Pre-Algebra Examples

$- x (- 3 y + 13) + 5 y = 11$

Step 1

Simplify.

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Step 1.1

Add $13 x$ to both sides of the equation.

$3 x y + 5 y = 11 + 13 x$

Step 1.2

Factor $y$ out of $3 x y + 5 y$ .

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Step 1.2.1

Factor $y$ out of $3 x y$ .

$y (3 x) + 5 y = 11 + 13 x$

Step 1.2.2

Factor $y$ out of $5 y$ .

$y (3 x) + y \cdot 5 = 11 + 13 x$

Step 1.2.3

Factor $y$ out of $y (3 x) + y \cdot 5$ .

$y (3 x + 5) = 11 + 13 x$

Step 1.3

Divide each term in $y (3 x + 5) = 11 + 13 x$ by $3 x + 5$ and simplify.

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Step 1.3.1

Divide each term in $y (3 x + 5) = 11 + 13 x$ by $3 x + 5$ .

$\frac{y (3 x + 5)}{3 x + 5} = \frac{11}{3 x + 5} + \frac{13 x}{3 x + 5}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $3 x + 5$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{y (3 x + 5)}{3 x + 5} = \frac{11}{3 x + 5} + \frac{13 x}{3 x + 5}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{11}{3 x + 5} + \frac{13 x}{3 x + 5}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Combine the numerators over the common denominator.

$y = \frac{11 + 13 x}{3 x + 5}$

Step 2

Find where the expression $\frac{11 + 13 x}{3 x + 5}$ is undefined.

$x = - \frac{5}{3}$

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 $a = 13$ and $b = 3$ .

$y = \frac{13}{3}$

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: $x = - \frac{5}{3}$

Horizontal Asymptotes: $y = \frac{13}{3}$

No Oblique Asymptotes

Step 8