Finite Math Examples

$3 x - 4 y = 8$ , $(6, - 2)$

Step 1

Solve $3 x - 4 y = 8$ .

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Step 1.1

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

$- 4 y = 8 - 3 x$

Step 1.2

Divide each term in $- 4 y = 8 - 3 x$ by $- 4$ and simplify.

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Step 1.2.1

Divide each term in $- 4 y = 8 - 3 x$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{8}{- 4} + \frac{- 3 x}{- 4}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{8}{- 4} + \frac{- 3 x}{- 4}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{8}{- 4} + \frac{- 3 x}{- 4}$

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

Divide $8$ by $- 4$ .

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

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

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

Step 2

Find the slope when $y = - 2 + \frac{3 x}{4}$ .

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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 $- 2$ and $\frac{3 x}{4}$ .

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

Step 2.1.3

Reorder terms.

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

Step 2.2

Using the slope-intercept form, the slope is $\frac{3}{4}$ .

$m = \frac{3}{4}$

Step 3

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

$m_{perpendicular} = - \frac{1}{\frac{3}{4}}$

Step 4

Simplify $- \frac{1}{\frac{3}{4}}$ to find the slope of the perpendicular line.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{4}{3}))$

Step 4.2

Multiply $\frac{4}{3}$ by $1$ .

$m_{perpendicular} = - \frac{4}{3}$

Step 5

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

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Step 5.1

Use the slope $- \frac{4}{3}$ and a given point $(6, - 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 2) = - \frac{4}{3} \cdot (x - (6))$

Step 5.2

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

$y + 2 = - \frac{4}{3} \cdot (x - 6)$

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 $- \frac{4}{3} \cdot (x - 6)$ .

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Step 6.1.1.1

Rewrite.

$y + 2 = 0 + 0 - \frac{4}{3} \cdot (x - 6)$

Step 6.1.1.2

Simplify terms.

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Step 6.1.1.2.1

Apply the distributive property.

$y + 2 = - \frac{4}{3} x - \frac{4}{3} \cdot - 6$

Step 6.1.1.2.2

Combine $x$ and $\frac{4}{3}$ .

$y + 2 = - \frac{x \cdot 4}{3} - \frac{4}{3} \cdot - 6$

Step 6.1.1.2.3

Cancel the common factor of $3$ .

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Step 6.1.1.2.3.1

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$y + 2 = - \frac{x \cdot 4}{3} + \frac{- 4}{3} \cdot - 6$

Step 6.1.1.2.3.2

Factor $3$ out of $- 6$ .

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

Step 6.1.1.2.3.3

Cancel the common factor.

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

Step 6.1.1.2.3.4

Rewrite the expression.

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

Step 6.1.1.2.4

Multiply $- 4$ by $- 2$ .

$y + 2 = - \frac{x \cdot 4}{3} + 8$

Step 6.1.1.3

Move $4$ to the left of $x$ .

$y + 2 = - \frac{4 x}{3} + 8$

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 $2$ from both sides of the equation.

$y = - \frac{4 x}{3} + 8 - 2$

Step 6.1.2.2

Subtract $2$ from $8$ .

$y = - \frac{4 x}{3} + 6$

Step 6.2

Reorder terms.

$y = - (\frac{4}{3} x) + 6$

Step 6.3

Remove parentheses.

$y = - \frac{4}{3} x + 6$

Step 7