Precalculus Examples

$\frac{a x - a y}{5 x^{2} y^{2}} \times (- \frac{5 x y}{b y - b x})$

Step 1

Set the denominator in $\frac{a x - a y}{5 x^{2} y^{2}}$ equal to $0$ to find where the expression is undefined.

$5 x^{2} y^{2} = 0$

Step 2

Solve for $a$ .

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Step 2.1

Divide each term in $5 x^{2} y^{2} = 0$ by $5 y^{2}$ and simplify.

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Step 2.1.1

Divide each term in $5 x^{2} y^{2} = 0$ by $5 y^{2}$ .

$\frac{5 x^{2} y^{2}}{5 y^{2}} = \frac{0}{5 y^{2}}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $5$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{5 x^{2} y^{2}}{5 y^{2}} = \frac{0}{5 y^{2}}$

Step 2.1.2.1.2

Rewrite the expression.

$\frac{x^{2} y^{2}}{y^{2}} = \frac{0}{5 y^{2}}$

Step 2.1.2.2

Cancel the common factor of $y^{2}$ .

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Step 2.1.2.2.1

Cancel the common factor.

$\frac{x^{2} y^{2}}{y^{2}} = \frac{0}{5 y^{2}}$

Step 2.1.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{0}{5 y^{2}}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Cancel the common factor of $0$ and $5$ .

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Step 2.1.3.1.1

Factor $5$ out of $0$ .

$x^{2} = \frac{5 (0)}{5 y^{2}}$

Step 2.1.3.1.2

Cancel the common factors.

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Step 2.1.3.1.2.1

Factor $5$ out of $5 y^{2}$ .

$x^{2} = \frac{5 (0)}{5 (y^{2})}$

Step 2.1.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{5 \cdot 0}{5 y^{2}}$

Step 2.1.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{0}{y^{2}}$

Step 2.1.3.2

Divide $0$ by $y^{2}$ .

$x^{2} = 0$

Step 2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 2.3

Simplify $\pm \sqrt{0}$ .

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Step 2.3.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 2.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3

Set the denominator in $\frac{5 x y}{b y - b x}$ equal to $0$ to find where the expression is undefined.

$b y - b x = 0$

Step 4

Solve for $a$ .

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Step 4.1

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

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Step 4.1.1

Factor $b$ out of $b y$ .

$b (y) - b x = 0$

Step 4.1.2

Factor $b$ out of $- b x$ .

$b (y) + b (- 1 x) = 0$

Step 4.1.3

Factor $b$ out of $b (y) + b (- 1 x)$ .

$b (y - 1 x) = 0$

Step 4.2

Rewrite $- 1 x$ as $- x$ .

$b (y - x) = 0$

Step 4.3

Divide each term in $b (y - x) = 0$ by $y - x$ and simplify.

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Step 4.3.1

Divide each term in $b (y - x) = 0$ by $y - x$ .

$\frac{b (y - x)}{y - x} = \frac{0}{y - x}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $y - x$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{b (y - x)}{y - x} = \frac{0}{y - x}$

Step 4.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{0}{y - x}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $0$ by $y - x$ .

$b = 0$

Step 5

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

Interval Notation:

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

Set-Builder Notation:

${a | a \neq 0}$