Finite Math Examples

$x - 4 y = - 4$ , $(- 4, 3)$

Step 1

Solve $x - 4 y = - 4$ .

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Step 1.1

Subtract $x$ from both sides of the equation.

$- 4 y = - 4 - x$

Step 1.2

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

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Step 1.2.1

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

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

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.

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

Step 1.2.2.1.2

Divide $y$ by $1$ .

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

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 $- 4$ by $- 4$ .

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

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

$y = 1 + \frac{x}{4}$

Step 2

Find the slope when $y = 1 + \frac{x}{4}$ .

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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 $1$ and $\frac{x}{4}$ .

$y = \frac{x}{4} + 1$

Step 2.1.3

Reorder terms.

$y = \frac{1}{4} x + 1$

Step 2.2

Using the slope-intercept form, the slope is $\frac{1}{4}$ .

$m = \frac{1}{4}$

Step 3

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

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

Step 4

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

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Multiply $- (1 \cdot 4)$ .

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Step 4.2.1

Multiply $4$ by $1$ .

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

Step 4.2.2

Multiply $- 1$ by $4$ .

$m_{perpendicular} = - 4$

Step 5

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

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Step 5.1

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

$y - (3) = - 4 \cdot (x - (- 4))$

Step 5.2

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

$y - 3 = - 4 \cdot (x + 4)$

Step 6

Solve for $y$ .

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Step 6.1

Simplify $- 4 \cdot (x + 4)$ .

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Step 6.1.1

Rewrite.

$y - 3 = 0 + 0 - 4 \cdot (x + 4)$

Step 6.1.2

Simplify by adding zeros.

$y - 3 = - 4 \cdot (x + 4)$

Step 6.1.3

Apply the distributive property.

$y - 3 = - 4 x - 4 \cdot 4$

Step 6.1.4

Multiply $- 4$ by $4$ .

$y - 3 = - 4 x - 16$

Step 6.2

Move all terms not containing $y$ to the right side of the equation.

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Step 6.2.1

Add $3$ to both sides of the equation.

$y = - 4 x - 16 + 3$

Step 6.2.2

Add $- 16$ and $3$ .

$y = - 4 x - 13$

Step 7