Precalculus Examples

$2 x + 7 y = 3$ , $(3, 9)$

Step 1

Solve $2 x + 7 y = 3$ .

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Step 1.1

Subtract $2 x$ from both sides of the equation.

$7 y = 3 - 2 x$

Step 1.2

Divide each term in $7 y = 3 - 2 x$ by $7$ and simplify.

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Step 1.2.1

Divide each term in $7 y = 3 - 2 x$ by $7$ .

$\frac{7 y}{7} = \frac{3}{7} + \frac{- 2 x}{7}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $7$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{7 y}{7} = \frac{3}{7} + \frac{- 2 x}{7}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{3}{7} + \frac{- 2 x}{7}$

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.

$y = \frac{3}{7} - \frac{2 x}{7}$

Step 2

Find the slope when $y = \frac{3}{7} - \frac{2 x}{7}$ .

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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 $\frac{3}{7}$ and $- \frac{2 x}{7}$ .

$y = - \frac{2 x}{7} + \frac{3}{7}$

Step 2.1.3

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

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Step 2.1.3.1

Reorder terms.

$y = - (\frac{2}{7} x) + \frac{3}{7}$

Step 2.1.3.2

Remove parentheses.

$y = - \frac{2}{7} x + \frac{3}{7}$

Step 2.2

Using the slope-intercept form, the slope is $- \frac{2}{7}$ .

$m = - \frac{2}{7}$

Step 3

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

$m_{perpendicular} = - \frac{1}{- \frac{2}{7}}$

Step 4

Simplify $- \frac{1}{- \frac{2}{7}}$ 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 $- 1 (- 1)$ .

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{2}{7}}$

Step 4.1.2

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{2}{7}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{7}{2})$

Step 4.3

Multiply $\frac{7}{2}$ by $1$ .

$m_{perpendicular} = \frac{7}{2}$

Step 4.4

Multiply $- - \frac{7}{2}$ .

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Step 4.4.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{7}{2})$

Step 4.4.2

Multiply $\frac{7}{2}$ by $1$ .

$m_{perpendicular} = \frac{7}{2}$

Step 5

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

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Step 5.1

Use the slope $\frac{7}{2}$ and a given point $(3, 9)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (9) = \frac{7}{2} \cdot (x - (3))$

Step 5.2

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

$y - 9 = \frac{7}{2} \cdot (x - 3)$

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 $\frac{7}{2} \cdot (x - 3)$ .

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Step 6.1.1.1

Rewrite.

$y - 9 = 0 + 0 + \frac{7}{2} \cdot (x - 3)$

Step 6.1.1.2

Simplify by adding zeros.

$y - 9 = \frac{7}{2} \cdot (x - 3)$

Step 6.1.1.3

Apply the distributive property.

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

Step 6.1.1.4

Combine $\frac{7}{2}$ and $x$ .

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

Step 6.1.1.5

Multiply $\frac{7}{2} \cdot - 3$ .

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Step 6.1.1.5.1

Combine $\frac{7}{2}$ and $- 3$ .

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

Step 6.1.1.5.2

Multiply $7$ by $- 3$ .

$y - 9 = \frac{7 x}{2} + \frac{- 21}{2}$

Step 6.1.1.6

Move the negative in front of the fraction.

$y - 9 = \frac{7 x}{2} - \frac{21}{2}$

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 $9$ to both sides of the equation.

$y = \frac{7 x}{2} - \frac{21}{2} + 9$

Step 6.1.2.2

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{7 x}{2} - \frac{21}{2} + 9 \cdot \frac{2}{2}$

Step 6.1.2.3

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

$y = \frac{7 x}{2} - \frac{21}{2} + \frac{9 \cdot 2}{2}$

Step 6.1.2.4

Combine the numerators over the common denominator.

$y = \frac{7 x}{2} + \frac{- 21 + 9 \cdot 2}{2}$

Step 6.1.2.5

Simplify the numerator.

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Step 6.1.2.5.1

Multiply $9$ by $2$ .

$y = \frac{7 x}{2} + \frac{- 21 + 18}{2}$

Step 6.1.2.5.2

Add $- 21$ and $18$ .

$y = \frac{7 x}{2} + \frac{- 3}{2}$

Step 6.1.2.6

Move the negative in front of the fraction.

$y = \frac{7 x}{2} - \frac{3}{2}$

Step 6.2

Reorder terms.

$y = \frac{7}{2} x - \frac{3}{2}$

Step 7