Basic Math Examples

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

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

Multiply $4$ by $- 2$ .

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

Step 1.1.4

Move the negative in front of the fraction.

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Multiply $5$ by $6$ .

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

Step 1.4.3

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

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

Step 1.4.4

Multiply $6$ by $5$ .

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 8$ by $6$ .

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

Step 1.6.2

Multiply $5$ by $5$ .

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

Step 1.6.3

Add $- 48$ and $25$ .

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

Step 1.7

Move the negative in front of the fraction.

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

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

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

Step 2.3

Cancel the common factor of $3$ .

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Step 2.3.1

Factor $3$ out of $6$ .

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

Step 2.3.2

Cancel the common factor.

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

Step 2.3.3

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.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Multiply $5$ by $6$ .

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

Step 3.4.3

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

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

Step 3.4.4

Multiply $6$ by $5$ .

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Combine the numerators over the common denominator.

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

Step 3.7

Simplify each term.

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Step 3.7.1

Multiply $6$ by $4$ .

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

Step 3.7.2

Multiply $5$ by $- 1$ .

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

Step 3.8

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.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $30$ .

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Step 5.1.1.1

Cancel the common factor.

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

Step 5.1.1.2

Rewrite the expression.

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

Step 5.2

Simplify the right side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Factor $2$ out of $30$ .

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

Step 5.2.1.2

Cancel the common factor.

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

Step 5.2.1.3

Rewrite the expression.

$19 z - 23 = 15$

Step 6

Solve for $z$ .

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Step 6.1

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

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Step 6.1.1

Add $23$ to both sides of the equation.

$19 z = 15 + 23$

Step 6.1.2

Add $15$ and $23$ .

$19 z = 38$

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $19$ .

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Step 6.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.1.2

Divide $z$ by $1$ .

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

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $38$ by $19$ .

$z = 2$