Algebra Examples

$(- 6, - 4)$ and is perpendicular to the line given by $y = \frac{3}{4} x - \frac{1}{2}$

Step 1

Find the slope when $y = \frac{3}{4} x - \frac{1}{2}$ .

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Step 1.1

Rewrite in slope-intercept form.

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Step 1.1.1

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 1.1.2

Simplify the right side.

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Step 1.1.2.1

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

$y = \frac{3 x}{4} - \frac{1}{2}$

Step 1.1.3

Reorder terms.

$y = \frac{3}{4} x - \frac{1}{2}$

Step 1.2

Using the slope-intercept form, the slope is $\frac{3}{4}$ .

$m = \frac{3}{4}$

Step 2

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

$m_{perpendicular} = - \frac{1}{\frac{3}{4}}$

Step 3

Simplify $- \frac{1}{\frac{3}{4}}$ to find the slope of the perpendicular line.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{4}{3}))$

Step 3.2

Multiply $\frac{4}{3}$ by $1$ .

$m_{perpendicular} = - \frac{4}{3}$

Step 4

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

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Step 4.1

Use the slope $- \frac{4}{3}$ and a given point $(- 6, - 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 4) = - \frac{4}{3} \cdot (x - (- 6))$

Step 4.2

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

$y + 4 = - \frac{4}{3} \cdot (x + 6)$

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 $- \frac{4}{3} \cdot (x + 6)$ .

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Step 5.1.1.1

Rewrite.

$y + 4 = 0 + 0 - \frac{4}{3} \cdot (x + 6)$

Step 5.1.1.2

Simplify terms.

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Step 5.1.1.2.1

Apply the distributive property.

$y + 4 = - \frac{4}{3} x - \frac{4}{3} \cdot 6$

Step 5.1.1.2.2

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

$y + 4 = - \frac{x \cdot 4}{3} - \frac{4}{3} \cdot 6$

Step 5.1.1.2.3

Cancel the common factor of $3$ .

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Step 5.1.1.2.3.1

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$y + 4 = - \frac{x \cdot 4}{3} + \frac{- 4}{3} \cdot 6$

Step 5.1.1.2.3.2

Factor $3$ out of $6$ .

$y + 4 = - \frac{x \cdot 4}{3} + \frac{- 4}{3} \cdot (3 (2))$

Step 5.1.1.2.3.3

Cancel the common factor.

$y + 4 = - \frac{x \cdot 4}{3} + \frac{- 4}{3} \cdot (3 \cdot 2)$

Step 5.1.1.2.3.4

Rewrite the expression.

$y + 4 = - \frac{x \cdot 4}{3} - 4 \cdot 2$

Step 5.1.1.2.4

Multiply $- 4$ by $2$ .

$y + 4 = - \frac{x \cdot 4}{3} - 8$

Step 5.1.1.3

Move $4$ to the left of $x$ .

$y + 4 = - \frac{4 x}{3} - 8$

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

Subtract $4$ from both sides of the equation.

$y = - \frac{4 x}{3} - 8 - 4$

Step 5.1.2.2

Subtract $4$ from $- 8$ .

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

Step 5.2

Reorder terms.

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

Step 5.3

Remove parentheses.

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

Step 6