Algebra Examples

$\frac{3}{y - 5} - \frac{5 y - 16}{y^{2} - 7 y + 10}$

Step 1

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

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Step 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 1.2

Write the factored form using these integers.

$\frac{3}{y - 5} - \frac{5 y - 16}{(y - 5) (y - 2)}$

Step 2

To write $\frac{3}{y - 5}$ as a fraction with a common denominator, multiply by $\frac{y - 2}{y - 2}$ .

$\frac{3}{y - 5} \cdot \frac{y - 2}{y - 2} - \frac{5 y - 16}{(y - 5) (y - 2)}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{3}{y - 5}$ by $\frac{y - 2}{y - 2}$ .

$\frac{3 (y - 2)}{(y - 5) (y - 2)} - \frac{5 y - 16}{(y - 5) (y - 2)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{3 (y - 2) - (5 y - 16)}{(y - 5) (y - 2)}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{3 y + 3 \cdot - 2 - (5 y - 16)}{(y - 5) (y - 2)}$

Step 4.2

Multiply $3$ by $- 2$ .

$\frac{3 y - 6 - (5 y - 16)}{(y - 5) (y - 2)}$

Step 4.3

Apply the distributive property.

$\frac{3 y - 6 - (5 y) - - 16}{(y - 5) (y - 2)}$

Step 4.4

Multiply $5$ by $- 1$ .

$\frac{3 y - 6 - 5 y - - 16}{(y - 5) (y - 2)}$

Step 4.5

Multiply $- 1$ by $- 16$ .

$\frac{3 y - 6 - 5 y + 16}{(y - 5) (y - 2)}$

Step 4.6

Subtract $5 y$ from $3 y$ .

$\frac{- 2 y - 6 + 16}{(y - 5) (y - 2)}$

Step 4.7

Add $- 6$ and $16$ .

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

Step 4.8

Factor $2$ out of $- 2 y + 10$ .

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Step 4.8.1

Factor $2$ out of $- 2 y$ .

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

Step 4.8.2

Factor $2$ out of $10$ .

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

Step 4.8.3

Factor $2$ out of $2 (- y) + 2 (5)$ .

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

Step 5

Cancel the common factor of $- y + 5$ and $y - 5$ .

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Step 5.1

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

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

Step 5.2

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{2 (- (y) - 1 (- 5))}{(y - 5) (y - 2)}$

Step 5.3

Factor $- 1$ out of $- (y) - 1 (- 5)$ .

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

Step 5.4

Rewrite $- (y - 5)$ as $- 1 (y - 5)$ .

$\frac{2 (- 1 (y - 5))}{(y - 5) (y - 2)}$

Step 5.5

Cancel the common factor.

$\frac{2 (- 1 (y - 5))}{(y - 5) (y - 2)}$

Step 5.6

Rewrite the expression.

$\frac{2 \cdot (- 1)}{y - 2}$

Step 6

Multiply $2$ by $- 1$ .

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

Step 7

Move the negative in front of the fraction.

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