Algebra Examples

(- 1, 4)

$(- 1, 4)$ and

x - 6 y = 12

$x - 6 y = 12$

Step 1

Solve

x - 6 y = 12

$x - 6 y = 12$ .

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Step 1.1

Subtract

x

$x$ from both sides of the equation.

- 6 y = 12 - x

$- 6 y = 12 - x$

Step 1.2

Divide each term in

- 6 y = 12 - x

$- 6 y = 12 - x$ by

- 6

$- 6$ and simplify.

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Step 1.2.1

Divide each term in

- 6 y = 12 - x

$- 6 y = 12 - x$ by

- 6

$- 6$ .

\frac{- 6 y}{- 6} = \frac{12}{- 6} + \frac{- x}{- 6}

$\frac{- 6 y}{- 6} = \frac{12}{- 6} + \frac{- x}{- 6}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of

- 6

$- 6$ .

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Step 1.2.2.1.1

Cancel the common factor.

\frac{- 6 y}{- 6} = \frac{12}{- 6} + \frac{- x}{- 6}

$\frac{- 6 y}{- 6} = \frac{12}{- 6} + \frac{- x}{- 6}$

Step 1.2.2.1.2

Divide

y

$y$ by

1

$1$ .

y = \frac{12}{- 6} + \frac{- x}{- 6}

$y = \frac{12}{- 6} + \frac{- x}{- 6}$

y = \frac{12}{- 6} + \frac{- x}{- 6}

$y = \frac{12}{- 6} + \frac{- x}{- 6}$

y = \frac{12}{- 6} + \frac{- x}{- 6}

$y = \frac{12}{- 6} + \frac{- x}{- 6}$

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

Divide

12

$12$ by

- 6

$- 6$ .

y = - 2 + \frac{- x}{- 6}

$y = - 2 + \frac{- x}{- 6}$

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

y = - 2 + \frac{x}{6}

$y = - 2 + \frac{x}{6}$

y = - 2 + \frac{x}{6}

$y = - 2 + \frac{x}{6}$

y = - 2 + \frac{x}{6}

$y = - 2 + \frac{x}{6}$

y = - 2 + \frac{x}{6}

$y = - 2 + \frac{x}{6}$

y = - 2 + \frac{x}{6}

$y = - 2 + \frac{x}{6}$

Step 2

Find the slope when

y = - 2 + \frac{x}{6}

$y = - 2 + \frac{x}{6}$ .

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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

$y = m x + b$ , where

m

$m$ is the slope and

b

$b$ is the y-intercept.

y = m x + b

$y = m x + b$

Step 2.1.2

Reorder

- 2

$- 2$ and

\frac{x}{6}

$\frac{x}{6}$ .

y = \frac{x}{6} - 2

$y = \frac{x}{6} - 2$

Step 2.1.3

Reorder terms.

y = \frac{1}{6} x - 2

$y = \frac{1}{6} x - 2$

y = \frac{1}{6} x - 2

$y = \frac{1}{6} x - 2$

Step 2.2

Using the slope-intercept form, the slope is

\frac{1}{6}

$\frac{1}{6}$ .

m = \frac{1}{6}

$m = \frac{1}{6}$

m = \frac{1}{6}

$m = \frac{1}{6}$

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}{6}}

$m_{perpendicular} = - \frac{1}{\frac{1}{6}}$

Step 4

Simplify

- \frac{1}{\frac{1}{6}}

$- \frac{1}{\frac{1}{6}}$ 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 6)

$m_{perpendicular} = - (1 \cdot 6)$

Step 4.2

Multiply

- (1 \cdot 6)

$- (1 \cdot 6)$ .

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Step 4.2.1

Multiply

6

$6$ by

1

$1$ .

m_{perpendicular} = - 1 \cdot 6

$m_{perpendicular} = - 1 \cdot 6$

Step 4.2.2

Multiply

- 1

$- 1$ by

6

$6$ .

m_{perpendicular} = - 6

$m_{perpendicular} = - 6$

m_{perpendicular} = - 6

$m_{perpendicular} = - 6$

m_{perpendicular} = - 6

$m_{perpendicular} = - 6$

Step 5

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

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Step 5.1

Use the slope

- 6

$- 6$ and a given point

(- 1, 4)

$(- 1, 4)$ 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 - (4) = - 6 \cdot (x - (- 1))

$y - (4) = - 6 \cdot (x - (- 1))$

Step 5.2

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

y - 4 = - 6 \cdot (x + 1)

$y - 4 = - 6 \cdot (x + 1)$

y - 4 = - 6 \cdot (x + 1)

$y - 4 = - 6 \cdot (x + 1)$

Step 6

Solve for

y

$y$ .

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Step 6.1

Simplify

- 6 \cdot (x + 1)

$- 6 \cdot (x + 1)$ .

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Step 6.1.1

Rewrite.

y - 4 = 0 + 0 - 6 \cdot (x + 1)

$y - 4 = 0 + 0 - 6 \cdot (x + 1)$

Step 6.1.2

Simplify by adding zeros.

y - 4 = - 6 \cdot (x + 1)

$y - 4 = - 6 \cdot (x + 1)$

Step 6.1.3

Apply the distributive property.

y - 4 = - 6 x - 6 \cdot 1

$y - 4 = - 6 x - 6 \cdot 1$

Step 6.1.4

Multiply

- 6

$- 6$ by

1

$1$ .

y - 4 = - 6 x - 6

$y - 4 = - 6 x - 6$

y - 4 = - 6 x - 6

$y - 4 = - 6 x - 6$

Step 6.2

Move all terms not containing

y

$y$ to the right side of the equation.

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Step 6.2.1

Add

4

$4$ to both sides of the equation.

y = - 6 x - 6 + 4

$y = - 6 x - 6 + 4$

Step 6.2.2

Add

- 6

$- 6$ and

4

$4$ .

y = - 6 x - 2

$y = - 6 x - 2$

y = - 6 x - 2

$y = - 6 x - 2$

y = - 6 x - 2

$y = - 6 x - 2$

Step 7