Solve for z z/2-3/5=-2/3z+1/6

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

Step 1

Simplify each term.

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

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

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

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

Step 2

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

Add $\frac{2 z}{3}$ to both sides of the equation.

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

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

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

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

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

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

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

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

Multiply $2$ by $3$ .

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

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

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

Multiply $3$ by $2$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

Simplify the numerator.

Factor $z$ out of $z \cdot 3 + 2 z \cdot 2$ .

Factor $z$ out of $z \cdot 3$ .

$\frac{z (3) + 2 z \cdot 2}{6} - \frac{3}{5} = \frac{1}{6}$

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

$\frac{z (3) + z (2 \cdot 2)}{6} - \frac{3}{5} = \frac{1}{6}$

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

$\frac{z (3 + 2 \cdot 2)}{6} - \frac{3}{5} = \frac{1}{6}$

Multiply $2$ by $2$ .

$\frac{z (3 + 4)}{6} - \frac{3}{5} = \frac{1}{6}$

Add $3$ and $4$ .

$\frac{z \cdot 7}{6} - \frac{3}{5} = \frac{1}{6}$

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

$\frac{7 z}{6} - \frac{3}{5} = \frac{1}{6}$

Step 3

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

Add $\frac{3}{5}$ to both sides of the equation.

$\frac{7 z}{6} = \frac{1}{6} + \frac{3}{5}$

To write $\frac{1}{6}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{7 z}{6} = \frac{1}{6} \cdot \frac{5}{5} + \frac{3}{5}$

To write $\frac{3}{5}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{7 z}{6} = \frac{1}{6} \cdot \frac{5}{5} + \frac{3}{5} \cdot \frac{6}{6}$

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{6}$ by $\frac{5}{5}$ .

$\frac{7 z}{6} = \frac{5}{6 \cdot 5} + \frac{3}{5} \cdot \frac{6}{6}$

Multiply $6$ by $5$ .

$\frac{7 z}{6} = \frac{5}{30} + \frac{3}{5} \cdot \frac{6}{6}$

Multiply $\frac{3}{5}$ by $\frac{6}{6}$ .

$\frac{7 z}{6} = \frac{5}{30} + \frac{3 \cdot 6}{5 \cdot 6}$

Multiply $5$ by $6$ .

$\frac{7 z}{6} = \frac{5}{30} + \frac{3 \cdot 6}{30}$

Combine the numerators over the common denominator.

$\frac{7 z}{6} = \frac{5 + 3 \cdot 6}{30}$

Simplify the numerator.

Multiply $3$ by $6$ .

$\frac{7 z}{6} = \frac{5 + 18}{30}$

Add $5$ and $18$ .

$\frac{7 z}{6} = \frac{23}{30}$

Step 4

Multiply both sides of the equation by $\frac{6}{7}$ .

$\frac{6}{7} \cdot \frac{7 z}{6} = \frac{6}{7} \cdot \frac{23}{30}$

Step 5

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{6}{7} \cdot \frac{7 z}{6}$ .

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6}{7} \cdot \frac{7 z}{6} = \frac{6}{7} \cdot \frac{23}{30}$

Rewrite the expression.

$\frac{1}{7} (7 z) = \frac{6}{7} \cdot \frac{23}{30}$

Cancel the common factor of $7$ .

Factor $7$ out of $7 z$ .

$\frac{1}{7} (7 (z)) = \frac{6}{7} \cdot \frac{23}{30}$

Cancel the common factor.

$\frac{1}{7} (7 z) = \frac{6}{7} \cdot \frac{23}{30}$

Rewrite the expression.

$z = \frac{6}{7} \cdot \frac{23}{30}$

Simplify the right side.

Simplify $\frac{6}{7} \cdot \frac{23}{30}$ .

Cancel the common factor of $6$ .

Factor $6$ out of $30$ .

$z = \frac{6}{7} \cdot \frac{23}{6 (5)}$

Cancel the common factor.

$z = \frac{6}{7} \cdot \frac{23}{6 \cdot 5}$

Rewrite the expression.

$z = \frac{1}{7} \cdot \frac{23}{5}$

Multiply $\frac{1}{7}$ by $\frac{23}{5}$ .

$z = \frac{23}{7 \cdot 5}$

Multiply $7$ by $5$ .

$z = \frac{23}{35}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$z = \frac{23}{35}$

Decimal Form:

$z = 0.6\overline{571428}$

Solve for z z/2-3/5=-2/3z+1/6

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