Basic Math Examples

$\frac{y - 2}{y^{2} - 5 y} \div \frac{1}{y} - \frac{3}{y^{2} - 10 y + 25} \cdot (y - 5)$

Step 1

Simplify each term.

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Step 1.1

To divide by a fraction, multiply by its reciprocal.

$\frac{y - 2}{y^{2} - 5 y} y - \frac{3}{y^{2} - 10 y + 25} \cdot (y - 5)$

Step 1.2

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

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Step 1.2.1

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

$\frac{y - 2}{y \cdot y - 5 y} y - \frac{3}{y^{2} - 10 y + 25} \cdot (y - 5)$

Step 1.2.2

Factor $y$ out of $- 5 y$ .

$\frac{y - 2}{y \cdot y + y \cdot - 5} y - \frac{3}{y^{2} - 10 y + 25} \cdot (y - 5)$

Step 1.2.3

Factor $y$ out of $y \cdot y + y \cdot - 5$ .

$\frac{y - 2}{y (y - 5)} y - \frac{3}{y^{2} - 10 y + 25} \cdot (y - 5)$

Step 1.3

Cancel the common factor of $y$ .

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Step 1.3.1

Cancel the common factor.

$\frac{y - 2}{y (y - 5)} y - \frac{3}{y^{2} - 10 y + 25} \cdot (y - 5)$

Step 1.3.2

Rewrite the expression.

$\frac{y - 2}{y - 5} - \frac{3}{y^{2} - 10 y + 25} \cdot (y - 5)$

Step 1.4

Factor using the perfect square rule.

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Step 1.4.1

Rewrite $25$ as $5^{2}$ .

$\frac{y - 2}{y - 5} - \frac{3}{y^{2} - 10 y + 5^{2}} \cdot (y - 5)$

Step 1.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 y = 2 \cdot y \cdot 5$

Step 1.4.3

Rewrite the polynomial.

$\frac{y - 2}{y - 5} - \frac{3}{y^{2} - 2 \cdot y \cdot 5 + 5^{2}} \cdot (y - 5)$

Step 1.4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = y$ and $b = 5$ .

$\frac{y - 2}{y - 5} - \frac{3}{{(y - 5)}^{2}} \cdot (y - 5)$

Step 1.5

Cancel the common factor of $y - 5$ .

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Step 1.5.1

Move the leading negative in $- \frac{3}{{(y - 5)}^{2}}$ into the numerator.

$\frac{y - 2}{y - 5} + \frac{- 3}{{(y - 5)}^{2}} \cdot (y - 5)$

Step 1.5.2

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

$\frac{y - 2}{y - 5} + \frac{- 3}{(y - 5) (y - 5)} \cdot (y - 5)$

Step 1.5.3

Cancel the common factor.

$\frac{y - 2}{y - 5} + \frac{- 3}{(y - 5) (y - 5)} \cdot (y - 5)$

Step 1.5.4

Rewrite the expression.

$\frac{y - 2}{y - 5} + \frac{- 3}{y - 5}$

Step 1.6

Move the negative in front of the fraction.

$\frac{y - 2}{y - 5} - \frac{3}{y - 5}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{y - 2 - 3}{y - 5}$

Step 2.2

Subtract $3$ from $- 2$ .

$\frac{y - 5}{y - 5}$

Step 2.3

Cancel the common factor of $y - 5$ .

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Step 2.3.1

Cancel the common factor.

$\frac{y - 5}{y - 5}$

Step 2.3.2

Rewrite the expression.

$1$