Precalculus Examples

$(\frac{6}{7}, 0)$ , $\frac{9}{2} x + \frac{2}{3} y = \frac{3}{4}$

Step 1

Solve $\frac{9}{2} x + \frac{2}{3} y = \frac{3}{4}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{9}{2}$ and $x$ .

$\frac{9 x}{2} + \frac{2}{3} y = \frac{3}{4}$

Step 1.1.2

Combine $\frac{2}{3}$ and $y$ .

$\frac{9 x}{2} + \frac{2 y}{3} = \frac{3}{4}$

Step 1.2

Subtract $\frac{9 x}{2}$ from both sides of the equation.

$\frac{2 y}{3} = \frac{3}{4} - \frac{9 x}{2}$

Step 1.3

Multiply both sides of the equation by $\frac{3}{2}$ .

$\frac{3}{2} \cdot \frac{2 y}{3} = \frac{3}{2} (\frac{3}{4} - \frac{9 x}{2})$

Step 1.4

Simplify both sides of the equation.

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Step 1.4.1

Simplify the left side.

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Step 1.4.1.1

Simplify $\frac{3}{2} \cdot \frac{2 y}{3}$ .

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Step 1.4.1.1.1

Cancel the common factor of $3$ .

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Step 1.4.1.1.1.1

Cancel the common factor.

$\frac{3}{2} \cdot \frac{2 y}{3} = \frac{3}{2} (\frac{3}{4} - \frac{9 x}{2})$

Step 1.4.1.1.1.2

Rewrite the expression.

$\frac{1}{2} (2 y) = \frac{3}{2} (\frac{3}{4} - \frac{9 x}{2})$

Step 1.4.1.1.2

Cancel the common factor of $2$ .

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Step 1.4.1.1.2.1

Factor $2$ out of $2 y$ .

$\frac{1}{2} (2 (y)) = \frac{3}{2} (\frac{3}{4} - \frac{9 x}{2})$

Step 1.4.1.1.2.2

Cancel the common factor.

$\frac{1}{2} (2 y) = \frac{3}{2} (\frac{3}{4} - \frac{9 x}{2})$

Step 1.4.1.1.2.3

Rewrite the expression.

$y = \frac{3}{2} (\frac{3}{4} - \frac{9 x}{2})$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify $\frac{3}{2} (\frac{3}{4} - \frac{9 x}{2})$ .

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Step 1.4.2.1.1

Apply the distributive property.

$y = \frac{3}{2} \cdot \frac{3}{4} + \frac{3}{2} (- \frac{9 x}{2})$

Step 1.4.2.1.2

Multiply $\frac{3}{2} \cdot \frac{3}{4}$ .

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Step 1.4.2.1.2.1

Multiply $\frac{3}{2}$ by $\frac{3}{4}$ .

$y = \frac{3 \cdot 3}{2 \cdot 4} + \frac{3}{2} (- \frac{9 x}{2})$

Step 1.4.2.1.2.2

Multiply $3$ by $3$ .

$y = \frac{9}{2 \cdot 4} + \frac{3}{2} (- \frac{9 x}{2})$

Step 1.4.2.1.2.3

Multiply $2$ by $4$ .

$y = \frac{9}{8} + \frac{3}{2} (- \frac{9 x}{2})$

Step 1.4.2.1.3

Multiply $\frac{3}{2} (- \frac{9 x}{2})$ .

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Step 1.4.2.1.3.1

Multiply $\frac{3}{2}$ by $\frac{9 x}{2}$ .

$y = \frac{9}{8} - \frac{3 (9 x)}{2 \cdot 2}$

Step 1.4.2.1.3.2

Multiply $9$ by $3$ .

$y = \frac{9}{8} - \frac{27 x}{2 \cdot 2}$

Step 1.4.2.1.3.3

Multiply $2$ by $2$ .

$y = \frac{9}{8} - \frac{27 x}{4}$

Step 1.5

Reorder $\frac{9}{8}$ and $- \frac{27 x}{4}$ .

