Algebra Examples

$\frac{3 m + 1}{2 m (m - 5) (3 m + 1)} + \frac{2 - m}{10 m (m - 2)}$

Step 1

Cancel the common factor of $3 m + 1$ .

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Step 1.1

Cancel the common factor.

$\frac{3 m + 1}{2 m (m - 5) (3 m + 1)} + \frac{2 - m}{10 m (m - 2)}$

Step 1.2

Rewrite the expression.

$\frac{1}{2 m (m - 5)} + \frac{2 - m}{10 m (m - 2)}$

Step 2

Cancel the common factor of $2 - m$ and $m - 2$ .

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Step 2.1

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{1}{2 m (m - 5)} + \frac{- 1 \cdot - 2 - m}{10 m (m - 2)}$

Step 2.2

Factor $- 1$ out of $- m$ .

$\frac{1}{2 m (m - 5)} + \frac{- 1 \cdot - 2 - (m)}{10 m (m - 2)}$

Step 2.3

Factor $- 1$ out of $- 1 (- 2) - (m)$ .

$\frac{1}{2 m (m - 5)} + \frac{- 1 (- 2 + m)}{10 m (m - 2)}$

Step 2.4

Reorder terms.

$\frac{1}{2 m (m - 5)} + \frac{- 1 (m - 2)}{10 m (m - 2)}$

Step 2.5

Cancel the common factor.

$\frac{1}{2 m (m - 5)} + \frac{- 1 (m - 2)}{10 m (m - 2)}$

Step 2.6

Rewrite the expression.

$\frac{1}{2 m (m - 5)} + \frac{- 1}{10 m}$

Step 3

Move the negative in front of the fraction.

$\frac{1}{2 m (m - 5)} - \frac{1}{10 m}$

Step 4

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 m (m - 5), 10 m$

Step 5

Since $2 m (m - 5), 10 m$ contains both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.

Steps to find the LCM for $2 m (m - 5), 10 m$ are:

1. Find the LCM for the numeric part $2, 10$ .

2. Find the LCM for the variable part $m^{1}, m^{1}$ .

3. Find the LCM for the compound variable part $m - 5$ .

4. Multiply each LCM together.

Step 6

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 7

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 8

$10$ has factors of $2$ and $5$ .

$2 \cdot 5$

Step 9

Multiply $2$ by $5$ .

$10$

Step 10

The factor for $m^{1}$ is $m$ itself.

$m^{1} = m$

$m$ occurs $1$ time.

Step 11

The LCM of $m^{1}, m^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$m$

Step 12

The factor for $m - 5$ is $m - 5$ itself.

$(m - 5) = m - 5$

$(m - 5)$ occurs $1$ time.

Step 13

The LCM of $m - 5$ is the result of multiplying all factors the greatest number of times they occur in either term.

$m - 5$

Step 14

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$10 m (m - 5)$