Algebra Examples

$\frac{4 y}{y^{2} - 10 y + 21} - \frac{1}{y - 7}$

Step 1

Factor $y^{2} - 10 y + 21$ 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 $21$ and whose sum is $- 10$ .

$- 7, - 3$

Step 1.2

Write the factored form using these integers.

$\frac{4 y}{(y - 7) (y - 3)} - \frac{1}{y - 7}$

Step 2

To write $- \frac{1}{y - 7}$ as a fraction with a common denominator, multiply by $\frac{y - 3}{y - 3}$ .

$\frac{4 y}{(y - 7) (y - 3)} - \frac{1}{y - 7} \cdot \frac{y - 3}{y - 3}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{1}{y - 7}$ by $\frac{y - 3}{y - 3}$ .

$\frac{4 y}{(y - 7) (y - 3)} - \frac{y - 3}{(y - 7) (y - 3)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{4 y - (y - 3)}{(y - 7) (y - 3)}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{4 y - y - - 3}{(y - 7) (y - 3)}$

Step 4.2

Multiply $- 1$ by $- 3$ .

$\frac{4 y - y + 3}{(y - 7) (y - 3)}$

Step 4.3

Subtract $y$ from $4 y$ .

$\frac{3 y + 3}{(y - 7) (y - 3)}$

Step 4.4

Factor $3$ out of $3 y + 3$ .

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Step 4.4.1

Factor $3$ out of $3 y$ .

$\frac{3 (y) + 3}{(y - 7) (y - 3)}$

Step 4.4.2

Factor $3$ out of $3$ .

$\frac{3 (y) + 3 (1)}{(y - 7) (y - 3)}$

Step 4.4.3

Factor $3$ out of $3 (y) + 3 (1)$ .

$\frac{3 (y + 1)}{(y - 7) (y - 3)}$