Algebra Examples

$4 x = - 3 y + x y$

Step 1

Rewrite the equation as $- 3 y + x y = 4 x$ .

$- 3 y + x y = 4 x$

Step 2

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

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Step 2.1

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

$y \cdot - 3 + x y = 4 x$

Step 2.2

Factor $y$ out of $x y$ .

$y \cdot - 3 + y x = 4 x$

Step 2.3

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

$y (- 3 + x) = 4 x$

Step 3

Divide each term in $y (- 3 + x) = 4 x$ by $- 3 + x$ and simplify.

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Step 3.1

Divide each term in $y (- 3 + x) = 4 x$ by $- 3 + x$ .

$\frac{y (- 3 + x)}{- 3 + x} = \frac{4 x}{- 3 + x}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Cancel the common factor.

$\frac{y (- 3 + x)}{- 3 + x} = \frac{4 x}{- 3 + x}$

Step 3.2.1.2

Divide $y$ by $1$ .

$y = \frac{4 x}{- 3 + x}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

$y = \frac{4 x}{- 1 (3) + x}$

Step 3.3.2

Factor $- 1$ out of $x$ .

$y = \frac{4 x}{- 1 (3) - 1 (- x)}$

Step 3.3.3

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

$y = \frac{4 x}{- 1 (3 - x)}$

Step 3.3.4

Move the negative in front of the fraction.

$y = - \frac{4 x}{3 - x}$

Step 4

Set the denominator in $\frac{4 x}{3 - x}$ equal to $0$ to find where the expression is undefined.

$3 - x = 0$

Step 5

Solve for $x$ .

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Step 5.1

Subtract $3$ from both sides of the equation.

$- x = - 3$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2.2

Divide $x$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 6

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 7

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

Interval Notation:

$(- \infty, 4) \cup (4, \infty)$

Set-Builder Notation:

${y | y \neq 4}$

Step 8

Determine the domain and range.

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

Range: $(- \infty, 4) \cup (4, \infty), {y | y \neq 4}$

Step 9