Algebra Examples

What is an equation of the line that passes through the point $(- 4, 3)$ and is perpendicular to the line $4 x - 5 y = 20$ ?

Step 1

Write the problem as a mathematical expression.

$(- 4, 3)$ , $4 x - 5 y = 20$

Step 2

Solve $4 x - 5 y = 20$ .

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Step 2.1

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

$- 5 y = 20 - 4 x$

Step 2.2

Divide each term in $- 5 y = 20 - 4 x$ by $- 5$ and simplify.

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Step 2.2.1

Divide each term in $- 5 y = 20 - 4 x$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{20}{- 5} + \frac{- 4 x}{- 5}$

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.

$\frac{- 5 y}{- 5} = \frac{20}{- 5} + \frac{- 4 x}{- 5}$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{20}{- 5} + \frac{- 4 x}{- 5}$

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 $20$ by $- 5$ .

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

Step 2.2.3.1.2

Dividing two negative values results in a positive value.

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

Step 3

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

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

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

Step 3.1.3

Reorder terms.

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

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 $(- 4, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = - \frac{5}{4} \cdot (x - (- 4))$

Step 6.2

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

$y - 3 = - \frac{5}{4} \cdot (x + 4)$

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.

$y - 3 = 0 + 0 - \frac{5}{4} \cdot (x + 4)$

Step 7.1.1.2

Simplify terms.

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Step 7.1.1.2.1

Apply the distributive property.

$y - 3 = - \frac{5}{4} x - \frac{5}{4} \cdot 4$

Step 7.1.1.2.2

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

$y - 3 = - \frac{x \cdot 5}{4} - \frac{5}{4} \cdot 4$

Step 7.1.1.2.3

Cancel the common factor of $4$ .

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Step 7.1.1.2.3.1

Move the leading negative in $- \frac{5}{4}$ into the numerator.

$y - 3 = - \frac{x \cdot 5}{4} + \frac{- 5}{4} \cdot 4$

Step 7.1.1.2.3.2

Cancel the common factor.

$y - 3 = - \frac{x \cdot 5}{4} + \frac{- 5}{4} \cdot 4$

Step 7.1.1.2.3.3

Rewrite the expression.

$y - 3 = - \frac{x \cdot 5}{4} - 5$

Step 7.1.1.3

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

$y - 3 = - \frac{5 x}{4} - 5$

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

Add $3$ to both sides of the equation.

$y = - \frac{5 x}{4} - 5 + 3$

Step 7.1.2.2

Add $- 5$ and $3$ .

$y = - \frac{5 x}{4} - 2$

Step 7.2

Reorder terms.

$y = - (\frac{5}{4} x) - 2$

Step 7.3

Remove parentheses.

$y = - \frac{5}{4} x - 2$

Step 8