Algebra Examples

$\frac{y + 7}{y^{2} - 7 y + 10} - \frac{7}{y^{2} - 25}$

Step 1

Simplify each term.

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Step 1.1

Factor $y^{2} - 7 y + 10$ using the AC method.

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Step 1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $10$ and whose sum is $- 7$ .

$- 5, - 2$

Step 1.1.2

Write the factored form using these integers.

$\frac{y + 7}{(y - 5) (y - 2)} - \frac{7}{y^{2} - 25}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

$\frac{y + 7}{(y - 5) (y - 2)} - \frac{7}{y^{2} - 5^{2}}$

Step 1.2.2

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 $b = 5$ .

$\frac{y + 7}{(y - 5) (y - 2)} - \frac{7}{(y + 5) (y - 5)}$

Step 2

To write $\frac{y + 7}{(y - 5) (y - 2)}$ as a fraction with a common denominator, multiply by $\frac{y + 5}{y + 5}$ .

$\frac{y + 7}{(y - 5) (y - 2)} \cdot \frac{y + 5}{y + 5} - \frac{7}{(y + 5) (y - 5)}$

Step 3

To write $- \frac{7}{(y + 5) (y - 5)}$ as a fraction with a common denominator, multiply by $\frac{y - 2}{y - 2}$ .

$\frac{y + 7}{(y - 5) (y - 2)} \cdot \frac{y + 5}{y + 5} - \frac{7}{(y + 5) (y - 5)} \cdot \frac{y - 2}{y - 2}$

Step 4

Write each expression with a common denominator of $(y - 5) (y - 2) (y + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{y + 7}{(y - 5) (y - 2)}$ by $\frac{y + 5}{y + 5}$ .

$\frac{(y + 7) (y + 5)}{(y - 5) (y - 2) (y + 5)} - \frac{7}{(y + 5) (y - 5)} \cdot \frac{y - 2}{y - 2}$

Step 4.2

Multiply $\frac{7}{(y + 5) (y - 5)}$ by $\frac{y - 2}{y - 2}$ .

$\frac{(y + 7) (y + 5)}{(y - 5) (y - 2) (y + 5)} - \frac{7 (y - 2)}{(y + 5) (y - 5) (y - 2)}$

Step 4.3

Reorder the factors of $(y - 5) (y - 2) (y + 5)$ .

$\frac{(y + 7) (y + 5)}{(y + 5) (y - 2) (y - 5)} - \frac{7 (y - 2)}{(y + 5) (y - 5) (y - 2)}$

Step 4.4

Reorder the factors of $(y + 5) (y - 5) (y - 2)$ .

$\frac{(y + 7) (y + 5)}{(y + 5) (y - 2) (y - 5)} - \frac{7 (y - 2)}{(y + 5) (y - 2) (y - 5)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(y + 7) (y + 5) - 7 (y - 2)}{(y + 5) (y - 2) (y - 5)}$

Step 6

Simplify the numerator.

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Step 6.1

Expand $(y + 7) (y + 5)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{y (y + 5) + 7 (y + 5) - 7 (y - 2)}{(y + 5) (y - 2) (y - 5)}$

Step 6.1.2

Apply the distributive property.

$\frac{y \cdot y + y \cdot 5 + 7 (y + 5) - 7 (y - 2)}{(y + 5) (y - 2) (y - 5)}$

Step 6.1.3

Apply the distributive property.

$\frac{y \cdot y + y \cdot 5 + 7 y + 7 \cdot 5 - 7 (y - 2)}{(y + 5) (y - 2) (y - 5)}$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $y$ by $y$ .

$\frac{y^{2} + y \cdot 5 + 7 y + 7 \cdot 5 - 7 (y - 2)}{(y + 5) (y - 2) (y - 5)}$

Step 6.2.1.2

Move $5$ to the left of $y$ .

$\frac{y^{2} + 5 \cdot y + 7 y + 7 \cdot 5 - 7 (y - 2)}{(y + 5) (y - 2) (y - 5)}$

Step 6.2.1.3

Multiply $7$ by $5$ .

$\frac{y^{2} + 5 y + 7 y + 35 - 7 (y - 2)}{(y + 5) (y - 2) (y - 5)}$

Step 6.2.2

Add $5 y$ and $7 y$ .

$\frac{y^{2} + 12 y + 35 - 7 (y - 2)}{(y + 5) (y - 2) (y - 5)}$

Step 6.3

Apply the distributive property.

$\frac{y^{2} + 12 y + 35 - 7 y - 7 \cdot - 2}{(y + 5) (y - 2) (y - 5)}$

Step 6.4

Multiply $- 7$ by $- 2$ .

$\frac{y^{2} + 12 y + 35 - 7 y + 14}{(y + 5) (y - 2) (y - 5)}$

Step 6.5

Subtract $7 y$ from $12 y$ .

$\frac{y^{2} + 5 y + 35 + 14}{(y + 5) (y - 2) (y - 5)}$

Step 6.6

Add $35$ and $14$ .

$\frac{y^{2} + 5 y + 49}{(y + 5) (y - 2) (y - 5)}$