Algebra Examples

What is an equation of the line that passes through the point $(- 4, - 6)$ and is perpendicular to the line $4 x + 5 y = 25$ ?

Step 1

Write the problem as a mathematical expression.

$(- 4, - 6)$ , $4 x + 5 y = 25$

Step 2

Solve $4 x + 5 y = 25$ .

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Step 2.1

Subtract $4 x$ from both sides of the equation.

$5 y = 25 - 4 x$

Step 2.2

Divide each term in $5 y = 25 - 4 x$ by $5$ and simplify.

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Step 2.2.1

Divide each term in $5 y = 25 - 4 x$ by $5$ .

$\frac{5 y}{5} = \frac{25}{5} + \frac{- 4 x}{5}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $5$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{25}{5} + \frac{- 4 x}{5}$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{25}{5} + \frac{- 4 x}{5}$

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 $25$ by $5$ .

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

Step 2.2.3.1.2

Move the negative in front of the fraction.

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

Step 3

Find the slope when $y = 5 - \frac{4 x}{5}$ .

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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 $5$ and $- \frac{4 x}{5}$ .

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

Step 3.1.3

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

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Step 3.1.3.1

Reorder terms.

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

Step 3.1.3.2

Remove parentheses.

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

Step 3.2

Using the slope-intercept form, the slope is $- \frac{4}{5}$ .

$m = - \frac{4}{5}$

Step 4

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

$m_{perpendicular} = - \frac{1}{- \frac{4}{5}}$

Step 5

Simplify $- \frac{1}{- \frac{4}{5}}$ 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 $- 1 (- 1)$ .

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{4}{5}}$

Step 5.1.2

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{4}{5}}$

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{5}{4})$

Step 5.3

Multiply $\frac{5}{4}$ by $1$ .

$m_{perpendicular} = \frac{5}{4}$

Step 5.4

Multiply $- - \frac{5}{4}$ .

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Step 5.4.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{5}{4})$

Step 5.4.2

Multiply $\frac{5}{4}$ by $1$ .

$m_{perpendicular} = \frac{5}{4}$

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

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 $\frac{5}{4} \cdot (x + 4)$ .

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Step 7.1.1.1

Rewrite.

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

Step 7.1.1.2

Simplify by adding zeros.

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

Step 7.1.1.3

Apply the distributive property.

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

Step 7.1.1.4

Combine $\frac{5}{4}$ and $x$ .

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

Step 7.1.1.5

Cancel the common factor of $4$ .

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Step 7.1.1.5.1

Cancel the common factor.

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

Step 7.1.1.5.2

Rewrite the expression.

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

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 $6$ from both sides of the equation.

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

Step 7.1.2.2

Subtract $6$ from $5$ .

$y = \frac{5 x}{4} - 1$

Step 7.2

Reorder terms.

$y = \frac{5}{4} x - 1$

Step 8