Basic Math Examples

$\frac{5}{3} y + \frac{1}{3} y = 5 z + \frac{70}{3} + \frac{5}{3} z$

Step 1

Rewrite the equation as $5 z + \frac{70}{3} + \frac{5}{3} z = \frac{5}{3} y + \frac{1}{3} y$ .

$5 z + \frac{70}{3} + \frac{5}{3} z = \frac{5}{3} y + \frac{1}{3} y$

Step 2

Simplify $5 z + \frac{70}{3} + \frac{5}{3} z$ .

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Step 2.1

Combine $\frac{5}{3}$ and $z$ .

$5 z + \frac{70}{3} + \frac{5 z}{3} = \frac{5}{3} y + \frac{1}{3} y$

Step 2.2

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

$5 z \cdot \frac{3}{3} + \frac{5 z}{3} + \frac{70}{3} = \frac{5}{3} y + \frac{1}{3} y$

Step 2.3

Combine $5 z$ and $\frac{3}{3}$ .

$\frac{5 z \cdot 3}{3} + \frac{5 z}{3} + \frac{70}{3} = \frac{5}{3} y + \frac{1}{3} y$

Step 2.4

Combine the numerators over the common denominator.

$\frac{5 z \cdot 3 + 5 z}{3} + \frac{70}{3} = \frac{5}{3} y + \frac{1}{3} y$

Step 2.5

Combine the numerators over the common denominator.

$\frac{5 z \cdot 3 + 5 z + 70}{3} = \frac{5}{3} y + \frac{1}{3} y$

Step 2.6

Multiply $3$ by $5$ .

$\frac{15 z + 5 z + 70}{3} = \frac{5}{3} y + \frac{1}{3} y$

Step 2.7

Add $15 z$ and $5 z$ .

$\frac{20 z + 70}{3} = \frac{5}{3} y + \frac{1}{3} y$

Step 2.8

Factor $10$ out of $20 z + 70$ .

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Step 2.8.1

Factor $10$ out of $20 z$ .

$\frac{10 (2 z) + 70}{3} = \frac{5}{3} y + \frac{1}{3} y$

Step 2.8.2

Factor $10$ out of $70$ .

$\frac{10 (2 z) + 10 (7)}{3} = \frac{5}{3} y + \frac{1}{3} y$

Step 2.8.3

Factor $10$ out of $10 (2 z) + 10 (7)$ .

$\frac{10 (2 z + 7)}{3} = \frac{5}{3} y + \frac{1}{3} y$

Step 3

Simplify $\frac{5}{3} y + \frac{1}{3} y$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Combine $\frac{5}{3}$ and $y$ .

$\frac{10 (2 z + 7)}{3} = \frac{5 y}{3} + \frac{1}{3} y$

Step 3.1.2

Combine $\frac{1}{3}$ and $y$ .

$\frac{10 (2 z + 7)}{3} = \frac{5 y}{3} + \frac{y}{3}$

Step 3.2

Simplify terms.

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Step 3.2.1

Combine the numerators over the common denominator.

$\frac{10 (2 z + 7)}{3} = \frac{5 y + y}{3}$

Step 3.2.2

Add $5 y$ and $y$ .

$\frac{10 (2 z + 7)}{3} = \frac{6 y}{3}$

Step 3.2.3

Cancel the common factor of $6$ and $3$ .

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Step 3.2.3.1

Factor $3$ out of $6 y$ .

$\frac{10 (2 z + 7)}{3} = \frac{3 (2 y)}{3}$

Step 3.2.3.2

Cancel the common factors.

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Step 3.2.3.2.1

Factor $3$ out of $3$ .

$\frac{10 (2 z + 7)}{3} = \frac{3 (2 y)}{3 (1)}$

Step 3.2.3.2.2

Cancel the common factor.

$\frac{10 (2 z + 7)}{3} = \frac{3 (2 y)}{3 \cdot 1}$

Step 3.2.3.2.3

Rewrite the expression.

$\frac{10 (2 z + 7)}{3} = \frac{2 y}{1}$

Step 3.2.3.2.4

Divide $2 y$ by $1$ .

$\frac{10 (2 z + 7)}{3} = 2 y$

Step 4

Multiply both sides by $3$ .

$\frac{10 (2 z + 7)}{3} \cdot 3 = 2 y \cdot 3$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $\frac{10 (2 z + 7)}{3} \cdot 3$ .

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Step 5.1.1.1

Cancel the common factor of $3$ .

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Step 5.1.1.1.1

Cancel the common factor.

$\frac{10 (2 z + 7)}{3} \cdot 3 = 2 y \cdot 3$

Step 5.1.1.1.2

Rewrite the expression.

$10 (2 z + 7) = 2 y \cdot 3$

Step 5.1.1.2

Apply the distributive property.

$10 (2 z) + 10 \cdot 7 = 2 y \cdot 3$

Step 5.1.1.3

Multiply.

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Step 5.1.1.3.1

Multiply $2$ by $10$ .

$20 z + 10 \cdot 7 = 2 y \cdot 3$

Step 5.1.1.3.2

Multiply $10$ by $7$ .

$20 z + 70 = 2 y \cdot 3$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $3$ by $2$ .

$20 z + 70 = 6 y$

Step 6

Solve for $z$ .

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Step 6.1

Subtract $70$ from both sides of the equation.

$20 z = 6 y - 70$

Step 6.2

Divide each term in $20 z = 6 y - 70$ by $20$ and simplify.

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Step 6.2.1

Divide each term in $20 z = 6 y - 70$ by $20$ .

$\frac{20 z}{20} = \frac{6 y}{20} + \frac{- 70}{20}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $20$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{20 z}{20} = \frac{6 y}{20} + \frac{- 70}{20}$

Step 6.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{6 y}{20} + \frac{- 70}{20}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Simplify each term.

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Step 6.2.3.1.1

Cancel the common factor of $6$ and $20$ .

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Step 6.2.3.1.1.1

Factor $2$ out of $6 y$ .

$z = \frac{2 (3 y)}{20} + \frac{- 70}{20}$

Step 6.2.3.1.1.2

Cancel the common factors.

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Step 6.2.3.1.1.2.1

Factor $2$ out of $20$ .

$z = \frac{2 (3 y)}{2 (10)} + \frac{- 70}{20}$

Step 6.2.3.1.1.2.2

Cancel the common factor.

$z = \frac{2 (3 y)}{2 \cdot 10} + \frac{- 70}{20}$

Step 6.2.3.1.1.2.3

Rewrite the expression.

$z = \frac{3 y}{10} + \frac{- 70}{20}$

Step 6.2.3.1.2

Cancel the common factor of $- 70$ and $20$ .

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Step 6.2.3.1.2.1

Factor $10$ out of $- 70$ .

$z = \frac{3 y}{10} + \frac{10 (- 7)}{20}$

Step 6.2.3.1.2.2

Cancel the common factors.

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Step 6.2.3.1.2.2.1

Factor $10$ out of $20$ .

$z = \frac{3 y}{10} + \frac{10 \cdot - 7}{10 \cdot 2}$

Step 6.2.3.1.2.2.2

Cancel the common factor.

$z = \frac{3 y}{10} + \frac{10 \cdot - 7}{10 \cdot 2}$

Step 6.2.3.1.2.2.3

Rewrite the expression.

$z = \frac{3 y}{10} + \frac{- 7}{2}$

Step 6.2.3.1.3

Move the negative in front of the fraction.

$z = \frac{3 y}{10} - \frac{7}{2}$