Algebra Examples

through $(8, - 3)$ ; perpendicular to $8 y = x - 16$

Step 1

Divide each term in $8 y = x - 16$ by $8$ and simplify.

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Step 1.1

Divide each term in $8 y = x - 16$ by $8$ .

$\frac{8 y}{8} = \frac{x}{8} + \frac{- 16}{8}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $8$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{8 y}{8} = \frac{x}{8} + \frac{- 16}{8}$

Step 1.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{8} + \frac{- 16}{8}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $- 16$ by $8$ .

$y = \frac{x}{8} - 2$

Step 2

Find the slope when $y = \frac{x}{8} - 2$ .

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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 terms.

$y = \frac{1}{8} x - 2$

Step 2.2

Using the slope-intercept form, the slope is $\frac{1}{8}$ .

$m = \frac{1}{8}$

Step 3

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

$m_{perpendicular} = - \frac{1}{\frac{1}{8}}$

Step 4

Simplify $- \frac{1}{\frac{1}{8}}$ to find the slope of the perpendicular line.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 \cdot 8)$

Step 4.2

Multiply $- (1 \cdot 8)$ .

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Step 4.2.1

Multiply $8$ by $1$ .

$m_{perpendicular} = - 1 \cdot 8$

Step 4.2.2

Multiply $- 1$ by $8$ .

$m_{perpendicular} = - 8$

Step 5

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

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Step 5.1

Use the slope $- 8$ and a given point $(8, - 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 3) = - 8 \cdot (x - (8))$

Step 5.2

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

$y + 3 = - 8 \cdot (x - 8)$

Step 6

Solve for $y$ .

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Step 6.1

Simplify $- 8 \cdot (x - 8)$ .

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Step 6.1.1

Rewrite.

$y + 3 = 0 + 0 - 8 \cdot (x - 8)$

Step 6.1.2

Simplify by adding zeros.

$y + 3 = - 8 \cdot (x - 8)$

Step 6.1.3

Apply the distributive property.

$y + 3 = - 8 x - 8 \cdot - 8$

Step 6.1.4

Multiply $- 8$ by $- 8$ .

$y + 3 = - 8 x + 64$

Step 6.2

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

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Step 6.2.1

Subtract $3$ from both sides of the equation.

$y = - 8 x + 64 - 3$

Step 6.2.2

Subtract $3$ from $64$ .

$y = - 8 x + 61$

Step 7