Trigonometry Examples

$y x + 3 y - 5 x = 0$

Step 1

Simplify.

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Step 1.1

Add $5 x$ to both sides of the equation.

$y x + 3 y = 5 x$

Step 1.2

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

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Step 1.2.1

Factor $y$ out of $y x$ .

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

Step 1.2.2

Factor $y$ out of $3 y$ .

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

Step 1.2.3

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

$y (x + 3) = 5 x$

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $x + 3$ .

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Step 1.3.2.1.1

Cancel the common factor.

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

Step 1.3.2.1.2

Divide $y$ by $1$ .

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

Step 2

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

$x = - 3$

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 $a = 5$ and $b = 1$ .

$y = 5$

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: $x = - 3$

Horizontal Asymptotes: $y = 5$

No Oblique Asymptotes

Step 8