Algebra Examples

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

Step 1

Write the problem as a mathematical expression.

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

Step 2

Solve $4 x - 5 y = - 1$ .

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Step 2.1

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

$- 5 y = - 1 - 4 x$

Step 2.2

Divide each term in $- 5 y = - 1 - 4 x$ by $- 5$ and simplify.

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Step 2.2.1

Divide each term in $- 5 y = - 1 - 4 x$ by $- 5$ .

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

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.

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

Step 2.2.2.1.2

Divide $y$ by $1$ .

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

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

Dividing two negative values results in a positive value.

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

Step 2.2.3.1.2

Dividing two negative values results in a positive value.

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

Step 3

Find the slope when $y = \frac{1}{5} + \frac{4 x}{5}$ .

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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 $\frac{1}{5}$ and $\frac{4 x}{5}$ .

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

Step 3.1.3

Reorder terms.

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

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

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2

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 $(5, - 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 6.2

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

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

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

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Step 7.1.1.1

Rewrite.

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

Step 7.1.1.2

Simplify by adding zeros.

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

Step 7.1.1.3

Apply the distributive property.

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

Step 7.1.1.4

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

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

Step 7.1.1.5

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

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Step 7.1.1.5.1

Multiply $- 5$ by $- 1$ .

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

Step 7.1.1.5.2

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

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

Step 7.1.1.5.3

Multiply $5$ by $5$ .

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

Step 7.1.1.6

Move $5$ to the left of $x$ .

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

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.

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

Step 7.1.2.2

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

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

Step 7.1.2.3

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

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

Step 7.1.2.4

Combine the numerators over the common denominator.

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

Step 7.1.2.5

Simplify the numerator.

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Step 7.1.2.5.1

Multiply $- 4$ by $4$ .

$y = - \frac{5 x}{4} + \frac{25 - 16}{4}$

Step 7.1.2.5.2

Subtract $16$ from $25$ .

$y = - \frac{5 x}{4} + \frac{9}{4}$

Step 7.2

Reorder terms.

$y = - (\frac{5}{4} x) + \frac{9}{4}$

Step 7.3

Remove parentheses.

$y = - \frac{5}{4} x + \frac{9}{4}$

Step 8