Algebra Examples

$\frac{1}{6 k^{2}} = \frac{1}{3 k^{2}} - \frac{1}{k}$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\frac{1}{3 k^{2}}$ from both sides of the equation.

$\frac{1}{6 k^{2}} - \frac{1}{3 k^{2}} = - \frac{1}{k}$

Step 1.2

Add $\frac{1}{k}$ to both sides of the equation.

$\frac{1}{6 k^{2}} - \frac{1}{3 k^{2}} + \frac{1}{k} = 0$

Step 2

Simplify $\frac{1}{6 k^{2}} - \frac{1}{3 k^{2}} + \frac{1}{k}$ .

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Step 2.1

To write $- \frac{1}{3 k^{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{6 k^{2}} - \frac{1}{3 k^{2}} \cdot \frac{2}{2} + \frac{1}{k} = 0$

Step 2.2

Write each expression with a common denominator of $6 k^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1

Multiply $\frac{1}{3 k^{2}}$ by $\frac{2}{2}$ .

$\frac{1}{6 k^{2}} - \frac{2}{3 k^{2} \cdot 2} + \frac{1}{k} = 0$

Step 2.2.2

Multiply $2$ by $3$ .

$\frac{1}{6 k^{2}} - \frac{2}{6 k^{2}} + \frac{1}{k} = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{1 - 2}{6 k^{2}} + \frac{1}{k} = 0$

Step 2.4

Subtract $2$ from $1$ .

$\frac{- 1}{6 k^{2}} + \frac{1}{k} = 0$

Step 2.5

Move the negative in front of the fraction.

$- \frac{1}{6 k^{2}} + \frac{1}{k} = 0$

Step 3

To write $\frac{1}{k}$ as a fraction with a common denominator, multiply by $\frac{6 k}{6 k}$ .

$- \frac{1}{6 k^{2}} + \frac{1}{k} \cdot \frac{6 k}{6 k} = 0$

Step 4

Write each expression with a common denominator of $6 k^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{1}{k}$ by $\frac{6 k}{6 k}$ .

$- \frac{1}{6 k^{2}} + \frac{6 k}{k (6 k)} = 0$

Step 4.2

Raise $k$ to the power of $1$ .

$- \frac{1}{6 k^{2}} + \frac{6 k}{6 (k \cdot k)} = 0$

Step 4.3

Raise $k$ to the power of $1$ .

$- \frac{1}{6 k^{2}} + \frac{6 k}{6 (k \cdot k)} = 0$

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{1}{6 k^{2}} + \frac{6 k}{6 k^{1 + 1}} = 0$

Step 4.5

Add $1$ and $1$ .

$- \frac{1}{6 k^{2}} + \frac{6 k}{6 k^{2}} = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{- 1 + 6 k}{6 k^{2}} = 0$

Step 6

Set the numerator equal to zero.

$- 1 + 6 k = 0$

Step 7

Solve the equation for $k$ .

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Step 7.1

Add $1$ to both sides of the equation.

$6 k = 1$

Step 7.2

Divide each term in $6 k = 1$ by $6$ and simplify.

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Step 7.2.1

Divide each term in $6 k = 1$ by $6$ .

$\frac{6 k}{6} = \frac{1}{6}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $6$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{6 k}{6} = \frac{1}{6}$

Step 7.2.2.1.2

Divide $k$ by $1$ .

$k = \frac{1}{6}$