

$y + 3 = - 9 \cdot (x - 9)$

Step 6

Solve for

$y$ .

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Step 6.1

Simplify

$- 9 \cdot (x - 9)$ .

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Step 6.1.1

Rewrite.

$y + 3 = 0 + 0 - 9 \cdot (x - 9)$

Step 6.1.2

Simplify by adding zeros.

$y + 3 = - 9 \cdot (x - 9)$

Step 6.1.3

Apply the distributive property.

$y + 3 = - 9 x - 9 \cdot - 9$

Step 6.1.4

Multiply

$- 9$ by

$- 9$ .

$y + 3 = - 9 x + 81$

Step 6.2

Move all terms not containing

$y$ to the right side of the equation.

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Step 6.2.1

Subtract

$3$ from both sides of the equation.

$y = - 9 x + 81 - 3$

Step 6.2.2

Subtract

$3$ from

$81$ .

$y = - 9 x + 78$

Step 7