$y = - \frac{27 x}{4} + \frac{9}{8}$

Step 2

Find the slope when $y = - \frac{27 x}{4} + \frac{9}{8}$ .

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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.

$y = - (\frac{27}{4} x) + \frac{9}{8}$

Step 2.1.2.2

Remove parentheses.

$y = - \frac{27}{4} x + \frac{9}{8}$

Step 2.2

Using the slope-intercept form, the slope is $- \frac{27}{4}$ .

$m = - \frac{27}{4}$

Step 3

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

$m_{perpendicular} = - \frac{1}{- \frac{27}{4}}$

Step 4

Simplify $- \frac{1}{- \frac{27}{4}}$ 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 $- 1 (- 1)$ .

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{27}{4}}$

Step 4.1.2

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{27}{4}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{4}{27})$

Step 4.3

Multiply $\frac{4}{27}$ by $1$ .

$m_{perpendicular} = \frac{4}{27}$

Step 4.4

Multiply $- - \frac{4}{27}$ .

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Step 4.4.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{4}{27})$

Step 4.4.2

Multiply $\frac{4}{27}$ by $1$ .

$m_{perpendicular} = \frac{4}{27}$

Step 5

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

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Step 5.1

Use the slope $\frac{4}{27}$ and a given point $(\frac{6}{7}, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = \frac{4}{27} \cdot (x - (\frac{6}{7}))$

Step 5.2

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

$y + 0 = \frac{4}{27} \cdot (x - \frac{6}{7})$

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

Add $y$ and $0$ .

$y = \frac{4}{27} \cdot (x - \frac{6}{7})$

Step 6.1.2

Simplify $\frac{4}{27} \cdot (x - \frac{6}{7})$ .

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Step 6.1.2.1

Apply the distributive property.

$y = \frac{4}{27} x + \frac{4}{27} (- \frac{6}{7})$

Step 6.1.2.2

Combine $\frac{4}{27}$ and $x$ .

$y = \frac{4 x}{27} + \frac{4}{27} (- \frac{6}{7})$

Step 6.1.2.3

Cancel the common factor of $3$ .

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Step 6.1.2.3.1

Move the leading negative in $- \frac{6}{7}$ into the numerator.

$y = \frac{4 x}{27} + \frac{4}{27} \cdot \frac{- 6}{7}$

Step 6.1.2.3.2

Factor $3$ out of $27$ .

$y = \frac{4 x}{27} + \frac{4}{3 (9)} \cdot \frac{- 6}{7}$

Step 6.1.2.3.3

Factor $3$ out of $- 6$ .

$y = \frac{4 x}{27} + \frac{4}{3 \cdot 9} \cdot \frac{3 \cdot - 2}{7}$

Step 6.1.2.3.4

Cancel the common factor.

$y = \frac{4 x}{27} + \frac{4}{3 \cdot 9} \cdot \frac{3 \cdot - 2}{7}$

Step 6.1.2.3.5

Rewrite the expression.

$y = \frac{4 x}{27} + \frac{4}{9} \cdot \frac{- 2}{7}$

Step 6.1.2.4

Multiply $\frac{4}{9}$ by $\frac{- 2}{7}$ .

$y = \frac{4 x}{27} + \frac{4 \cdot - 2}{9 \cdot 7}$

Step 6.1.2.5

Simplify the expression.

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Step 6.1.2.5.1

Multiply $4$ by $- 2$ .

$y = \frac{4 x}{27} + \frac{- 8}{9 \cdot 7}$

Step 6.1.2.5.2

Multiply $9$ by $7$ .

$y = \frac{4 x}{27} + \frac{- 8}{63}$

Step 6.1.2.5.3

Move the negative in front of the fraction.

$y = \frac{4 x}{27} - \frac{8}{63}$

Step 6.2

Reorder terms.

$y = \frac{4}{27} x - \frac{8}{63}$

Step 7