Basic Math Examples

$\frac{8}{13} y + \frac{1}{3} = \frac{2}{5} - \frac{5}{13} y + \frac{2}{3}$

Step 1

Combine $\frac{8}{13}$ and $y$ .

$\frac{8 y}{13} + \frac{1}{3} = \frac{2}{5} - \frac{5}{13} y + \frac{2}{3}$

Step 2

Simplify $\frac{2}{5} - \frac{5}{13} y + \frac{2}{3}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $y$ and $\frac{5}{13}$ .

$\frac{8 y}{13} + \frac{1}{3} = \frac{2}{5} - \frac{y \cdot 5}{13} + \frac{2}{3}$

Step 2.1.2

Move $5$ to the left of $y$ .

$\frac{8 y}{13} + \frac{1}{3} = \frac{2}{5} - \frac{5 y}{13} + \frac{2}{3}$

Step 2.2

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

$\frac{8 y}{13} + \frac{1}{3} = - \frac{5 y}{13} + \frac{2}{5} \cdot \frac{3}{3} + \frac{2}{3}$

Step 2.3

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

$\frac{8 y}{13} + \frac{1}{3} = - \frac{5 y}{13} + \frac{2}{5} \cdot \frac{3}{3} + \frac{2}{3} \cdot \frac{5}{5}$

Step 2.4

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

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

$\frac{8 y}{13} + \frac{1}{3} = - \frac{5 y}{13} + \frac{2 \cdot 3}{5 \cdot 3} + \frac{2}{3} \cdot \frac{5}{5}$

Step 2.4.2

Multiply $5$ by $3$ .

$\frac{8 y}{13} + \frac{1}{3} = - \frac{5 y}{13} + \frac{2 \cdot 3}{15} + \frac{2}{3} \cdot \frac{5}{5}$

Step 2.4.3

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

$\frac{8 y}{13} + \frac{1}{3} = - \frac{5 y}{13} + \frac{2 \cdot 3}{15} + \frac{2 \cdot 5}{3 \cdot 5}$

Step 2.4.4

Multiply $3$ by $5$ .

$\frac{8 y}{13} + \frac{1}{3} = - \frac{5 y}{13} + \frac{2 \cdot 3}{15} + \frac{2 \cdot 5}{15}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{8 y}{13} + \frac{1}{3} = - \frac{5 y}{13} + \frac{2 \cdot 3 + 2 \cdot 5}{15}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $2$ by $3$ .

$\frac{8 y}{13} + \frac{1}{3} = - \frac{5 y}{13} + \frac{6 + 2 \cdot 5}{15}$

Step 2.6.2

Multiply $2$ by $5$ .

$\frac{8 y}{13} + \frac{1}{3} = - \frac{5 y}{13} + \frac{6 + 10}{15}$

Step 2.6.3

Add $6$ and $10$ .

$\frac{8 y}{13} + \frac{1}{3} = - \frac{5 y}{13} + \frac{16}{15}$

Step 3

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

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Step 3.1

Add $\frac{5 y}{13}$ to both sides of the equation.

$\frac{8 y}{13} + \frac{1}{3} + \frac{5 y}{13} = \frac{16}{15}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{8 y + 5 y}{13} + \frac{1}{3} = \frac{16}{15}$

Step 3.3

Add $8 y$ and $5 y$ .

$\frac{13 y}{13} + \frac{1}{3} = \frac{16}{15}$

Step 3.4

Cancel the common factor of $13$ .

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Step 3.4.1

Cancel the common factor.

$\frac{13 y}{13} + \frac{1}{3} = \frac{16}{15}$

Step 3.4.2

Divide $y$ by $1$ .

$y + \frac{1}{3} = \frac{16}{15}$

Step 4

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

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Step 4.1

Subtract $\frac{1}{3}$ from both sides of the equation.

$y = \frac{16}{15} - \frac{1}{3}$

Step 4.2

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{16}{15} - \frac{1}{3} \cdot \frac{5}{5}$

Step 4.3

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{1}{3}$ by $\frac{5}{5}$ .

$y = \frac{16}{15} - \frac{5}{3 \cdot 5}$

Step 4.3.2

Multiply $3$ by $5$ .

$y = \frac{16}{15} - \frac{5}{15}$

Step 4.4

Combine the numerators over the common denominator.

$y = \frac{16 - 5}{15}$

Step 4.5

Subtract $5$ from $16$ .

$y = \frac{11}{15}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = \frac{11}{15}$

Decimal Form:

$y = 0.7\overline{3}$