Algebra Examples

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

Step 1

Simplify $5 - \frac{2}{5} f + \frac{3}{2} f$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $f$ and $\frac{2}{5}$ .

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

Step 1.1.2

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

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

Step 1.1.3

Combine $\frac{3}{2}$ and $f$ .

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

Step 1.2

To write $- \frac{2 f}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.3

To write $\frac{3 f}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 1.4

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{2 f}{5}$ by $\frac{2}{2}$ .

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

Step 1.4.2

Multiply $5$ by $2$ .

$5 - \frac{2 f \cdot 2}{10} + \frac{3 f}{2} \cdot \frac{5}{5} = f - \frac{1}{2} f + 4$

Step 1.4.3

Multiply $\frac{3 f}{2}$ by $\frac{5}{5}$ .

$5 - \frac{2 f \cdot 2}{10} + \frac{3 f \cdot 5}{2 \cdot 5} = f - \frac{1}{2} f + 4$

Step 1.4.4

Multiply $2$ by $5$ .

$5 - \frac{2 f \cdot 2}{10} + \frac{3 f \cdot 5}{10} = f - \frac{1}{2} f + 4$

Step 1.5

Combine the numerators over the common denominator.

$5 + \frac{- 2 f \cdot 2 + 3 f \cdot 5}{10} = f - \frac{1}{2} f + 4$

Step 1.6

Simplify each term.

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Factor $f$ out of $- 2 f \cdot 2 + 3 f \cdot 5$ .

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Step 1.6.1.1.1

Factor $f$ out of $- 2 f \cdot 2$ .

$5 + \frac{f (- 2 \cdot 2) + 3 f \cdot 5}{10} = f - \frac{1}{2} f + 4$

Step 1.6.1.1.2

Factor $f$ out of $3 f \cdot 5$ .

$5 + \frac{f (- 2 \cdot 2) + f (3 \cdot 5)}{10} = f - \frac{1}{2} f + 4$

Step 1.6.1.1.3

Factor $f$ out of $f (- 2 \cdot 2) + f (3 \cdot 5)$ .

$5 + \frac{f (- 2 \cdot 2 + 3 \cdot 5)}{10} = f - \frac{1}{2} f + 4$

Step 1.6.1.2

Multiply $- 2$ by $2$ .

$5 + \frac{f (- 4 + 3 \cdot 5)}{10} = f - \frac{1}{2} f + 4$

Step 1.6.1.3

Multiply $3$ by $5$ .

$5 + \frac{f (- 4 + 15)}{10} = f - \frac{1}{2} f + 4$

Step 1.6.1.4

Add $- 4$ and $15$ .

$5 + \frac{f \cdot 11}{10} = f - \frac{1}{2} f + 4$

Step 1.6.2

Move $11$ to the left of $f$ .

$5 + \frac{11 f}{10} = f - \frac{1}{2} f + 4$

Step 2

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

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Step 2.1

Combine $f$ and $\frac{1}{2}$ .

$5 + \frac{11 f}{10} = f - \frac{f}{2} + 4$

Step 2.2

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

$5 + \frac{11 f}{10} = f \cdot \frac{2}{2} - \frac{f}{2} + 4$

Step 2.3

Combine $f$ and $\frac{2}{2}$ .

$5 + \frac{11 f}{10} = \frac{f \cdot 2}{2} - \frac{f}{2} + 4$

Step 2.4

Combine the numerators over the common denominator.

$5 + \frac{11 f}{10} = \frac{f \cdot 2 - f}{2} + 4$

Step 2.5

Subtract $f$ from $f \cdot 2$ .

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Step 2.5.1

Reorder $f$ and $2$ .

$5 + \frac{11 f}{10} = \frac{2 \cdot f - f}{2} + 4$

Step 2.5.2

Subtract $f$ from $2 \cdot f$ .

$5 + \frac{11 f}{10} = \frac{f}{2} + 4$

Step 3

Move all terms containing $f$ to the left side of the equation.

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Step 3.1

Subtract $\frac{f}{2}$ from both sides of the equation.

