Find the Domain xy+3y=2x+1

$x y + 3 y = 2 x + 1$

Step 1

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

Factor $y$ out of $x y$ .

$y x + 3 y = 2 x + 1$

Factor $y$ out of $3 y$ .

$y x + y \cdot 3 = 2 x + 1$

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

$y (x + 3) = 2 x + 1$

Step 2

Divide each term in $y (x + 3) = 2 x + 1$ by $x + 3$ and simplify.

Divide each term in $y (x + 3) = 2 x + 1$ by $x + 3$ .

$\frac{y (x + 3)}{x + 3} = \frac{2 x}{x + 3} + \frac{1}{x + 3}$

Simplify the left side.

Cancel the common factor of $x + 3$ .

Cancel the common factor.

$\frac{y (x + 3)}{x + 3} = \frac{2 x}{x + 3} + \frac{1}{x + 3}$

Divide $y$ by $1$ .

$y = \frac{2 x}{x + 3} + \frac{1}{x + 3}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{2 x + 1}{x + 3}$

Step 3

Set the denominator in $\frac{2 x + 1}{x + 3}$ equal to $0$ to find where the expression is undefined.

$x + 3 = 0$

Step 4

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 5

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 3) \cup (- 3, \infty)$

Set-Builder Notation:

${x | x \neq - 3}$

Step 6