Find the Domain and Range 3y-xy=2

$3 y - x y = 2$

Step 1

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

Factor $y$ out of $3 y$ .

$y \cdot 3 - x y = 2$

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

$y \cdot 3 + y (- x) = 2$

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

$y (3 - x) = 2$

Step 2

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

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

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

Simplify the left side.

Cancel the common factor of $3 - x$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

$y = \frac{2}{3 - x}$

Step 3

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

$3 - x = 0$

Step 4

Solve for $x$ .

Subtract $3$ from both sides of the equation.

$- x = - 3$

Divide each term in $- x = - 3$ by $- 1$ and simplify.

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 3}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Simplify the right side.

Divide $- 3$ by $- 1$ .

$x = 3$

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

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${y | y \neq 0}$

Step 7

Determine the domain and range.

Domain: $(- \infty, 3) \cup (3, \infty), {x | x \neq 3}$

Range: $(- \infty, 0) \cup (0, \infty), {y | y \neq 0}$

Step 8

Find the Domain and Range 3y-xy=2

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