Solve for z 4/5(z-2)+5/6=1/6(z+3)

$\frac{4}{5} \cdot (z - 2) + \frac{5}{6} = \frac{1}{6} \cdot (z + 3)$

Step 1

Simplify $\frac{4}{5} \cdot (z - 2) + \frac{5}{6}$ .

Simplify each term.

Apply the distributive property.

$\frac{4}{5} z + \frac{4}{5} \cdot - 2 + \frac{5}{6} = \frac{1}{6} \cdot (z + 3)$

Combine $\frac{4}{5}$ and $z$ .

$\frac{4 z}{5} + \frac{4}{5} \cdot - 2 + \frac{5}{6} = \frac{1}{6} \cdot (z + 3)$

Multiply $\frac{4}{5} \cdot - 2$ .

Combine $\frac{4}{5}$ and $- 2$ .

$\frac{4 z}{5} + \frac{4 \cdot - 2}{5} + \frac{5}{6} = \frac{1}{6} \cdot (z + 3)$

Multiply $4$ by $- 2$ .

$\frac{4 z}{5} + \frac{- 8}{5} + \frac{5}{6} = \frac{1}{6} \cdot (z + 3)$

Move the negative in front of the fraction.

$\frac{4 z}{5} - \frac{8}{5} + \frac{5}{6} = \frac{1}{6} \cdot (z + 3)$

To write $- \frac{8}{5}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{4 z}{5} - \frac{8}{5} \cdot \frac{6}{6} + \frac{5}{6} = \frac{1}{6} \cdot (z + 3)$

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

$\frac{4 z}{5} - \frac{8}{5} \cdot \frac{6}{6} + \frac{5}{6} \cdot \frac{5}{5} = \frac{1}{6} \cdot (z + 3)$

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

Multiply $\frac{8}{5}$ by $\frac{6}{6}$ .

$\frac{4 z}{5} - \frac{8 \cdot 6}{5 \cdot 6} + \frac{5}{6} \cdot \frac{5}{5} = \frac{1}{6} \cdot (z + 3)$

Multiply $5$ by $6$ .

$\frac{4 z}{5} - \frac{8 \cdot 6}{30} + \frac{5}{6} \cdot \frac{5}{5} = \frac{1}{6} \cdot (z + 3)$

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

$\frac{4 z}{5} - \frac{8 \cdot 6}{30} + \frac{5 \cdot 5}{6 \cdot 5} = \frac{1}{6} \cdot (z + 3)$

Multiply $6$ by $5$ .

$\frac{4 z}{5} - \frac{8 \cdot 6}{30} + \frac{5 \cdot 5}{30} = \frac{1}{6} \cdot (z + 3)$

Combine the numerators over the common denominator.

$\frac{4 z}{5} + \frac{- 8 \cdot 6 + 5 \cdot 5}{30} = \frac{1}{6} \cdot (z + 3)$

Simplify the numerator.

Multiply $- 8$ by $6$ .

$\frac{4 z}{5} + \frac{- 48 + 5 \cdot 5}{30} = \frac{1}{6} \cdot (z + 3)$

Multiply $5$ by $5$ .

$\frac{4 z}{5} + \frac{- 48 + 25}{30} = \frac{1}{6} \cdot (z + 3)$

Add $- 48$ and $25$ .

$\frac{4 z}{5} + \frac{- 23}{30} = \frac{1}{6} \cdot (z + 3)$

Move the negative in front of the fraction.

$\frac{4 z}{5} - \frac{23}{30} = \frac{1}{6} \cdot (z + 3)$

Step 2

Simplify $\frac{1}{6} \cdot (z + 3)$ .

Apply the distributive property.

$\frac{4 z}{5} - \frac{23}{30} = \frac{1}{6} z + \frac{1}{6} \cdot 3$

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

$\frac{4 z}{5} - \frac{23}{30} = \frac{z}{6} + \frac{1}{6} \cdot 3$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$\frac{4 z}{5} - \frac{23}{30} = \frac{z}{6} + \frac{1}{3 (2)} \cdot 3$

Cancel the common factor.

