Algebra Examples

$\frac{w + 1}{2} - \frac{w + 2}{4} = 7$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

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

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Step 1.1.1.1

Simplify each term.

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Step 1.1.1.1.1

Split the fraction $\frac{w + 1}{2}$ into two fractions.

$\frac{w}{2} + \frac{1}{2} - \frac{w + 2}{4} = 7$

Step 1.1.1.1.2

Split the fraction $\frac{w + 2}{4}$ into two fractions.

$\frac{w}{2} + \frac{1}{2} - (\frac{w}{4} + \frac{2}{4}) = 7$

Step 1.1.1.1.3

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

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Step 1.1.1.1.3.1

Factor $2$ out of $2$ .

$\frac{w}{2} + \frac{1}{2} - (\frac{w}{4} + \frac{2 (1)}{4}) = 7$

Step 1.1.1.1.3.2

Cancel the common factors.

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Step 1.1.1.1.3.2.1

Factor $2$ out of $4$ .

$\frac{w}{2} + \frac{1}{2} - (\frac{w}{4} + \frac{2 \cdot 1}{2 \cdot 2}) = 7$

Step 1.1.1.1.3.2.2

Cancel the common factor.

$\frac{w}{2} + \frac{1}{2} - (\frac{w}{4} + \frac{2 \cdot 1}{2 \cdot 2}) = 7$

Step 1.1.1.1.3.2.3

Rewrite the expression.

$\frac{w}{2} + \frac{1}{2} - (\frac{w}{4} + \frac{1}{2}) = 7$

Step 1.1.1.1.4

Apply the distributive property.

$\frac{w}{2} + \frac{1}{2} - \frac{w}{4} - \frac{1}{2} = 7$

Step 1.1.1.2

Combine the opposite terms in $\frac{w}{2} + \frac{1}{2} - \frac{w}{4} - \frac{1}{2}$ .

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Step 1.1.1.2.1

Combine the numerators over the common denominator.

$\frac{w}{2} - \frac{w}{4} + \frac{1 - 1}{2} = 7$

Step 1.1.1.2.2

Subtract $1$ from $1$ .

$\frac{w}{2} - \frac{w}{4} + \frac{0}{2} = 7$

Step 1.1.1.2.3

Divide $0$ by $2$ .

$\frac{w}{2} - \frac{w}{4} + 0 = 7$

Step 1.1.1.2.4

Add $\frac{w}{2} - \frac{w}{4}$ and $0$ .

$\frac{w}{2} - \frac{w}{4} = 7$

Step 1.2

Subtract $7$ from both sides of the equation.

$\frac{w}{2} - \frac{w}{4} - 7 = 0$

Step 2

Simplify $\frac{w}{2} - \frac{w}{4} - 7$ .

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Step 2.1

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

$\frac{w}{2} \cdot \frac{2}{2} - \frac{w}{4} - 7 = 0$

Step 2.2

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

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Step 2.2.1

Multiply $\frac{w}{2}$ by $\frac{2}{2}$ .

$\frac{w \cdot 2}{2 \cdot 2} - \frac{w}{4} - 7 = 0$

Step 2.2.2

Multiply $2$ by $2$ .

$\frac{w \cdot 2}{4} - \frac{w}{4} - 7 = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{w \cdot 2 - w}{4} - 7 = 0$

Step 2.4

Subtract $w$ from $w \cdot 2$ .

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Step 2.4.1

Reorder $w$ and $2$ .

$\frac{2 \cdot w - w}{4} - 7 = 0$

Step 2.4.2

Subtract $w$ from $2 \cdot w$ .

$\frac{w}{4} - 7 = 0$

Step 3

Add $7$ to both sides of the equation.

$\frac{w}{4} = 7$

Step 4

Multiply both sides of the equation by $4$ .

$4 \frac{w}{4} = 4 \cdot 7$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $4$ .

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Step 5.1.1.1

Cancel the common factor.

$4 \frac{w}{4} = 4 \cdot 7$

Step 5.1.1.2

Rewrite the expression.

$w = 4 \cdot 7$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $4$ by $7$ .

$w = 28$