Algebra Examples

$a_{n} = 4 n - 2$

Step 1

Solve the equation for $n$ .

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Step 1.1

Subtract $4 n$ from both sides of the equation.

$a_{n} - 4 n = - 2$

Step 1.2

Subtract $a_{n}$ from both sides of the equation.

$- 4 n = - 2 - a_{n}$

Step 1.3

Divide each term in $- 4 n = - 2 - a_{n}$ by $- 4$ and simplify.

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Step 1.3.1

Divide each term in $- 4 n = - 2 - a_{n}$ by $- 4$ .

$\frac{- 4 n}{- 4} = \frac{- 2}{- 4} + \frac{- a_{n}}{- 4}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $- 4$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{- 4 n}{- 4} = \frac{- 2}{- 4} + \frac{- a_{n}}{- 4}$

Step 1.3.2.1.2

Divide $n$ by $1$ .

$n = \frac{- 2}{- 4} + \frac{- a_{n}}{- 4}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Cancel the common factor of $- 2$ and $- 4$ .

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Step 1.3.3.1.1.1

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

$n = \frac{- 2 (1)}{- 4} + \frac{- a_{n}}{- 4}$

Step 1.3.3.1.1.2

Cancel the common factors.

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Step 1.3.3.1.1.2.1

Factor $- 2$ out of $- 4$ .

$n = \frac{- 2 \cdot 1}{- 2 \cdot 2} + \frac{- a_{n}}{- 4}$

Step 1.3.3.1.1.2.2

Cancel the common factor.

$n = \frac{- 2 \cdot 1}{- 2 \cdot 2} + \frac{- a_{n}}{- 4}$

Step 1.3.3.1.1.2.3

Rewrite the expression.

$n = \frac{1}{2} + \frac{- a_{n}}{- 4}$

Step 1.3.3.1.2

Dividing two negative values results in a positive value.

$n = \frac{1}{2} + \frac{a_{n}}{4}$

Step 2

Simplify the right side.

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Step 2.1

Simplify $\frac{1}{2} + \frac{4 n - 2}{4}$ .

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Step 2.1.1

Cancel the common factor of $4 n - 2$ and $4$ .

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Step 2.1.1.1

Factor $2$ out of $4 n$ .

$n = \frac{1}{2} + \frac{2 (2 n) - 2}{4}$

Step 2.1.1.2

Factor $2$ out of $- 2$ .

$n = \frac{1}{2} + \frac{2 (2 n) + 2 (- 1)}{4}$

Step 2.1.1.3

Factor $2$ out of $2 (2 n) + 2 (- 1)$ .

$n = \frac{1}{2} + \frac{2 (2 n - 1)}{4}$

Step 2.1.1.4

Cancel the common factors.

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Step 2.1.1.4.1

Factor $2$ out of $4$ .

$n = \frac{1}{2} + \frac{2 (2 n - 1)}{2 (2)}$

Step 2.1.1.4.2

Cancel the common factor.

$n = \frac{1}{2} + \frac{2 (2 n - 1)}{2 \cdot 2}$

Step 2.1.1.4.3

Rewrite the expression.

$n = \frac{1}{2} + \frac{2 n - 1}{2}$

Step 2.1.2

Combine the numerators over the common denominator.

$n = \frac{1 + 2 n - 1}{2}$

Step 2.1.3

Combine the opposite terms in $1 + 2 n - 1$ .

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Step 2.1.3.1

Subtract $1$ from $1$ .

$n = \frac{2 n + 0}{2}$

Step 2.1.3.2

Add $2 n$ and $0$ .

$n = \frac{2 n}{2}$

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Cancel the common factor.

$n = \frac{2 n}{2}$

Step 2.1.4.2

Divide $n$ by $1$ .

$n = n$