Pre-Algebra Examples

$\frac{5 x^{2} - 5 y^{2}}{6 y^{2} - 6 x^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

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

$\frac{5 (x^{2}) - 5 y^{2}}{6 y^{2} - 6 x^{2}}$

Step 1.1.2

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

$\frac{5 (x^{2}) + 5 (- y^{2})}{6 y^{2} - 6 x^{2}}$

Step 1.1.3

Factor $5$ out of $5 (x^{2}) + 5 (- y^{2})$ .

$\frac{5 (x^{2} - y^{2})}{6 y^{2} - 6 x^{2}}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = y$ .

$\frac{5 (x + y) (x - y)}{6 y^{2} - 6 x^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $6$ out of $6 y^{2} - 6 x^{2}$ .

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Step 2.1.1

Factor $6$ out of $6 y^{2}$ .

$\frac{5 (x + y) (x - y)}{6 (y^{2}) - 6 x^{2}}$

Step 2.1.2

Factor $6$ out of $- 6 x^{2}$ .

$\frac{5 (x + y) (x - y)}{6 (y^{2}) + 6 (- x^{2})}$

Step 2.1.3

Factor $6$ out of $6 (y^{2}) + 6 (- x^{2})$ .

$\frac{5 (x + y) (x - y)}{6 (y^{2} - x^{2})}$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = x$ .

$\frac{5 (x + y) (x - y)}{6 (y + x) (y - x)}$

Step 3

Cancel the common factor of $x + y$ and $y + x$ .

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Step 3.1

Reorder terms.

$\frac{5 (x + y) (x - y)}{6 (x + y) (y - x)}$

Step 3.2

Cancel the common factor.

$\frac{5 (x + y) (x - y)}{6 (x + y) (y - x)}$

Step 3.3

Rewrite the expression.

$\frac{5 (x - y)}{6 (y - x)}$

Step 4

Cancel the common factor of $x - y$ and $y - x$ .

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Step 4.1

Factor $- 1$ out of $x$ .

$\frac{5 (- 1 (- x) - y)}{6 (y - x)}$

Step 4.2

Factor $- 1$ out of $- y$ .

$\frac{5 (- 1 (- x) - (y))}{6 (y - x)}$

Step 4.3

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

$\frac{5 (- 1 (- x + y))}{6 (y - x)}$

Step 4.4

Reorder terms.

$\frac{5 (- 1 (- x + y))}{6 (- x + y)}$

Step 4.5

Cancel the common factor.

$\frac{5 (- 1 (- x + y))}{6 (- x + y)}$

Step 4.6

Rewrite the expression.

$\frac{5 \cdot (- 1)}{6}$

Step 5

Multiply $5$ by $- 1$ .

$\frac{- 5}{6}$

Step 6

Move the negative in front of the fraction.

$- \frac{5}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{5}{6}$

Decimal Form:

$- 0.8\overline{3}$