Algebra Examples

$(- 3, 3)$ and is perpendicular to the line $2 y = 8 x - 6$

Step 1

Divide each term in $2 y = 8 x - 6$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 y = 8 x - 6$ by $2$ .

$\frac{2 y}{2} = \frac{8 x}{2} + \frac{- 6}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{8 x}{2} + \frac{- 6}{2}$

Step 1.2.1.2

Divide $y$ by $1$ .

$y = \frac{8 x}{2} + \frac{- 6}{2}$

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

Cancel the common factor of $8$ and $2$ .

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Step 1.3.1.1.1

Factor $2$ out of $8 x$ .

$y = \frac{2 (4 x)}{2} + \frac{- 6}{2}$

Step 1.3.1.1.2

Cancel the common factors.

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Step 1.3.1.1.2.1

Factor $2$ out of $2$ .

$y = \frac{2 (4 x)}{2 (1)} + \frac{- 6}{2}$

Step 1.3.1.1.2.2

Cancel the common factor.

$y = \frac{2 (4 x)}{2 \cdot 1} + \frac{- 6}{2}$

Step 1.3.1.1.2.3

Rewrite the expression.

$y = \frac{4 x}{1} + \frac{- 6}{2}$

Step 1.3.1.1.2.4

Divide $4 x$ by $1$ .

$y = 4 x + \frac{- 6}{2}$

Step 1.3.1.2

Divide $- 6$ by $2$ .

$y = 4 x - 3$

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 $4$ .

$m = 4$

Step 3

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

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

Step 4

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

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Step 4.1

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

$y - (3) = - \frac{1}{4} \cdot (x - (- 3))$

Step 4.2

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

$y - 3 = - \frac{1}{4} \cdot (x + 3)$

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 $- \frac{1}{4} \cdot (x + 3)$ .

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Step 5.1.1.1

Rewrite.

$y - 3 = 0 + 0 - \frac{1}{4} \cdot (x + 3)$

Step 5.1.1.2

Simplify by adding zeros.

$y - 3 = - \frac{1}{4} \cdot (x + 3)$

Step 5.1.1.3

Apply the distributive property.

$y - 3 = - \frac{1}{4} x - \frac{1}{4} \cdot 3$

Step 5.1.1.4

Combine $x$ and $\frac{1}{4}$ .

$y - 3 = - \frac{x}{4} - \frac{1}{4} \cdot 3$

Step 5.1.1.5

Multiply $- \frac{1}{4} \cdot 3$ .

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Step 5.1.1.5.1

Multiply $3$ by $- 1$ .

$y - 3 = - \frac{x}{4} - 3 (\frac{1}{4})$

Step 5.1.1.5.2

Combine $- 3$ and $\frac{1}{4}$ .

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

Step 5.1.1.6

Move the negative in front of the fraction.

$y - 3 = - \frac{x}{4} - \frac{3}{4}$

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 $3$ to both sides of the equation.

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

Step 5.1.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 5.1.2.3

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

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

Step 5.1.2.4

Combine the numerators over the common denominator.

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

Step 5.1.2.5

Simplify the numerator.

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Step 5.1.2.5.1

Multiply $3$ by $4$ .

$y = - \frac{x}{4} + \frac{- 3 + 12}{4}$

Step 5.1.2.5.2

Add $- 3$ and $12$ .

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

Step 5.2

Reorder terms.

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

Step 5.3

Remove parentheses.

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

Step 6