Solve for d -1/2d=3(2/3d-1)+2(1/6d+2)

$- \frac{1}{2} d = 3 (\frac{2}{3} d - 1) + 2 (\frac{1}{6} d + 2)$

Step 1

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

$3 (\frac{2}{3} d - 1) + 2 (\frac{1}{6} d + 2) = - \frac{1}{2} d$

Step 2

Simplify $3 (\frac{2}{3} d - 1) + 2 (\frac{1}{6} d + 2)$ .

Simplify each term.

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

$3 (\frac{2 d}{3} - 1) + 2 (\frac{1}{6} d + 2) = - \frac{1}{2} d$

Apply the distributive property.

$3 \frac{2 d}{3} + 3 \cdot - 1 + 2 (\frac{1}{6} d + 2) = - \frac{1}{2} d$

Cancel the common factor of $3$ .

Cancel the common factor.

$3 \frac{2 d}{3} + 3 \cdot - 1 + 2 (\frac{1}{6} d + 2) = - \frac{1}{2} d$

Rewrite the expression.

$2 d + 3 \cdot - 1 + 2 (\frac{1}{6} d + 2) = - \frac{1}{2} d$

Multiply $3$ by $- 1$ .

$2 d - 3 + 2 (\frac{1}{6} d + 2) = - \frac{1}{2} d$

Combine $\frac{1}{6}$ and $d$ .

$2 d - 3 + 2 (\frac{d}{6} + 2) = - \frac{1}{2} d$

Apply the distributive property.

$2 d - 3 + 2 \frac{d}{6} + 2 \cdot 2 = - \frac{1}{2} d$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$2 d - 3 + 2 \frac{d}{2 (3)} + 2 \cdot 2 = - \frac{1}{2} d$

Cancel the common factor.

$2 d - 3 + 2 \frac{d}{2 \cdot 3} + 2 \cdot 2 = - \frac{1}{2} d$

Rewrite the expression.

$2 d - 3 + \frac{d}{3} + 2 \cdot 2 = - \frac{1}{2} d$

Multiply $2$ by $2$ .

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

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

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

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

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

Combine the numerators over the common denominator.

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

Find the common denominator.

Write $- 3$ as a fraction with denominator $1$ .

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

Multiply $\frac{- 3}{1}$ by $\frac{3}{3}$ .

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

Multiply $\frac{- 3}{1}$ by $\frac{3}{3}$ .

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

Write $4$ as a fraction with denominator $1$ .

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

Multiply $\frac{4}{1}$ by $\frac{3}{3}$ .

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

Multiply $\frac{4}{1}$ by $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

Multiply $3$ by $2$ .

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

Multiply $- 3$ by $3$ .

$\frac{6 d + d - 9 + 4 \cdot 3}{3} = - \frac{1}{2} d$

Multiply $4$ by $3$ .

$\frac{6 d + d - 9 + 12}{3} = - \frac{1}{2} d$

Simplify by adding terms.

Add $6 d$ and $d$ .

$\frac{7 d - 9 + 12}{3} = - \frac{1}{2} d$

Add $- 9$ and $12$ .

$\frac{7 d + 3}{3} = - \frac{1}{2} d$

Step 3

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

$\frac{7 d + 3}{3} = - \frac{d}{2}$

Step 4

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

Add $\frac{d}{2}$ to both sides of the equation.

$\frac{7 d + 3}{3} + \frac{d}{2} = 0$

To write $\frac{7 d + 3}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{7 d + 3}{3} \cdot \frac{2}{2} + \frac{d}{2} = 0$

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

$\frac{7 d + 3}{3} \cdot \frac{2}{2} + \frac{d}{2} \cdot \frac{3}{3} = 0$

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

Multiply $\frac{7 d + 3}{3}$ by $\frac{2}{2}$ .

$\frac{(7 d + 3) \cdot 2}{3 \cdot 2} + \frac{d}{2} \cdot \frac{3}{3} = 0$

Multiply $3$ by $2$ .

$\frac{(7 d + 3) \cdot 2}{6} + \frac{d}{2} \cdot \frac{3}{3} = 0$

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

$\frac{(7 d + 3) \cdot 2}{6} + \frac{d \cdot 3}{2 \cdot 3} = 0$

Multiply $2$ by $3$ .

$\frac{(7 d + 3) \cdot 2}{6} + \frac{d \cdot 3}{6} = 0$

Combine the numerators over the common denominator.

$\frac{(7 d + 3) \cdot 2 + d \cdot 3}{6} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{7 d \cdot 2 + 3 \cdot 2 + d \cdot 3}{6} = 0$

Multiply $2$ by $7$ .

$\frac{14 d + 3 \cdot 2 + d \cdot 3}{6} = 0$

Multiply $3$ by $2$ .

$\frac{14 d + 6 + d \cdot 3}{6} = 0$

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

$\frac{14 d + 6 + 3 \cdot d}{6} = 0$

Add $14 d$ and $3 d$ .

$\frac{17 d + 6}{6} = 0$

Step 5

Set the numerator equal to zero.

$17 d + 6 = 0$

Step 6

Solve the equation for $d$ .

Subtract $6$ from both sides of the equation.

$17 d = - 6$

Divide each term in $17 d = - 6$ by $17$ and simplify.

Divide each term in $17 d = - 6$ by $17$ .

$\frac{17 d}{17} = \frac{- 6}{17}$

Simplify the left side.

Cancel the common factor of $17$ .

Cancel the common factor.

$\frac{17 d}{17} = \frac{- 6}{17}$

Divide $d$ by $1$ .

$d = \frac{- 6}{17}$

Simplify the right side.

Move the negative in front of the fraction.

$d = - \frac{6}{17}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$d = - \frac{6}{17}$

Decimal Form:

$d = - 0.35294117 \dots$

Solve for d -1/2d=3(2/3d-1)+2(1/6d+2)

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