$\frac{4 z}{5} - \frac{23}{30} = \frac{z}{6} + \frac{1}{3 \cdot 2} \cdot 3$

Rewrite the expression.

$\frac{4 z}{5} - \frac{23}{30} = \frac{z}{6} + \frac{1}{2}$

Step 3

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

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

$\frac{4 z}{5} - \frac{23}{30} - \frac{z}{6} = \frac{1}{2}$

To write $\frac{4 z}{5}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{4 z}{5} \cdot \frac{6}{6} - \frac{z}{6} - \frac{23}{30} = \frac{1}{2}$

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

$\frac{4 z}{5} \cdot \frac{6}{6} - \frac{z}{6} \cdot \frac{5}{5} - \frac{23}{30} = \frac{1}{2}$

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

Multiply $\frac{4 z}{5}$ by $\frac{6}{6}$ .

$\frac{4 z \cdot 6}{5 \cdot 6} - \frac{z}{6} \cdot \frac{5}{5} - \frac{23}{30} = \frac{1}{2}$

Multiply $5$ by $6$ .

$\frac{4 z \cdot 6}{30} - \frac{z}{6} \cdot \frac{5}{5} - \frac{23}{30} = \frac{1}{2}$

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

$\frac{4 z \cdot 6}{30} - \frac{z \cdot 5}{6 \cdot 5} - \frac{23}{30} = \frac{1}{2}$

Multiply $6$ by $5$ .

$\frac{4 z \cdot 6}{30} - \frac{z \cdot 5}{30} - \frac{23}{30} = \frac{1}{2}$

Combine the numerators over the common denominator.

$\frac{4 z \cdot 6 - z \cdot 5}{30} - \frac{23}{30} = \frac{1}{2}$

Combine the numerators over the common denominator.

$\frac{4 z \cdot 6 - z \cdot 5 - 23}{30} = \frac{1}{2}$

Simplify each term.

Multiply $6$ by $4$ .

$\frac{24 z - z \cdot 5 - 23}{30} = \frac{1}{2}$

Multiply $5$ by $- 1$ .

$\frac{24 z - 5 z - 23}{30} = \frac{1}{2}$

Subtract $5 z$ from $24 z$ .

$\frac{19 z - 23}{30} = \frac{1}{2}$

Step 4

Multiply both sides by $30$ .

$\frac{19 z - 23}{30} \cdot 30 = \frac{1}{2} \cdot 30$

Step 5

Simplify.

Simplify the left side.

Cancel the common factor of $30$ .

Cancel the common factor.

$\frac{19 z - 23}{30} \cdot 30 = \frac{1}{2} \cdot 30$

Rewrite the expression.

$19 z - 23 = \frac{1}{2} \cdot 30$

Simplify the right side.

Cancel the common factor of $2$ .

Factor $2$ out of $30$ .

$19 z - 23 = \frac{1}{2} \cdot (2 (15))$

Cancel the common factor.

$19 z - 23 = \frac{1}{2} \cdot (2 \cdot 15)$

Rewrite the expression.

$19 z - 23 = 15$

Step 6

Solve for $z$ .

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

Add $23$ to both sides of the equation.

$19 z = 15 + 23$

Add $15$ and $23$ .

$19 z = 38$

Divide each term in $19 z = 38$ by $19$ and simplify.

Divide each term in $19 z = 38$ by $19$ .

$\frac{19 z}{19} = \frac{38}{19}$

Simplify the left side.

Cancel the common factor of $19$ .

Cancel the common factor.

$\frac{19 z}{19} = \frac{38}{19}$

Divide $z$ by $1$ .

$z = \frac{38}{19}$

Simplify the right side.

Divide $38$ by $19$ .

$z = 2$

Solve for z 4/5*(z-2)+5/6=1/6*(z+3)

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Solve for z 4/5(z-2)+5/6=1/6(z+3)