Solve for z 4/15z+5/6z+1/2=1 3/5

$\frac{4}{15} z + \frac{5}{6} z + \frac{1}{2} = 1 \frac{3}{5}$

Step 1

Simplify the left side.

Simplify $\frac{4}{15} z + \frac{5}{6} z + \frac{1}{2}$ .

Simplify each term.

Combine $\frac{4}{15}$ and $z$ .

$\frac{4 z}{15} + \frac{5}{6} z + \frac{1}{2} = 1 \frac{3}{5}$

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

$\frac{4 z}{15} + \frac{5 z}{6} + \frac{1}{2} = 1 \frac{3}{5}$

To write $\frac{4 z}{15}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{4 z}{15} \cdot \frac{2}{2} + \frac{5 z}{6} + \frac{1}{2} = 1 \frac{3}{5}$

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

$\frac{4 z}{15} \cdot \frac{2}{2} + \frac{5 z}{6} \cdot \frac{5}{5} + \frac{1}{2} = 1 \frac{3}{5}$

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

Multiply $\frac{4 z}{15}$ by $\frac{2}{2}$ .

$\frac{4 z \cdot 2}{15 \cdot 2} + \frac{5 z}{6} \cdot \frac{5}{5} + \frac{1}{2} = 1 \frac{3}{5}$

Multiply $15$ by $2$ .

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

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

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

Multiply $6$ by $5$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

Simplify the numerator.

Factor $z$ out of $4 z \cdot 2 + 5 z \cdot 5$ .

Factor $z$ out of $4 z \cdot 2$ .

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

Factor $z$ out of $5 z \cdot 5$ .

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

Factor $z$ out of $z (4 \cdot 2) + z (5 \cdot 5)$ .

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

Multiply $4$ by $2$ .

$\frac{z (8 + 5 \cdot 5)}{30} + \frac{1}{2} = 1 \frac{3}{5}$

Multiply $5$ by $5$ .

$\frac{z (8 + 25)}{30} + \frac{1}{2} = 1 \frac{3}{5}$

Add $8$ and $25$ .

$\frac{z \cdot 33}{30} + \frac{1}{2} = 1 \frac{3}{5}$

Cancel the common factor of $33$ and $30$ .

Factor $3$ out of $z \cdot 33$ .

$\frac{3 (z \cdot 11)}{30} + \frac{1}{2} = 1 \frac{3}{5}$

Cancel the common factors.

Factor $3$ out of $30$ .

$\frac{3 (z \cdot 11)}{3 (10)} + \frac{1}{2} = 1 \frac{3}{5}$

Cancel the common factor.

$\frac{3 (z \cdot 11)}{3 \cdot 10} + \frac{1}{2} = 1 \frac{3}{5}$

Rewrite the expression.

$\frac{z \cdot 11}{10} + \frac{1}{2} = 1 \frac{3}{5}$

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

$\frac{11 z}{10} + \frac{1}{2} = 1 \frac{3}{5}$

Step 2

Simplify the right side.

Convert $1 \frac{3}{5}$ to an improper fraction.

A mixed number is an addition of its whole and fractional parts.

$\frac{11 z}{10} + \frac{1}{2} = 1 + \frac{3}{5}$

Add $1$ and $\frac{3}{5}$ .

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

$\frac{11 z}{10} + \frac{1}{2} = \frac{5}{5} + \frac{3}{5}$

Combine the numerators over the common denominator.

$\frac{11 z}{10} + \frac{1}{2} = \frac{5 + 3}{5}$

Add $5$ and $3$ .

$\frac{11 z}{10} + \frac{1}{2} = \frac{8}{5}$

Step 3

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

Subtract $\frac{1}{2}$ from both sides of the equation.

$\frac{11 z}{10} = \frac{8}{5} - \frac{1}{2}$

To write $\frac{8}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{11 z}{10} = \frac{8}{5} \cdot \frac{2}{2} - \frac{1}{2}$

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{11 z}{10} = \frac{8}{5} \cdot \frac{2}{2} - \frac{1}{2} \cdot \frac{5}{5}$

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{8}{5}$ by $\frac{2}{2}$ .

$\frac{11 z}{10} = \frac{8 \cdot 2}{5 \cdot 2} - \frac{1}{2} \cdot \frac{5}{5}$

Multiply $5$ by $2$ .

$\frac{11 z}{10} = \frac{8 \cdot 2}{10} - \frac{1}{2} \cdot \frac{5}{5}$

Multiply $\frac{1}{2}$ by $\frac{5}{5}$ .

$\frac{11 z}{10} = \frac{8 \cdot 2}{10} - \frac{5}{2 \cdot 5}$

Multiply $2$ by $5$ .

$\frac{11 z}{10} = \frac{8 \cdot 2}{10} - \frac{5}{10}$

Combine the numerators over the common denominator.

$\frac{11 z}{10} = \frac{8 \cdot 2 - 5}{10}$

Simplify the numerator.

Multiply $8$ by $2$ .

$\frac{11 z}{10} = \frac{16 - 5}{10}$

Subtract $5$ from $16$ .

$\frac{11 z}{10} = \frac{11}{10}$

Step 4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$11 z = 11$

Step 5

Divide each term in $11 z = 11$ by $11$ and simplify.

Divide each term in $11 z = 11$ by $11$ .

$\frac{11 z}{11} = \frac{11}{11}$

Simplify the left side.

Cancel the common factor of $11$ .

Cancel the common factor.

$\frac{11 z}{11} = \frac{11}{11}$

Divide $z$ by $1$ .

$z = \frac{11}{11}$

Simplify the right side.

Divide $11$ by $11$ .

$z = 1$

Solve for z 4/15z+5/6z+1/2=1 3/5

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