Solve for g 5/4(2/5g-4)=2g+1/3(3-1/2g)

$\frac{5}{4} (\frac{2}{5} g - 4) = 2 g + \frac{1}{3} (3 - \frac{1}{2} g)$

Step 1

Since $g$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 2

Simplify $2 g + \frac{1}{3} \cdot (3 - \frac{1}{2} g)$ .

Simplify each term.

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

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

Apply the distributive property.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{1}{3} (- \frac{g}{2})$ .

Multiply $\frac{1}{3}$ by $\frac{g}{2}$ .

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

Multiply $3$ by $2$ .

$2 g + 1 - \frac{g}{6} = \frac{5}{4} \cdot (\frac{2}{5} g - 4)$

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

$2 g \cdot \frac{6}{6} - \frac{g}{6} + 1 = \frac{5}{4} \cdot (\frac{2}{5} g - 4)$

Simplify terms.

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

$\frac{2 g \cdot 6}{6} - \frac{g}{6} + 1 = \frac{5}{4} \cdot (\frac{2}{5} g - 4)$

Combine the numerators over the common denominator.

$\frac{2 g \cdot 6 - g}{6} + 1 = \frac{5}{4} \cdot (\frac{2}{5} g - 4)$

Simplify each term.

Simplify the numerator.

Factor $g$ out of $2 g \cdot 6 - g$ .

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

$\frac{g (2 \cdot 6) - g}{6} + 1 = \frac{5}{4} \cdot (\frac{2}{5} g - 4)$

Factor $g$ out of $- g$ .

$\frac{g (2 \cdot 6) + g \cdot - 1}{6} + 1 = \frac{5}{4} \cdot (\frac{2}{5} g - 4)$

Factor $g$ out of $g (2 \cdot 6) + g \cdot - 1$ .

$\frac{g (2 \cdot 6 - 1)}{6} + 1 = \frac{5}{4} \cdot (\frac{2}{5} g - 4)$

Multiply $2$ by $6$ .

$\frac{g (12 - 1)}{6} + 1 = \frac{5}{4} \cdot (\frac{2}{5} g - 4)$

Subtract $1$ from $12$ .

$\frac{g \cdot 11}{6} + 1 = \frac{5}{4} \cdot (\frac{2}{5} g - 4)$

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

$\frac{11 g}{6} + 1 = \frac{5}{4} \cdot (\frac{2}{5} g - 4)$

Step 3

Simplify $\frac{5}{4} \cdot (\frac{2}{5} g - 4)$ .

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

$\frac{11 g}{6} + 1 = \frac{5}{4} \cdot (\frac{2 g}{5} - 4)$

Apply the distributive property.

$\frac{11 g}{6} + 1 = \frac{5}{4} \cdot \frac{2 g}{5} + \frac{5}{4} \cdot - 4$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{11 g}{6} + 1 = \frac{5}{4} \cdot \frac{2 g}{5} + \frac{5}{4} \cdot - 4$

Rewrite the expression.

$\frac{11 g}{6} + 1 = \frac{1}{4} (2 g) + \frac{5}{4} \cdot - 4$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$\frac{11 g}{6} + 1 = \frac{1}{2 (2)} (2 g) + \frac{5}{4} \cdot - 4$

Factor $2$ out of $2 g$ .

$\frac{11 g}{6} + 1 = \frac{1}{2 (2)} (2 (g)) + \frac{5}{4} \cdot - 4$

Cancel the common factor.

$\frac{11 g}{6} + 1 = \frac{1}{2 \cdot 2} (2 g) + \frac{5}{4} \cdot - 4$

Rewrite the expression.

$\frac{11 g}{6} + 1 = \frac{1}{2} g + \frac{5}{4} \cdot - 4$

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

$\frac{11 g}{6} + 1 = \frac{g}{2} + \frac{5}{4} \cdot - 4$

Cancel the common factor of $4$ .

Factor $4$ out of $- 4$ .

$\frac{11 g}{6} + 1 = \frac{g}{2} + \frac{5}{4} \cdot (4 (- 1))$

Cancel the common factor.

