Solve for f -3/2f-3+1/3f=4+2f-5/6f

$- \frac{3}{2} f - 3 + \frac{1}{3} f = 4 + 2 f - \frac{5}{6} f$

Step 1

Simplify $- \frac{3}{2} f - 3 + \frac{1}{3} f$ .

Simplify each term.

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

$- \frac{f \cdot 3}{2} - 3 + \frac{1}{3} f = 4 + 2 f - \frac{5}{6} f$

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

$- \frac{3 \cdot f}{2} - 3 + \frac{1}{3} f = 4 + 2 f - \frac{5}{6} f$

Combine $\frac{1}{3}$ and $f$ .

$- \frac{3 f}{2} - 3 + \frac{f}{3} = 4 + 2 f - \frac{5}{6} f$

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

$- \frac{3 f}{2} \cdot \frac{3}{3} + \frac{f}{3} - 3 = 4 + 2 f - \frac{5}{6} f$

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

$- \frac{3 f}{2} \cdot \frac{3}{3} + \frac{f}{3} \cdot \frac{2}{2} - 3 = 4 + 2 f - \frac{5}{6} f$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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

$- \frac{3 f \cdot 3}{2 \cdot 3} + \frac{f}{3} \cdot \frac{2}{2} - 3 = 4 + 2 f - \frac{5}{6} f$

Multiply $2$ by $3$ .

$- \frac{3 f \cdot 3}{6} + \frac{f}{3} \cdot \frac{2}{2} - 3 = 4 + 2 f - \frac{5}{6} f$

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

$- \frac{3 f \cdot 3}{6} + \frac{f \cdot 2}{3 \cdot 2} - 3 = 4 + 2 f - \frac{5}{6} f$

Multiply $3$ by $2$ .

$- \frac{3 f \cdot 3}{6} + \frac{f \cdot 2}{6} - 3 = 4 + 2 f - \frac{5}{6} f$

Combine the numerators over the common denominator.

$\frac{- 3 f \cdot 3 + f \cdot 2}{6} - 3 = 4 + 2 f - \frac{5}{6} f$

Simplify each term.

Simplify the numerator.

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

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

$\frac{f (- 3 \cdot 3) + f \cdot 2}{6} - 3 = 4 + 2 f - \frac{5}{6} f$

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

$\frac{f (- 3 \cdot 3) + f (2)}{6} - 3 = 4 + 2 f - \frac{5}{6} f$

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

$\frac{f (- 3 \cdot 3 + 2)}{6} - 3 = 4 + 2 f - \frac{5}{6} f$

Multiply $- 3$ by $3$ .

$\frac{f (- 9 + 2)}{6} - 3 = 4 + 2 f - \frac{5}{6} f$

Add $- 9$ and $2$ .

$\frac{f \cdot - 7}{6} - 3 = 4 + 2 f - \frac{5}{6} f$

Move $- 7$ to the left of $f$ .

$\frac{- 7 \cdot f}{6} - 3 = 4 + 2 f - \frac{5}{6} f$

Move the negative in front of the fraction.

$- \frac{7 f}{6} - 3 = 4 + 2 f - \frac{5}{6} f$

Step 2

Simplify $4 + 2 f - \frac{5}{6} f$ .

Simplify each term.

Combine $f$ and $\frac{5}{6}$ .

$- \frac{7 f}{6} - 3 = 4 + 2 f - \frac{f \cdot 5}{6}$

Move $5$ to the left of $f$ .

$- \frac{7 f}{6} - 3 = 4 + 2 f - \frac{5 f}{6}$

To write $2 f$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{7 f}{6} - 3 = 4 + 2 f \cdot \frac{6}{6} - \frac{5 f}{6}$

Simplify terms.

Combine $2 f$ and $\frac{6}{6}$ .

$- \frac{7 f}{6} - 3 = 4 + \frac{2 f \cdot 6}{6} - \frac{5 f}{6}$

Combine the numerators over the common denominator.

$- \frac{7 f}{6} - 3 = 4 + \frac{2 f \cdot 6 - 5 f}{6}$

Simplify each term.

