Algebra Examples

$\frac{k + 1}{2} + \frac{k + 3}{4} = \frac{1}{2}$

Step 1

Simplify $\frac{k + 1}{2} + \frac{k + 3}{4}$ .

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Step 1.1

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

$\frac{k + 1}{2} \cdot \frac{2}{2} + \frac{k + 3}{4} = \frac{1}{2}$

Step 1.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.1

Multiply $\frac{k + 1}{2}$ by $\frac{2}{2}$ .

$\frac{(k + 1) \cdot 2}{2 \cdot 2} + \frac{k + 3}{4} = \frac{1}{2}$

Step 1.2.2

Multiply $2$ by $2$ .

$\frac{(k + 1) \cdot 2}{4} + \frac{k + 3}{4} = \frac{1}{2}$

Step 1.3

Combine the numerators over the common denominator.

$\frac{(k + 1) \cdot 2 + k + 3}{4} = \frac{1}{2}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Apply the distributive property.

$\frac{k \cdot 2 + 1 \cdot 2 + k + 3}{4} = \frac{1}{2}$

Step 1.4.2

Move $2$ to the left of $k$ .

$\frac{2 \cdot k + 1 \cdot 2 + k + 3}{4} = \frac{1}{2}$

Step 1.4.3

Multiply $2$ by $1$ .

$\frac{2 \cdot k + 2 + k + 3}{4} = \frac{1}{2}$

Step 1.4.4

Add $2 k$ and $k$ .

$\frac{3 k + 2 + 3}{4} = \frac{1}{2}$

Step 1.4.5

Add $2$ and $3$ .

$\frac{3 k + 5}{4} = \frac{1}{2}$

Step 2

Multiply both sides by $4$ .

$\frac{3 k + 5}{4} \cdot 4 = \frac{1}{2} \cdot 4$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $4$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{3 k + 5}{4} \cdot 4 = \frac{1}{2} \cdot 4$

Step 3.1.1.2

Rewrite the expression.

$3 k + 5 = \frac{1}{2} \cdot 4$

Step 3.2

Simplify the right side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Factor $2$ out of $4$ .

$3 k + 5 = \frac{1}{2} \cdot (2 (2))$

Step 3.2.1.2

Cancel the common factor.

$3 k + 5 = \frac{1}{2} \cdot (2 \cdot 2)$

Step 3.2.1.3

Rewrite the expression.

$3 k + 5 = 2$

Step 4

Solve for $k$ .

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Step 4.1

Move all terms not containing $k$ to the right side of the equation.

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Step 4.1.1

Subtract $5$ from both sides of the equation.

$3 k = 2 - 5$

Step 4.1.2

Subtract $5$ from $2$ .

$3 k = - 3$

Step 4.2

Divide each term in $3 k = - 3$ by $3$ and simplify.

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Step 4.2.1

Divide each term in $3 k = - 3$ by $3$ .

$\frac{3 k}{3} = \frac{- 3}{3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{3 k}{3} = \frac{- 3}{3}$

Step 4.2.2.1.2

Divide $k$ by $1$ .

$k = \frac{- 3}{3}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 3$ by $3$ .

$k = - 1$