Finite Math Examples

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

Step 1

Multiply both sides by $3$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(\frac{3 z}{5} - \frac{z - 3}{2}) \cdot 3$ .

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Step 2.1.1.1

To write $\frac{3 z}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.1.1.2

To write $- \frac{z - 3}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 2.1.1.3

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

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Step 2.1.1.3.1

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

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

Step 2.1.1.3.2

Multiply $5$ by $2$ .

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

Step 2.1.1.3.3

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

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

Step 2.1.1.3.4

Multiply $2$ by $5$ .

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

Step 2.1.1.4

Combine the numerators over the common denominator.

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

Step 2.1.1.5

Simplify the numerator.

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Step 2.1.1.5.1

Multiply $2$ by $3$ .

$\frac{6 z - (z - 3) \cdot 5}{10} \cdot 3 = \frac{z + 2}{3} \cdot 3$

Step 2.1.1.5.2

Apply the distributive property.

$\frac{6 z + (- z - - 3) \cdot 5}{10} \cdot 3 = \frac{z + 2}{3} \cdot 3$

Step 2.1.1.5.3

Multiply $- 1$ by $- 3$ .

$\frac{6 z + (- z + 3) \cdot 5}{10} \cdot 3 = \frac{z + 2}{3} \cdot 3$

Step 2.1.1.5.4

Apply the distributive property.

$\frac{6 z - z \cdot 5 + 3 \cdot 5}{10} \cdot 3 = \frac{z + 2}{3} \cdot 3$

Step 2.1.1.5.5

Multiply $5$ by $- 1$ .

$\frac{6 z - 5 z + 3 \cdot 5}{10} \cdot 3 = \frac{z + 2}{3} \cdot 3$

Step 2.1.1.5.6

Multiply $3$ by $5$ .

$\frac{6 z - 5 z + 15}{10} \cdot 3 = \frac{z + 2}{3} \cdot 3$

Step 2.1.1.5.7

Subtract $5 z$ from $6 z$ .

$\frac{z + 15}{10} \cdot 3 = \frac{z + 2}{3} \cdot 3$

Step 2.1.1.6

Combine fractions.

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Step 2.1.1.6.1

Combine $\frac{z + 15}{10}$ and $3$ .

$\frac{(z + 15) \cdot 3}{10} = \frac{z + 2}{3} \cdot 3$

Step 2.1.1.6.2

Move $3$ to the left of $z + 15$ .

$\frac{3 (z + 15)}{10} = \frac{z + 2}{3} \cdot 3$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 (z + 15)}{10} = \frac{z + 2}{3} \cdot 3$

Step 2.2.1.2

Rewrite the expression.

$\frac{3 (z + 15)}{10} = z + 2$

Step 3

Solve for $z$ .

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Step 3.1

Multiply both sides by $10$ .

$\frac{3 (z + 15)}{10} \cdot 10 = (z + 2) \cdot 10$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{3 (z + 15)}{10} \cdot 10$ .

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Step 3.2.1.1.1

Cancel the common factor of $10$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{3 (z + 15)}{10} \cdot 10 = (z + 2) \cdot 10$

Step 3.2.1.1.1.2

Rewrite the expression.

$3 (z + 15) = (z + 2) \cdot 10$

Step 3.2.1.1.2

Apply the distributive property.

$3 z + 3 \cdot 15 = (z + 2) \cdot 10$

Step 3.2.1.1.3

Multiply $3$ by $15$ .

$3 z + 45 = (z + 2) \cdot 10$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $(z + 2) \cdot 10$ .

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Step 3.2.2.1.1

Apply the distributive property.

$3 z + 45 = z \cdot 10 + 2 \cdot 10$

Step 3.2.2.1.2

Simplify the expression.

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Step 3.2.2.1.2.1

Move $10$ to the left of $z$ .

$3 z + 45 = 10 \cdot z + 2 \cdot 10$

Step 3.2.2.1.2.2

Multiply $2$ by $10$ .

$3 z + 45 = 10 z + 20$

Step 3.3

Solve for $z$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $10 z$ from both sides of the equation.

$3 z + 45 - 10 z = 20$

Step 3.3.1.2

Subtract $10 z$ from $3 z$ .

$- 7 z + 45 = 20$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $45$ from both sides of the equation.

$- 7 z = 20 - 45$

Step 3.3.2.2

Subtract $45$ from $20$ .

$- 7 z = - 25$

Step 3.3.3

Divide each term in $- 7 z = - 25$ by $- 7$ and simplify.

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Step 3.3.3.1

Divide each term in $- 7 z = - 25$ by $- 7$ .

$\frac{- 7 z}{- 7} = \frac{- 25}{- 7}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 7$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 7 z}{- 7} = \frac{- 25}{- 7}$

Step 3.3.3.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 25}{- 7}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Dividing two negative values results in a positive value.

$z = \frac{25}{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$z = \frac{25}{7}$

Decimal Form:

$z = 3.\overline{571428}$

Mixed Number Form:

$z = 3 \frac{4}{7}$