Basic Math Examples

$\frac{y - 6}{2} + \frac{y + 5}{21} = \frac{13}{6}$

Step 1

Simplify $\frac{y - 6}{2} + \frac{y + 5}{21}$ .

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Step 1.1

To write $\frac{y - 6}{2}$ as a fraction with a common denominator, multiply by $\frac{21}{21}$ .

$\frac{y - 6}{2} \cdot \frac{21}{21} + \frac{y + 5}{21} = \frac{13}{6}$

Step 1.2

To write $\frac{y + 5}{21}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{y - 6}{2} \cdot \frac{21}{21} + \frac{y + 5}{21} \cdot \frac{2}{2} = \frac{13}{6}$

Step 1.3

Write each expression with a common denominator of $42$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{y - 6}{2}$ by $\frac{21}{21}$ .

$\frac{(y - 6) \cdot 21}{2 \cdot 21} + \frac{y + 5}{21} \cdot \frac{2}{2} = \frac{13}{6}$

Step 1.3.2

Multiply $2$ by $21$ .

$\frac{(y - 6) \cdot 21}{42} + \frac{y + 5}{21} \cdot \frac{2}{2} = \frac{13}{6}$

Step 1.3.3

Multiply $\frac{y + 5}{21}$ by $\frac{2}{2}$ .

$\frac{(y - 6) \cdot 21}{42} + \frac{(y + 5) \cdot 2}{21 \cdot 2} = \frac{13}{6}$

Step 1.3.4

Multiply $21$ by $2$ .

$\frac{(y - 6) \cdot 21}{42} + \frac{(y + 5) \cdot 2}{42} = \frac{13}{6}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{(y - 6) \cdot 21 + (y + 5) \cdot 2}{42} = \frac{13}{6}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{y \cdot 21 - 6 \cdot 21 + (y + 5) \cdot 2}{42} = \frac{13}{6}$

Step 1.5.2

Move $21$ to the left of $y$ .

$\frac{21 \cdot y - 6 \cdot 21 + (y + 5) \cdot 2}{42} = \frac{13}{6}$

Step 1.5.3

Multiply $- 6$ by $21$ .

$\frac{21 \cdot y - 126 + (y + 5) \cdot 2}{42} = \frac{13}{6}$

Step 1.5.4

Apply the distributive property.

$\frac{21 y - 126 + y \cdot 2 + 5 \cdot 2}{42} = \frac{13}{6}$

Step 1.5.5

Move $2$ to the left of $y$ .

$\frac{21 y - 126 + 2 \cdot y + 5 \cdot 2}{42} = \frac{13}{6}$

Step 1.5.6

Multiply $5$ by $2$ .

$\frac{21 y - 126 + 2 \cdot y + 10}{42} = \frac{13}{6}$

Step 1.5.7

Add $21 y$ and $2 y$ .

$\frac{23 y - 126 + 10}{42} = \frac{13}{6}$

Step 1.5.8

Add $- 126$ and $10$ .

$\frac{23 y - 116}{42} = \frac{13}{6}$

Step 2

Multiply both sides by $42$ .

$\frac{23 y - 116}{42} \cdot 42 = \frac{13}{6} \cdot 42$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $42$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{23 y - 116}{42} \cdot 42 = \frac{13}{6} \cdot 42$

Step 3.1.1.2

Rewrite the expression.

$23 y - 116 = \frac{13}{6} \cdot 42$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{13}{6} \cdot 42$ .

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Step 3.2.1.1

Cancel the common factor of $6$ .

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Step 3.2.1.1.1

Factor $6$ out of $42$ .

$23 y - 116 = \frac{13}{6} \cdot (6 (7))$

Step 3.2.1.1.2

Cancel the common factor.

$23 y - 116 = \frac{13}{6} \cdot (6 \cdot 7)$

Step 3.2.1.1.3

Rewrite the expression.

$23 y - 116 = 13 \cdot 7$

Step 3.2.1.2

Multiply $13$ by $7$ .

$23 y - 116 = 91$

Step 4

Solve for $y$ .

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Step 4.1

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

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Step 4.1.1

Add $116$ to both sides of the equation.

$23 y = 91 + 116$

Step 4.1.2

Add $91$ and $116$ .

$23 y = 207$

Step 4.2

Divide each term in $23 y = 207$ by $23$ and simplify.

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Step 4.2.1

Divide each term in $23 y = 207$ by $23$ .

$\frac{23 y}{23} = \frac{207}{23}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $23$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{23 y}{23} = \frac{207}{23}$

Step 4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{207}{23}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $207$ by $23$ .

$y = 9$