$5 + \frac{11 f}{10} - \frac{f}{2} = 4$

Step 3.2

To write $- \frac{f}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$5 + \frac{11 f}{10} - \frac{f}{2} \cdot \frac{5}{5} = 4$

Step 3.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{f}{2}$ by $\frac{5}{5}$ .

$5 + \frac{11 f}{10} - \frac{f \cdot 5}{2 \cdot 5} = 4$

Step 3.3.2

Multiply $2$ by $5$ .

$5 + \frac{11 f}{10} - \frac{f \cdot 5}{10} = 4$

Step 3.4

Combine the numerators over the common denominator.

$5 + \frac{11 f - f \cdot 5}{10} = 4$

Step 3.5

Simplify each term.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Factor $f$ out of $11 f - f \cdot 5$ .

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Step 3.5.1.1.1

Factor $f$ out of $11 f$ .

$5 + \frac{f \cdot 11 - f \cdot 5}{10} = 4$

Step 3.5.1.1.2

Factor $f$ out of $- f \cdot 5$ .

$5 + \frac{f \cdot 11 + f (- 1 \cdot 5)}{10} = 4$

Step 3.5.1.1.3

Factor $f$ out of $f \cdot 11 + f (- 1 \cdot 5)$ .

$5 + \frac{f (11 - 1 \cdot 5)}{10} = 4$

Step 3.5.1.2

Multiply $- 1$ by $5$ .

$5 + \frac{f (11 - 5)}{10} = 4$

Step 3.5.1.3

Subtract $5$ from $11$ .

$5 + \frac{f \cdot 6}{10} = 4$

Step 3.5.2

Cancel the common factor of $6$ and $10$ .

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Step 3.5.2.1

Factor $2$ out of $f \cdot 6$ .

$5 + \frac{2 (f \cdot 3)}{10} = 4$

Step 3.5.2.2

Cancel the common factors.

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Step 3.5.2.2.1

Factor $2$ out of $10$ .

$5 + \frac{2 (f \cdot 3)}{2 (5)} = 4$

Step 3.5.2.2.2

Cancel the common factor.

$5 + \frac{2 (f \cdot 3)}{2 \cdot 5} = 4$

Step 3.5.2.2.3

Rewrite the expression.

$5 + \frac{f \cdot 3}{5} = 4$

Step 3.5.3

Move $3$ to the left of $f$ .

$5 + \frac{3 f}{5} = 4$

Step 4

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

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Step 4.1

Subtract $5$ from both sides of the equation.

$\frac{3 f}{5} = 4 - 5$

Step 4.2

Subtract $5$ from $4$ .

$\frac{3 f}{5} = - 1$

Step 5

Multiply both sides of the equation by $\frac{5}{3}$ .

$\frac{5}{3} \cdot \frac{3 f}{5} = \frac{5}{3} \cdot - 1$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $\frac{5}{3} \cdot \frac{3 f}{5}$ .

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Step 6.1.1.1

Cancel the common factor of $5$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{5}{3} \cdot \frac{3 f}{5} = \frac{5}{3} \cdot - 1$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 f) = \frac{5}{3} \cdot - 1$

Step 6.1.1.2

Cancel the common factor of $3$ .

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Step 6.1.1.2.1

Factor $3$ out of $3 f$ .

$\frac{1}{3} (3 (f)) = \frac{5}{3} \cdot - 1$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 f) = \frac{5}{3} \cdot - 1$

Step 6.1.1.2.3

Rewrite the expression.

$f = \frac{5}{3} \cdot - 1$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{5}{3} \cdot - 1$ .

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Step 6.2.1.1

Multiply $\frac{5}{3} \cdot - 1$ .

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Step 6.2.1.1.1

Combine $\frac{5}{3}$ and $- 1$ .

$f = \frac{5 \cdot - 1}{3}$

Step 6.2.1.1.2

Multiply $5$ by $- 1$ .

$f = \frac{- 5}{3}$

Step 6.2.1.2

Move the negative in front of the fraction.

$f = - \frac{5}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$f = - \frac{5}{3}$

Decimal Form:

$f = - 1.\overline{6}$

Mixed Number Form:

$f = - 1 \frac{2}{3}$