Basic Math Examples

$\frac{2 z + 1}{3} - \frac{z - 2}{5} - 5 = \frac{z + 1}{10} + 4$

Step 1

Simplify $\frac{2 z + 1}{3} - \frac{z - 2}{5} - 5$ .

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Step 1.1

Find the common denominator.

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Step 1.1.1

Multiply $\frac{2 z + 1}{3}$ by $\frac{5}{5}$ .

$\frac{2 z + 1}{3} \cdot \frac{5}{5} - \frac{z - 2}{5} - 5 = \frac{z + 1}{10} + 4$

Step 1.1.2

Multiply $\frac{2 z + 1}{3}$ by $\frac{5}{5}$ .

$\frac{(2 z + 1) \cdot 5}{3 \cdot 5} - \frac{z - 2}{5} - 5 = \frac{z + 1}{10} + 4$

Step 1.1.3

Multiply $\frac{z - 2}{5}$ by $\frac{3}{3}$ .

$\frac{(2 z + 1) \cdot 5}{3 \cdot 5} - (\frac{z - 2}{5} \cdot \frac{3}{3}) - 5 = \frac{z + 1}{10} + 4$

Step 1.1.4

Multiply $\frac{z - 2}{5}$ by $\frac{3}{3}$ .

$\frac{(2 z + 1) \cdot 5}{3 \cdot 5} - \frac{(z - 2) \cdot 3}{5 \cdot 3} - 5 = \frac{z + 1}{10} + 4$

Step 1.1.5

Write $- 5$ as a fraction with denominator $1$ .

$\frac{(2 z + 1) \cdot 5}{3 \cdot 5} - \frac{(z - 2) \cdot 3}{5 \cdot 3} + \frac{- 5}{1} = \frac{z + 1}{10} + 4$

Step 1.1.6

Multiply $\frac{- 5}{1}$ by $\frac{15}{15}$ .

$\frac{(2 z + 1) \cdot 5}{3 \cdot 5} - \frac{(z - 2) \cdot 3}{5 \cdot 3} + \frac{- 5}{1} \cdot \frac{15}{15} = \frac{z + 1}{10} + 4$

Step 1.1.7

Multiply $\frac{- 5}{1}$ by $\frac{15}{15}$ .

$\frac{(2 z + 1) \cdot 5}{3 \cdot 5} - \frac{(z - 2) \cdot 3}{5 \cdot 3} + \frac{- 5 \cdot 15}{15} = \frac{z + 1}{10} + 4$

Step 1.1.8

Multiply $3$ by $5$ .

$\frac{(2 z + 1) \cdot 5}{15} - \frac{(z - 2) \cdot 3}{5 \cdot 3} + \frac{- 5 \cdot 15}{15} = \frac{z + 1}{10} + 4$

Step 1.1.9

Reorder the factors of $5 \cdot 3$ .

$\frac{(2 z + 1) \cdot 5}{15} - \frac{(z - 2) \cdot 3}{3 \cdot 5} + \frac{- 5 \cdot 15}{15} = \frac{z + 1}{10} + 4$

Step 1.1.10

Multiply $3$ by $5$ .

$\frac{(2 z + 1) \cdot 5}{15} - \frac{(z - 2) \cdot 3}{15} + \frac{- 5 \cdot 15}{15} = \frac{z + 1}{10} + 4$

Step 1.2

Combine the numerators over the common denominator.

$\frac{(2 z + 1) \cdot 5 - (z - 2) \cdot 3 - 5 \cdot 15}{15} = \frac{z + 1}{10} + 4$

Step 1.3

Simplify each term.

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Step 1.3.1

Apply the distributive property.

$\frac{2 z \cdot 5 + 1 \cdot 5 - (z - 2) \cdot 3 - 5 \cdot 15}{15} = \frac{z + 1}{10} + 4$

Step 1.3.2

Multiply $5$ by $2$ .

$\frac{10 z + 1 \cdot 5 - (z - 2) \cdot 3 - 5 \cdot 15}{15} = \frac{z + 1}{10} + 4$

Step 1.3.3

Multiply $5$ by $1$ .

$\frac{10 z + 5 - (z - 2) \cdot 3 - 5 \cdot 15}{15} = \frac{z + 1}{10} + 4$

Step 1.3.4

Apply the distributive property.

$\frac{10 z + 5 + (- z - - 2) \cdot 3 - 5 \cdot 15}{15} = \frac{z + 1}{10} + 4$

Step 1.3.5

Multiply $- 1$ by $- 2$ .

$\frac{10 z + 5 + (- z + 2) \cdot 3 - 5 \cdot 15}{15} = \frac{z + 1}{10} + 4$

Step 1.3.6

Apply the distributive property.

$\frac{10 z + 5 - z \cdot 3 + 2 \cdot 3 - 5 \cdot 15}{15} = \frac{z + 1}{10} + 4$

Step 1.3.7

Multiply $3$ by $- 1$ .

$\frac{10 z + 5 - 3 z + 2 \cdot 3 - 5 \cdot 15}{15} = \frac{z + 1}{10} + 4$

Step 1.3.8

Multiply $2$ by $3$ .

$\frac{10 z + 5 - 3 z + 6 - 5 \cdot 15}{15} = \frac{z + 1}{10} + 4$

Step 1.3.9

Multiply $- 5$ by $15$ .

