Algebra Examples

what is an equation of the line that passes through the point $(- 5, - 4)$ and is perpendicular to the line $5 x + 6 y = 36$

Step 1

Write the problem as a mathematical expression.

$(- 5, - 4)$ , $5 x + 6 y = 36$

Step 2

Solve $5 x + 6 y = 36$ .

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Step 2.1

Subtract $5 x$ from both sides of the equation.

$6 y = 36 - 5 x$

Step 2.2

Divide each term in $6 y = 36 - 5 x$ by $6$ and simplify.

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Step 2.2.1

Divide each term in $6 y = 36 - 5 x$ by $6$ .

$\frac{6 y}{6} = \frac{36}{6} + \frac{- 5 x}{6}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $6$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{36}{6} + \frac{- 5 x}{6}$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{36}{6} + \frac{- 5 x}{6}$

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 $36$ by $6$ .

$y = 6 + \frac{- 5 x}{6}$

Step 2.2.3.1.2

Move the negative in front of the fraction.

$y = 6 - \frac{5 x}{6}$

Step 3

Find the slope when $y = 6 - \frac{5 x}{6}$ .

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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 $6$ and $- \frac{5 x}{6}$ .

$y = - \frac{5 x}{6} + 6$

Step 3.1.3

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

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Step 3.1.3.1

Reorder terms.

$y = - (\frac{5}{6} x) + 6$

Step 3.1.3.2

Remove parentheses.

$y = - \frac{5}{6} x + 6$

Step 3.2

Using the slope-intercept form, the slope is $- \frac{5}{6}$ .

$m = - \frac{5}{6}$

Step 4

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

$m_{perpendicular} = - \frac{1}{- \frac{5}{6}}$

Step 5

Simplify $- \frac{1}{- \frac{5}{6}}$ to find the slope of the perpendicular line.

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Step 5.1

Cancel the common factor of $1$ and $- 1$ .

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Step 5.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{5}{6}}$

Step 5.1.2

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{5}{6}}$

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{6}{5})$

Step 5.3

Multiply $\frac{6}{5}$ by $1$ .

$m_{perpendicular} = \frac{6}{5}$

Step 5.4

Multiply $- - \frac{6}{5}$ .

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Step 5.4.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{6}{5})$

Step 5.4.2

Multiply $\frac{6}{5}$ by $1$ .

$m_{perpendicular} = \frac{6}{5}$

Step 6

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

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Step 6.1

Use the slope $\frac{6}{5}$ and a given point $(- 5, - 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 4) = \frac{6}{5} \cdot (x - (- 5))$

Step 6.2

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

$y + 4 = \frac{6}{5} \cdot (x + 5)$

Step 7

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

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Step 7.1

Solve for $y$ .

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Step 7.1.1

Simplify $\frac{6}{5} \cdot (x + 5)$ .

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Step 7.1.1.1

Rewrite.

$y + 4 = 0 + 0 + \frac{6}{5} \cdot (x + 5)$

Step 7.1.1.2

Simplify by adding zeros.

$y + 4 = \frac{6}{5} \cdot (x + 5)$

Step 7.1.1.3

Apply the distributive property.

$y + 4 = \frac{6}{5} x + \frac{6}{5} \cdot 5$

Step 7.1.1.4

Combine $\frac{6}{5}$ and $x$ .

$y + 4 = \frac{6 x}{5} + \frac{6}{5} \cdot 5$

Step 7.1.1.5

Cancel the common factor of $5$ .

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Step 7.1.1.5.1

Cancel the common factor.

$y + 4 = \frac{6 x}{5} + \frac{6}{5} \cdot 5$

Step 7.1.1.5.2

Rewrite the expression.

$y + 4 = \frac{6 x}{5} + 6$

Step 7.1.2

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

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Step 7.1.2.1

Subtract $4$ from both sides of the equation.

$y = \frac{6 x}{5} + 6 - 4$

Step 7.1.2.2

Subtract $4$ from $6$ .

$y = \frac{6 x}{5} + 2$

Step 7.2

Reorder terms.

$y = \frac{6}{5} x + 2$

Step 8