Precalculus Examples

$3 x + 5 y - 8 = 0$ , $(- 8, 1)$

Step 1

Solve $3 x + 5 y - 8 = 0$ .

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Step 1.1

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

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Step 1.1.1

Subtract $3 x$ from both sides of the equation.

$5 y - 8 = - 3 x$

Step 1.1.2

Add $8$ to both sides of the equation.

$5 y = - 3 x + 8$

Step 1.2

Divide each term in $5 y = - 3 x + 8$ by $5$ and simplify.

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Step 1.2.1

Divide each term in $5 y = - 3 x + 8$ by $5$ .

$\frac{5 y}{5} = \frac{- 3 x}{5} + \frac{8}{5}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{- 3 x}{5} + \frac{8}{5}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 3 x}{5} + \frac{8}{5}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$y = - \frac{3 x}{5} + \frac{8}{5}$

Step 2

Find the slope when $y = - \frac{3 x}{5} + \frac{8}{5}$ .

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Step 2.1

Rewrite in slope-intercept form.

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Step 2.1.1

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 2.1.2

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

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Step 2.1.2.1

Reorder terms.

$y = - (\frac{3}{5} x) + \frac{8}{5}$

Step 2.1.2.2

Remove parentheses.

$y = - \frac{3}{5} x + \frac{8}{5}$

Step 2.2

Using the slope-intercept form, the slope is $- \frac{3}{5}$ .

$m = - \frac{3}{5}$

Step 3

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

$m_{perpendicular} = - \frac{1}{- \frac{3}{5}}$

Step 4

Simplify $- \frac{1}{- \frac{3}{5}}$ to find the slope of the perpendicular line.

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Step 4.1

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

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Step 4.1.1

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

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{3}{5}}$

Step 4.1.2

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{3}{5}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{5}{3})$

Step 4.3

Multiply $\frac{5}{3}$ by $1$ .

$m_{perpendicular} = \frac{5}{3}$

Step 4.4

Multiply $- - \frac{5}{3}$ .

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Step 4.4.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{5}{3})$

Step 4.4.2

Multiply $\frac{5}{3}$ by $1$ .

$m_{perpendicular} = \frac{5}{3}$

Step 5

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

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Step 5.1

Use the slope $\frac{5}{3}$ and a given point $(- 8, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = \frac{5}{3} \cdot (x - (- 8))$

Step 5.2

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

$y - 1 = \frac{5}{3} \cdot (x + 8)$

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 $\frac{5}{3} \cdot (x + 8)$ .

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Step 6.1.1.1

Rewrite.

$y - 1 = 0 + 0 + \frac{5}{3} \cdot (x + 8)$

Step 6.1.1.2

Simplify by adding zeros.

$y - 1 = \frac{5}{3} \cdot (x + 8)$

Step 6.1.1.3

Apply the distributive property.

$y - 1 = \frac{5}{3} x + \frac{5}{3} \cdot 8$

Step 6.1.1.4

Combine $\frac{5}{3}$ and $x$ .

$y - 1 = \frac{5 x}{3} + \frac{5}{3} \cdot 8$

Step 6.1.1.5

Multiply $\frac{5}{3} \cdot 8$ .

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Step 6.1.1.5.1

Combine $\frac{5}{3}$ and $8$ .

$y - 1 = \frac{5 x}{3} + \frac{5 \cdot 8}{3}$

Step 6.1.1.5.2

Multiply $5$ by $8$ .

$y - 1 = \frac{5 x}{3} + \frac{40}{3}$

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.

$y = \frac{5 x}{3} + \frac{40}{3} + 1$

Step 6.1.2.2

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

$y = \frac{5 x}{3} + \frac{40}{3} + \frac{3}{3}$

Step 6.1.2.3

Combine the numerators over the common denominator.

$y = \frac{5 x}{3} + \frac{40 + 3}{3}$

Step 6.1.2.4

Add $40$ and $3$ .

$y = \frac{5 x}{3} + \frac{43}{3}$

Step 6.2

Reorder terms.

$y = \frac{5}{3} x + \frac{43}{3}$

Step 7