Algebra Examples

What is an equation of the line that passes through the point $(- 6, - 1)$ and is perpendicular to the line $6 x - y = 7$ ?

Step 1

Write the problem as a mathematical expression.

$(- 6, - 1)$ , $6 x - y = 7$

Step 2

Solve $6 x - y = 7$ .

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Step 2.1

Subtract $6 x$ from both sides of the equation.

$- y = 7 - 6 x$

Step 2.2

Divide each term in $- y = 7 - 6 x$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- y = 7 - 6 x$ by $- 1$ .

$\frac{- y}{- 1} = \frac{7}{- 1} + \frac{- 6 x}{- 1}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{7}{- 1} + \frac{- 6 x}{- 1}$

Step 2.2.2.2

Divide $y$ by $1$ .

$y = \frac{7}{- 1} + \frac{- 6 x}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Divide $7$ by $- 1$ .

$y = - 7 + \frac{- 6 x}{- 1}$

Step 2.2.3.1.2

Move the negative one from the denominator of $\frac{- 6 x}{- 1}$ .

$y = - 7 - 1 \cdot (- 6 x)$

Step 2.2.3.1.3

Rewrite $- 1 \cdot (- 6 x)$ as $- (- 6 x)$ .

$y = - 7 - (- 6 x)$

Step 2.2.3.1.4

Multiply $- 6$ by $- 1$ .

$y = - 7 + 6 x$

Step 3

Find the slope when $y = - 7 + 6 x$ .

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Step 3.1

Rewrite in slope-intercept form.

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Step 3.1.1

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 3.1.2

Reorder $- 7$ and $6 x$ .

$y = 6 x - 7$

Step 3.2

Using the slope-intercept form, the slope is $6$ .

$m = 6$

Step 4

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

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

Step 5

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

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Step 5.1

Use the slope $- \frac{1}{6}$ and a given point $(- 6, - 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 1) = - \frac{1}{6} \cdot (x - (- 6))$

Step 5.2

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

$y + 1 = - \frac{1}{6} \cdot (x + 6)$

Step 6

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

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Step 6.1

Solve for $y$ .

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Step 6.1.1

Simplify $- \frac{1}{6} \cdot (x + 6)$ .

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Step 6.1.1.1

Rewrite.

$y + 1 = 0 + 0 - \frac{1}{6} \cdot (x + 6)$

Step 6.1.1.2

Simplify by adding zeros.

$y + 1 = - \frac{1}{6} \cdot (x + 6)$

Step 6.1.1.3

Apply the distributive property.

$y + 1 = - \frac{1}{6} x - \frac{1}{6} \cdot 6$

Step 6.1.1.4

Combine $x$ and $\frac{1}{6}$ .

$y + 1 = - \frac{x}{6} - \frac{1}{6} \cdot 6$

Step 6.1.1.5

Cancel the common factor of $6$ .

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Step 6.1.1.5.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$y + 1 = - \frac{x}{6} + \frac{- 1}{6} \cdot 6$

Step 6.1.1.5.2

Cancel the common factor.

$y + 1 = - \frac{x}{6} + \frac{- 1}{6} \cdot 6$

Step 6.1.1.5.3

Rewrite the expression.

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

Step 6.1.2

Move all terms not containing $y$ to the right side of the equation.

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Step 6.1.2.1

Subtract $1$ from both sides of the equation.

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

Step 6.1.2.2

Subtract $1$ from $- 1$ .

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

Step 6.2

Reorder terms.

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

Step 6.3

Remove parentheses.

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

Step 7