Pre-Algebra Examples

$x y - 6 y + 3 x - 8$

Step 1

Simplify.

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Step 1.1

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

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Step 1.1.1

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

$x y - 6 y - 8 = - 3 x$

Step 1.1.2

Add $8$ to both sides of the equation.

$x y - 6 y = - 3 x + 8$

Step 1.2

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

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Step 1.2.1

Factor $y$ out of $x y$ .

$y x - 6 y = - 3 x + 8$

Step 1.2.2

Factor $y$ out of $- 6 y$ .

$y x + y \cdot - 6 = - 3 x + 8$

Step 1.2.3

Factor $y$ out of $y x + y \cdot - 6$ .

$y (x - 6) = - 3 x + 8$

Step 1.3

Divide each term in $y (x - 6) = - 3 x + 8$ by $x - 6$ and simplify.

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Step 1.3.1

Divide each term in $y (x - 6) = - 3 x + 8$ by $x - 6$ .

$\frac{y (x - 6)}{x - 6} = \frac{- 3 x}{x - 6} + \frac{8}{x - 6}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $x - 6$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{y (x - 6)}{x - 6} = \frac{- 3 x}{x - 6} + \frac{8}{x - 6}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 3 x}{x - 6} + \frac{8}{x - 6}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Combine the numerators over the common denominator.

$y = \frac{- 3 x + 8}{x - 6}$

Step 1.3.3.2

Factor $- 1$ out of $- 3 x$ .

$y = \frac{- (3 x) + 8}{x - 6}$

Step 1.3.3.3

Rewrite $8$ as $- 1 (- 8)$ .

$y = \frac{- (3 x) - 1 (- 8)}{x - 6}$

Step 1.3.3.4

Factor $- 1$ out of $- (3 x) - 1 (- 8)$ .

$y = \frac{- (3 x - 8)}{x - 6}$

Step 1.3.3.5

Simplify the expression.

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Step 1.3.3.5.1

Rewrite $- (3 x - 8)$ as $- 1 (3 x - 8)$ .

$y = \frac{- 1 (3 x - 8)}{x - 6}$

Step 1.3.3.5.2

Move the negative in front of the fraction.

$y = - \frac{3 x - 8}{x - 6}$

Step 2

Find where the expression $- \frac{3 x - 8}{x - 6}$ is undefined.

$x = 6$

Step 3

Consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 4

Find $n$ and $m$ .

$n = 1$

$m = 1$

Step 5

Since $n = m$ , the horizontal asymptote is the line $y = \frac{a}{b}$ where $a = - 3$ and $b = 1$ .

$y = - 3$

Step 6

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

Step 7

This is the set of all asymptotes.

Vertical Asymptotes: $x = 6$

Horizontal Asymptotes: $y = - 3$

No Oblique Asymptotes

Step 8