Finite Math Examples

$\frac{3 n - 3}{5 n - 1} + \frac{6 n}{6}$

Step 1

Simplify each term.

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Step 1.1

Factor $3$ out of $3 n - 3$ .

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Step 1.1.1

Factor $3$ out of $3 n$ .

$\frac{3 (n) - 3}{5 n - 1} + \frac{6 n}{6}$

Step 1.1.2

Factor $3$ out of $- 3$ .

$\frac{3 (n) + 3 (- 1)}{5 n - 1} + \frac{6 n}{6}$

Step 1.1.3

Factor $3$ out of $3 (n) + 3 (- 1)$ .

$\frac{3 (n - 1)}{5 n - 1} + \frac{6 n}{6}$

Step 1.2

Cancel the common factor of $6$ .

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Step 1.2.1

Cancel the common factor.

$\frac{3 (n - 1)}{5 n - 1} + \frac{6 n}{6}$

Step 1.2.2

Divide $n$ by $1$ .

$\frac{3 (n - 1)}{5 n - 1} + n$

Step 2

To write $n$ as a fraction with a common denominator, multiply by $\frac{5 n - 1}{5 n - 1}$ .

$\frac{3 (n - 1)}{5 n - 1} + \frac{n (5 n - 1)}{5 n - 1}$

Step 3

Combine the numerators over the common denominator.

$\frac{3 (n - 1) + n (5 n - 1)}{5 n - 1}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{3 n + 3 \cdot - 1 + n (5 n - 1)}{5 n - 1}$

Step 4.2

Multiply $3$ by $- 1$ .

$\frac{3 n - 3 + n (5 n - 1)}{5 n - 1}$

Step 4.3

Apply the distributive property.

$\frac{3 n - 3 + n (5 n) + n \cdot - 1}{5 n - 1}$

Step 4.4

Rewrite using the commutative property of multiplication.

$\frac{3 n - 3 + 5 n \cdot n + n \cdot - 1}{5 n - 1}$

Step 4.5

Move $- 1$ to the left of $n$ .

$\frac{3 n - 3 + 5 n \cdot n - 1 \cdot n}{5 n - 1}$

Step 4.6

Simplify each term.

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Step 4.6.1

Multiply $n$ by $n$ by adding the exponents.

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Step 4.6.1.1

Move $n$ .

$\frac{3 n - 3 + 5 (n \cdot n) - 1 \cdot n}{5 n - 1}$

Step 4.6.1.2

Multiply $n$ by $n$ .

$\frac{3 n - 3 + 5 n^{2} - 1 \cdot n}{5 n - 1}$

Step 4.6.2

Rewrite $- 1 n$ as $- n$ .

$\frac{3 n - 3 + 5 n^{2} - n}{5 n - 1}$

Step 4.7

Subtract $n$ from $3 n$ .

$\frac{2 n - 3 + 5 n^{2}}{5 n - 1}$

Step 4.8

Reorder terms.

$\frac{5 n^{2} + 2 n - 3}{5 n - 1}$

Step 4.9

Factor by grouping.

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Step 4.9.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 3 = - 15$ and whose sum is $b = 2$ .

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Step 4.9.1.1

Factor $2$ out of $2 n$ .

$\frac{5 n^{2} + 2 (n) - 3}{5 n - 1}$

Step 4.9.1.2

Rewrite $2$ as $- 3$ plus $5$

$\frac{5 n^{2} + (- 3 + 5) n - 3}{5 n - 1}$

Step 4.9.1.3

Apply the distributive property.

$\frac{5 n^{2} - 3 n + 5 n - 3}{5 n - 1}$

Step 4.9.2

Factor out the greatest common factor from each group.

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Step 4.9.2.1

Group the first two terms and the last two terms.

$\frac{(5 n^{2} - 3 n) + 5 n - 3}{5 n - 1}$

Step 4.9.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{n (5 n - 3) + 1 (5 n - 3)}{5 n - 1}$

Step 4.9.3

Factor the polynomial by factoring out the greatest common factor, $5 n - 3$ .

$\frac{(5 n - 3) (n + 1)}{5 n - 1}$