Simplify the numerator.

Factor $f$ out of $2 f \cdot 6 - 5 f$ .

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

$- \frac{7 f}{6} - 3 = 4 + \frac{f (2 \cdot 6) - 5 f}{6}$

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

$- \frac{7 f}{6} - 3 = 4 + \frac{f (2 \cdot 6) + f \cdot - 5}{6}$

Factor $f$ out of $f (2 \cdot 6) + f \cdot - 5$ .

$- \frac{7 f}{6} - 3 = 4 + \frac{f (2 \cdot 6 - 5)}{6}$

Multiply $2$ by $6$ .

$- \frac{7 f}{6} - 3 = 4 + \frac{f (12 - 5)}{6}$

Subtract $5$ from $12$ .

$- \frac{7 f}{6} - 3 = 4 + \frac{f \cdot 7}{6}$

Move $7$ to the left of $f$ .

$- \frac{7 f}{6} - 3 = 4 + \frac{7 f}{6}$

Step 3

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

Subtract $\frac{7 f}{6}$ from both sides of the equation.

$- \frac{7 f}{6} - 3 - \frac{7 f}{6} = 4$

Combine the numerators over the common denominator.

$- 3 + \frac{- 7 f - 7 f}{6} = 4$

Subtract $7 f$ from $- 7 f$ .

$- 3 + \frac{- 14 f}{6} = 4$

Simplify each term.

Cancel the common factor of $- 14$ and $6$ .

Factor $2$ out of $- 14 f$ .

$- 3 + \frac{2 (- 7 f)}{6} = 4$

Cancel the common factors.

Factor $2$ out of $6$ .

$- 3 + \frac{2 (- 7 f)}{2 (3)} = 4$

Cancel the common factor.

$- 3 + \frac{2 (- 7 f)}{2 \cdot 3} = 4$

Rewrite the expression.

$- 3 + \frac{- 7 f}{3} = 4$

Move the negative in front of the fraction.

$- 3 - \frac{7 f}{3} = 4$

Step 4

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

Add $3$ to both sides of the equation.

$- \frac{7 f}{3} = 4 + 3$

Add $4$ and $3$ .

$- \frac{7 f}{3} = 7$

Step 5

Multiply both sides of the equation by $- \frac{3}{7}$ .

$- \frac{3}{7} (- \frac{7 f}{3}) = - \frac{3}{7} \cdot 7$

Step 6

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{3}{7} (- \frac{7 f}{3})$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{3}{7}$ into the numerator.

$\frac{- 3}{7} (- \frac{7 f}{3}) = - \frac{3}{7} \cdot 7$

Move the leading negative in $- \frac{7 f}{3}$ into the numerator.

$\frac{- 3}{7} \cdot \frac{- 7 f}{3} = - \frac{3}{7} \cdot 7$

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{7} \cdot \frac{- 7 f}{3} = - \frac{3}{7} \cdot 7$

Cancel the common factor.

$\frac{3 \cdot - 1}{7} \cdot \frac{- 7 f}{3} = - \frac{3}{7} \cdot 7$

Rewrite the expression.

$\frac{- 1}{7} (- 7 f) = - \frac{3}{7} \cdot 7$

Cancel the common factor of $7$ .

Factor $7$ out of $- 7 f$ .

$\frac{- 1}{7} (7 (- f)) = - \frac{3}{7} \cdot 7$

Cancel the common factor.

$\frac{- 1}{7} (7 (- f)) = - \frac{3}{7} \cdot 7$

Rewrite the expression.

$- - f = - \frac{3}{7} \cdot 7$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 f = - \frac{3}{7} \cdot 7$

Multiply $f$ by $1$ .

$f = - \frac{3}{7} \cdot 7$

Simplify the right side.

Cancel the common factor of $7$ .

Move the leading negative in $- \frac{3}{7}$ into the numerator.

$f = \frac{- 3}{7} \cdot 7$

Cancel the common factor.

$f = \frac{- 3}{7} \cdot 7$

Rewrite the expression.

$f = - 3$

Solve for f -3/2f-3+1/3f=4+2f-5/6f

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