Algebra Examples

$- y = 6 x + 2$ ; $P = (1, - 4)$

Step 1

Divide each term in $- y = 6 x + 2$ by $- 1$ and simplify.

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Step 1.1

Divide each term in $- y = 6 x + 2$ by $- 1$ .

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.2

Divide $y$ by $1$ .

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

Step 1.3

Simplify the right side.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

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

Step 1.3.1.2

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

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

Step 1.3.1.3

Multiply $6$ by $- 1$ .

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

Step 1.3.1.4

Divide $2$ by $- 1$ .

$y = - 6 x - 2$

Step 2

Use the slope-intercept form to find the slope.

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Step 2.1

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 2.2

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

$m = - 6$

Step 3

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

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

Step 4

Simplify $- \frac{1}{- 6}$ to find the slope of the perpendicular line.

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Step 4.1

Move the negative in front of the fraction.

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

Step 4.2

Multiply $- - \frac{1}{6}$ .

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Step 4.2.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{1}{6})$

Step 4.2.2

Multiply $\frac{1}{6}$ by $1$ .

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

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 $P = (1, - 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 5.2

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

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

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

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Step 6.1.1.1

Rewrite.

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

Step 6.1.1.2

Simplify by adding zeros.

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

Step 6.1.1.3

Apply the distributive property.

$y + 4 = \frac{1}{6} x + \frac{1}{6} \cdot - 1$

Step 6.1.1.4

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

$y + 4 = \frac{x}{6} + \frac{1}{6} \cdot - 1$

Step 6.1.1.5

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

$y + 4 = \frac{x}{6} + \frac{- 1}{6}$

Step 6.1.1.6

Move the negative in front of the fraction.

$y + 4 = \frac{x}{6} - \frac{1}{6}$

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 $4$ from both sides of the equation.

$y = \frac{x}{6} - \frac{1}{6} - 4$

Step 6.1.2.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$y = \frac{x}{6} - \frac{1}{6} - 4 \cdot \frac{6}{6}$

Step 6.1.2.3

Combine $- 4$ and $\frac{6}{6}$ .

$y = \frac{x}{6} - \frac{1}{6} + \frac{- 4 \cdot 6}{6}$

Step 6.1.2.4

Combine the numerators over the common denominator.

$y = \frac{x}{6} + \frac{- 1 - 4 \cdot 6}{6}$

Step 6.1.2.5

Simplify the numerator.

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Step 6.1.2.5.1

Multiply $- 4$ by $6$ .

$y = \frac{x}{6} + \frac{- 1 - 24}{6}$

Step 6.1.2.5.2

Subtract $24$ from $- 1$ .

$y = \frac{x}{6} + \frac{- 25}{6}$

Step 6.1.2.6

Move the negative in front of the fraction.

$y = \frac{x}{6} - \frac{25}{6}$

Step 6.2

Reorder terms.

$y = \frac{1}{6} x - \frac{25}{6}$

Step 7