$\frac{10 z + 5 - 3 z + 6 - 75}{15} = \frac{z + 1}{10} + 4$

Step 1.4

Simplify by adding terms.

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Step 1.4.1

Subtract $3 z$ from $10 z$ .

$\frac{7 z + 5 + 6 - 75}{15} = \frac{z + 1}{10} + 4$

Step 1.4.2

Simplify by adding and subtracting.

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Step 1.4.2.1

Add $5$ and $6$ .

$\frac{7 z + 11 - 75}{15} = \frac{z + 1}{10} + 4$

Step 1.4.2.2

Subtract $75$ from $11$ .

$\frac{7 z - 64}{15} = \frac{z + 1}{10} + 4$

Step 2

Simplify $\frac{z + 1}{10} + 4$ .

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Step 2.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{7 z - 64}{15} = \frac{z + 1}{10} + 4 \cdot \frac{10}{10}$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine $4$ and $\frac{10}{10}$ .

$\frac{7 z - 64}{15} = \frac{z + 1}{10} + \frac{4 \cdot 10}{10}$

Step 2.2.2

Combine the numerators over the common denominator.

$\frac{7 z - 64}{15} = \frac{z + 1 + 4 \cdot 10}{10}$

Step 2.3

Simplify the numerator.

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Step 2.3.1

Multiply $4$ by $10$ .

$\frac{7 z - 64}{15} = \frac{z + 1 + 40}{10}$

Step 2.3.2

Add $1$ and $40$ .

$\frac{7 z - 64}{15} = \frac{z + 41}{10}$

Step 3

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

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Step 3.1

Subtract $\frac{z + 41}{10}$ from both sides of the equation.

$\frac{7 z - 64}{15} - \frac{z + 41}{10} = 0$

Step 3.2

To write $\frac{7 z - 64}{15}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{7 z - 64}{15} \cdot \frac{2}{2} - \frac{z + 41}{10} = 0$

Step 3.3

To write $- \frac{z + 41}{10}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{7 z - 64}{15} \cdot \frac{2}{2} - \frac{z + 41}{10} \cdot \frac{3}{3} = 0$

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 $\frac{7 z - 64}{15}$ by $\frac{2}{2}$ .

$\frac{(7 z - 64) \cdot 2}{15 \cdot 2} - \frac{z + 41}{10} \cdot \frac{3}{3} = 0$

Step 3.4.2

Multiply $15$ by $2$ .

$\frac{(7 z - 64) \cdot 2}{30} - \frac{z + 41}{10} \cdot \frac{3}{3} = 0$

Step 3.4.3

Multiply $\frac{z + 41}{10}$ by $\frac{3}{3}$ .

$\frac{(7 z - 64) \cdot 2}{30} - \frac{(z + 41) \cdot 3}{10 \cdot 3} = 0$

Step 3.4.4

Multiply $10$ by $3$ .

$\frac{(7 z - 64) \cdot 2}{30} - \frac{(z + 41) \cdot 3}{30} = 0$

Step 3.5

Combine the numerators over the common denominator.

$\frac{(7 z - 64) \cdot 2 - (z + 41) \cdot 3}{30} = 0$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Apply the distributive property.

$\frac{7 z \cdot 2 - 64 \cdot 2 - (z + 41) \cdot 3}{30} = 0$

Step 3.6.2

Multiply $2$ by $7$ .

$\frac{14 z - 64 \cdot 2 - (z + 41) \cdot 3}{30} = 0$

Step 3.6.3

Multiply $- 64$ by $2$ .

$\frac{14 z - 128 - (z + 41) \cdot 3}{30} = 0$

Step 3.6.4

Apply the distributive property.

$\frac{14 z - 128 + (- z - 1 \cdot 41) \cdot 3}{30} = 0$

Step 3.6.5

Multiply $- 1$ by $41$ .

$\frac{14 z - 128 + (- z - 41) \cdot 3}{30} = 0$

Step 3.6.6

Apply the distributive property.

$\frac{14 z - 128 - z \cdot 3 - 41 \cdot 3}{30} = 0$

Step 3.6.7

Multiply $3$ by $- 1$ .

$\frac{14 z - 128 - 3 z - 41 \cdot 3}{30} = 0$

Step 3.6.8

Multiply $- 41$ by $3$ .

$\frac{14 z - 128 - 3 z - 123}{30} = 0$

Step 3.6.9

Subtract $3 z$ from $14 z$ .

$\frac{11 z - 128 - 123}{30} = 0$

Step 3.6.10

Subtract $123$ from $- 128$ .

$\frac{11 z - 251}{30} = 0$

Step 4

Set the numerator equal to zero.

$11 z - 251 = 0$

Step 5

Solve the equation for $z$ .

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Step 5.1

Add $251$ to both sides of the equation.

$11 z = 251$

Step 5.2

Divide each term in $11 z = 251$ by $11$ and simplify.

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Step 5.2.1

Divide each term in $11 z = 251$ by $11$ .

$\frac{11 z}{11} = \frac{251}{11}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $11$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{11 z}{11} = \frac{251}{11}$

Step 5.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{251}{11}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$z = \frac{251}{11}$

Decimal Form:

$z = 22.\overline{81}$

Mixed Number Form:

$z = 22 \frac{9}{11}$