Algebra Examples

find the equation of a line perpendicular to $- 4 x + y = - 5$ that contains the point $(- 5, 1)$

Step 1

Write the problem as a mathematical expression.

$- 4 x + y = - 5$ , $(- 5, 1)$

Step 2

Add $4 x$ to both sides of the equation.

$y = - 5 + 4 x$

Step 3

Find the slope when $y = - 5 + 4 x$ .

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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 $4 x$ .

$y = 4 x - 5$

Step 3.2

Using the slope-intercept form, the slope is $4$ .

$m = 4$

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

$y - 1 = - \frac{1}{4} \cdot (x + 5)$

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 $- \frac{1}{4} \cdot (x + 5)$ .

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Step 6.1.1.1

Rewrite.

$y - 1 = 0 + 0 - \frac{1}{4} \cdot (x + 5)$

Step 6.1.1.2

Simplify by adding zeros.

$y - 1 = - \frac{1}{4} \cdot (x + 5)$

Step 6.1.1.3

Apply the distributive property.

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

Step 6.1.1.4

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

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

Step 6.1.1.5

Multiply $- \frac{1}{4} \cdot 5$ .

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Step 6.1.1.5.1

Multiply $5$ by $- 1$ .

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

Step 6.1.1.5.2

Combine $- 5$ and $\frac{1}{4}$ .

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

Step 6.1.1.6

Move the negative in front of the fraction.

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

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

Add $1$ to both sides of the equation.

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

Step 6.1.2.2

Write $1$ as a fraction with a common denominator.

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

Step 6.1.2.3

Combine the numerators over the common denominator.

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

Step 6.1.2.4

Add $- 5$ and $4$ .

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

Step 6.1.2.5

Move the negative in front of the fraction.

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

Step 6.2

Reorder terms.

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

Step 6.3

Remove parentheses.

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

Step 7