$\frac{11 g}{6} + 1 = \frac{g}{2} + \frac{5}{4} \cdot (4 \cdot - 1)$

Rewrite the expression.

$\frac{11 g}{6} + 1 = \frac{g}{2} + 5 \cdot - 1$

Multiply $5$ by $- 1$ .

$\frac{11 g}{6} + 1 = \frac{g}{2} - 5$

Step 4

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

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

$\frac{11 g}{6} + 1 - \frac{g}{2} = - 5$

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

$\frac{11 g}{6} - \frac{g}{2} \cdot \frac{3}{3} + 1 = - 5$

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

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

$\frac{11 g}{6} - \frac{g \cdot 3}{2 \cdot 3} + 1 = - 5$

Multiply $2$ by $3$ .

$\frac{11 g}{6} - \frac{g \cdot 3}{6} + 1 = - 5$

Combine the numerators over the common denominator.

$\frac{11 g - g \cdot 3}{6} + 1 = - 5$

Simplify each term.

Simplify the numerator.

Factor $g$ out of $11 g - g \cdot 3$ .

Factor $g$ out of $11 g$ .

$\frac{g \cdot 11 - g \cdot 3}{6} + 1 = - 5$

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

$\frac{g \cdot 11 + g (- 1 \cdot 3)}{6} + 1 = - 5$

Factor $g$ out of $g \cdot 11 + g (- 1 \cdot 3)$ .

$\frac{g (11 - 1 \cdot 3)}{6} + 1 = - 5$

Multiply $- 1$ by $3$ .

$\frac{g (11 - 3)}{6} + 1 = - 5$

Subtract $3$ from $11$ .

$\frac{g \cdot 8}{6} + 1 = - 5$

Cancel the common factor of $8$ and $6$ .

Factor $2$ out of $g \cdot 8$ .

$\frac{2 (g \cdot 4)}{6} + 1 = - 5$

Cancel the common factors.

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move $4$ to the left of $g$ .

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

Step 5

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

Subtract $1$ from both sides of the equation.

$\frac{4 g}{3} = - 5 - 1$

Subtract $1$ from $- 5$ .

$\frac{4 g}{3} = - 6$

Step 6

Multiply both sides of the equation by $\frac{3}{4}$ .

$\frac{3}{4} \cdot \frac{4 g}{3} = \frac{3}{4} \cdot - 6$

Step 7

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{3}{4} \cdot \frac{4 g}{3}$ .

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3}{4} \cdot \frac{4 g}{3} = \frac{3}{4} \cdot - 6$

Rewrite the expression.

$\frac{1}{4} (4 g) = \frac{3}{4} \cdot - 6$

Cancel the common factor of $4$ .

Factor $4$ out of $4 g$ .

$\frac{1}{4} (4 (g)) = \frac{3}{4} \cdot - 6$

Cancel the common factor.

$\frac{1}{4} (4 g) = \frac{3}{4} \cdot - 6$

Rewrite the expression.

$g = \frac{3}{4} \cdot - 6$

Simplify the right side.

Simplify $\frac{3}{4} \cdot - 6$ .

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$g = \frac{3}{2 (2)} \cdot - 6$

Factor $2$ out of $- 6$ .

$g = \frac{3}{2 \cdot 2} \cdot (2 \cdot - 3)$

Cancel the common factor.

$g = \frac{3}{2 \cdot 2} \cdot (2 \cdot - 3)$

Rewrite the expression.

$g = \frac{3}{2} \cdot - 3$

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

$g = \frac{3 \cdot - 3}{2}$

Simplify the expression.

Multiply $3$ by $- 3$ .

$g = \frac{- 9}{2}$

Move the negative in front of the fraction.

$g = - \frac{9}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$g = - \frac{9}{2}$

Decimal Form:

$g = - 4.5$

Mixed Number Form:

$g = - 4 \frac{1}{2}$

Solve for g 5/4(2/5g-4)=2g+1/3(3-1/2g)

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