Precalculus Examples

$6 x + 2 y = 9$ , $(- 3.9, - 1.4)$

Step 1

Solve $6 x + 2 y = 9$ .

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Step 1.1

Subtract $6 x$ from both sides of the equation.

$2 y = 9 - 6 x$

Step 1.2

Divide each term in $2 y = 9 - 6 x$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 y = 9 - 6 x$ by $2$ .

$\frac{2 y}{2} = \frac{9}{2} + \frac{- 6 x}{2}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{9}{2} + \frac{- 6 x}{2}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{9}{2} + \frac{- 6 x}{2}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $- 6$ and $2$ .

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Step 1.2.3.1.1

Factor $2$ out of $- 6 x$ .

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

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Factor $2$ out of $2$ .

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

Step 1.2.3.1.2.2

Cancel the common factor.

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

Step 1.2.3.1.2.3

Rewrite the expression.

$y = \frac{9}{2} + \frac{- 3 x}{1}$

Step 1.2.3.1.2.4

Divide $- 3 x$ by $1$ .

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

Step 2

Find the slope when $y = \frac{9}{2} - 3 x$ .

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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 $\frac{9}{2}$ and $- 3 x$ .

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

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

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

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Step 4.1

Move the negative in front of the fraction.

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

Step 4.2

Multiply $- - \frac{1}{3}$ .

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Step 4.2.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2.2

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

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

Step 5

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

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Step 5.1

Use the slope $\frac{1}{3}$ and a given point $(- 3.9, - 1.4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 1.4) = \frac{1}{3} \cdot (x - (- 3.9))$

Step 5.2

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

$y + 1.4 = \frac{1}{3} \cdot (x + 3.9)$

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 $\frac{1}{3} \cdot (x + 3.9)$ .

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Step 6.1.1.1

Rewrite.

$y + 1.4 = 0 + 0 + \frac{1}{3} \cdot (x + 3.9)$

Step 6.1.1.2

Simplify by adding zeros.

$y + 1.4 = \frac{1}{3} \cdot (x + 3.9)$

Step 6.1.1.3

Apply the distributive property.

$y + 1.4 = \frac{1}{3} x + \frac{1}{3} \cdot 3.9$

Step 6.1.1.4

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

$y + 1.4 = \frac{x}{3} + \frac{1}{3} \cdot 3.9$

Step 6.1.1.5

Combine $\frac{1}{3}$ and $3.9$ .

$y + 1.4 = \frac{x}{3} + \frac{3.9}{3}$

Step 6.1.1.6

Divide $3.9$ by $3$ .

$y + 1.4 = \frac{x}{3} + 1.3$

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

Subtract $1.4$ from both sides of the equation.

$y = \frac{x}{3} + 1.3 - 1.4$

Step 6.1.2.2

Subtract $1.4$ from $1.3$ .

$y = \frac{x}{3} - 0.1$

Step 6.2

Reorder terms.

$y = \frac{1}{3} x - 0.1$

Step 7