Linear Algebra Examples

$a n = 14 + (n - 1) (- 3)$

Step 1

Divide each term in $a n = 14 + (n - 1) \cdot - 3$ by $n$ and simplify.

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Step 1.1

Divide each term in $a n = 14 + (n - 1) \cdot - 3$ by $n$ .

$\frac{a n}{n} = \frac{14}{n} + \frac{(n - 1) \cdot - 3}{n}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $n$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{a n}{n} = \frac{14}{n} + \frac{(n - 1) \cdot - 3}{n}$

Step 1.2.1.2

Divide $a$ by $1$ .

$a = \frac{14}{n} + \frac{(n - 1) \cdot - 3}{n}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Combine the numerators over the common denominator.

$a = \frac{14 + (n - 1) \cdot - 3}{n}$

Step 1.3.2

Simplify each term.

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Step 1.3.2.1

Apply the distributive property.

$a = \frac{14 + n \cdot - 3 - 1 \cdot - 3}{n}$

Step 1.3.2.2

Move $- 3$ to the left of $n$ .

$a = \frac{14 - 3 \cdot n - 1 \cdot - 3}{n}$

Step 1.3.2.3

Multiply $- 1$ by $- 3$ .

$a = \frac{14 - 3 n + 3}{n}$

Step 1.3.3

Simplify with factoring out.

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Step 1.3.3.1

Add $14$ and $3$ .

$a = \frac{- 3 n + 17}{n}$

Step 1.3.3.2

Factor $- 1$ out of $- 3 n$ .

$a = \frac{- (3 n) + 17}{n}$

Step 1.3.3.3

Rewrite $17$ as $- 1 (- 17)$ .

$a = \frac{- (3 n) - 1 (- 17)}{n}$

Step 1.3.3.4

Factor $- 1$ out of $- (3 n) - 1 (- 17)$ .

$a = \frac{- (3 n - 17)}{n}$

Step 1.3.3.5

Simplify the expression.

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Step 1.3.3.5.1

Rewrite $- (3 n - 17)$ as $- 1 (3 n - 17)$ .

$a = \frac{- 1 (3 n - 17)}{n}$

Step 1.3.3.5.2

Move the negative in front of the fraction.

$a = - \frac{3 n - 17}{n}$

Step 2

Set the denominator in $\frac{3 n - 17}{n}$ equal to $0$ to find where the expression is undefined.

$n = 0$

Step 3

The domain is all values of $n$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${n | n \neq 0}$

Step 4