Precalculus Examples

$15 x - 5 y = 7$ , $(0, - 4)$

Step 1

Solve $15 x - 5 y = 7$ .

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Step 1.1

Subtract $15 x$ from both sides of the equation.

$- 5 y = 7 - 15 x$

Step 1.2

Divide each term in $- 5 y = 7 - 15 x$ by $- 5$ and simplify.

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Step 1.2.1

Divide each term in $- 5 y = 7 - 15 x$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{7}{- 5} + \frac{- 15 x}{- 5}$

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.

$\frac{- 5 y}{- 5} = \frac{7}{- 5} + \frac{- 15 x}{- 5}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{7}{- 5} + \frac{- 15 x}{- 5}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{7}{5} + \frac{- 15 x}{- 5}$

Step 1.2.3.1.2

Cancel the common factor of $- 15$ and $- 5$ .

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Step 1.2.3.1.2.1

Factor $- 5$ out of $- 15 x$ .

$y = - \frac{7}{5} + \frac{- 5 (3 x)}{- 5}$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $- 5$ out of $- 5$ .

$y = - \frac{7}{5} + \frac{- 5 (3 x)}{- 5 (1)}$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$y = - \frac{7}{5} + \frac{- 5 (3 x)}{- 5 \cdot 1}$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$y = - \frac{7}{5} + \frac{3 x}{1}$

Step 1.2.3.1.2.2.4

Divide $3 x$ by $1$ .

$y = - \frac{7}{5} + 3 x$

Step 2

Find the slope when $y = - \frac{7}{5} + 3 x$ .

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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

Reorder $- \frac{7}{5}$ and $3 x$ .

$y = 3 x - \frac{7}{5}$

Step 2.2

Using the slope-intercept form, the slope is $3$ .

$m = 3$

Step 3

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

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

Step 4

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

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Step 4.1

Use the slope $- \frac{1}{3}$ and a given point $(0, - 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 4) = - \frac{1}{3} \cdot (x - (0))$

Step 4.2

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

$y + 4 = - \frac{1}{3} \cdot (x + 0)$

Step 5

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

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Step 5.1

Solve for $y$ .

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Step 5.1.1

Simplify $- \frac{1}{3} \cdot (x + 0)$ .

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Step 5.1.1.1

Add $x$ and $0$ .

$y + 4 = - \frac{1}{3} \cdot x$

Step 5.1.1.2

Combine $x$ and $\frac{1}{3}$ .

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

Step 5.1.2

Subtract $4$ from both sides of the equation.

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

Step 5.2

Reorder terms.

$y = - (\frac{1}{3} x) - 4$

Step 5.3

Remove parentheses.

$y = - \frac{1}{3} x - 4$

Step 6