Calculus Examples

$\frac{x y - x^{2}}{x - 9} = 7$

Step 1

Multiply both sides by $x - 9$ .

$\frac{x y - x^{2}}{x - 9} (x - 9) = 7 (x - 9)$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Simplify the left side.

Tap for more steps...

Step 2.1.1

Cancel the common factor of $x - 9$ .

Tap for more steps...

Step 2.1.1.1

Cancel the common factor.

$\frac{x y - x^{2}}{x - 9} (x - 9) = 7 (x - 9)$

Step 2.1.1.2

Rewrite the expression.

$x y - x^{2} = 7 (x - 9)$

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

Simplify $7 (x - 9)$ .

Tap for more steps...

Step 2.2.1.1

Apply the distributive property.

$x y - x^{2} = 7 x + 7 \cdot - 9$

Step 2.2.1.2

Multiply $7$ by $- 9$ .

$x y - x^{2} = 7 x - 63$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Add $x^{2}$ to both sides of the equation.

$x y = 7 x - 63 + x^{2}$

Step 3.2

Divide each term in $x y = 7 x - 63 + x^{2}$ by $x$ and simplify.

Tap for more steps...

Step 3.2.1

Divide each term in $x y = 7 x - 63 + x^{2}$ by $x$ .

$\frac{x y}{x} = \frac{7 x}{x} + \frac{- 63}{x} + \frac{x^{2}}{x}$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{x y}{x} = \frac{7 x}{x} + \frac{- 63}{x} + \frac{x^{2}}{x}$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{7 x}{x} + \frac{- 63}{x} + \frac{x^{2}}{x}$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Simplify each term.

Tap for more steps...

Step 3.2.3.1.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.2.3.1.1.1

Cancel the common factor.

$y = \frac{7 x}{x} + \frac{- 63}{x} + \frac{x^{2}}{x}$

Step 3.2.3.1.1.2

Divide $7$ by $1$ .

$y = 7 + \frac{- 63}{x} + \frac{x^{2}}{x}$

Step 3.2.3.1.2

Move the negative in front of the fraction.

$y = 7 - \frac{63}{x} + \frac{x^{2}}{x}$

Step 3.2.3.1.3

Cancel the common factor of $x^{2}$ and $x$ .

Tap for more steps...

Step 3.2.3.1.3.1

Factor $x$ out of $x^{2}$ .

$y = 7 - \frac{63}{x} + \frac{x \cdot x}{x}$

Step 3.2.3.1.3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.3.1.3.2.1

Raise $x$ to the power of $1$ .

$y = 7 - \frac{63}{x} + \frac{x \cdot x}{x^{1}}$

Step 3.2.3.1.3.2.2

Factor $x$ out of $x^{1}$ .

$y = 7 - \frac{63}{x} + \frac{x \cdot x}{x \cdot 1}$

Step 3.2.3.1.3.2.3

Cancel the common factor.

$y = 7 - \frac{63}{x} + \frac{x \cdot x}{x \cdot 1}$

Step 3.2.3.1.3.2.4

Rewrite the expression.

$y = 7 - \frac{63}{x} + \frac{x}{1}$

Step 3.2.3.1.3.2.5

Divide $x$ by $1$ .

$y = 7 - \frac{63}{x} + x$

Step 4

Set the denominator in $\frac{63}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 5

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 0}$

Step 6