Basic Math Examples

$\frac{y}{2} \cdot (y + 6) - y^{2} + 5 = 7 - y \cdot (3 + \frac{y}{2})$

Step 1

Simplify $\frac{y}{2} \cdot (y + 6) - y^{2} + 5$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{y}{2} y + \frac{y}{2} \cdot 6 - y^{2} + 5 = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.1.2

Multiply $\frac{y}{2} y$ .

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Step 1.1.2.1

Combine $\frac{y}{2}$ and $y$ .

$\frac{y \cdot y}{2} + \frac{y}{2} \cdot 6 - y^{2} + 5 = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.1.2.2

Raise $y$ to the power of $1$ .

$\frac{y^{1} y}{2} + \frac{y}{2} \cdot 6 - y^{2} + 5 = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.1.2.3

Raise $y$ to the power of $1$ .

$\frac{y^{1} y^{1}}{2} + \frac{y}{2} \cdot 6 - y^{2} + 5 = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.1.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{y^{1 + 1}}{2} + \frac{y}{2} \cdot 6 - y^{2} + 5 = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.1.2.5

Add $1$ and $1$ .

$\frac{y^{2}}{2} + \frac{y}{2} \cdot 6 - y^{2} + 5 = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.1.3

Cancel the common factor of $2$ .

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Step 1.1.3.1

Factor $2$ out of $6$ .

$\frac{y^{2}}{2} + \frac{y}{2} \cdot (2 (3)) - y^{2} + 5 = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.1.3.2

Cancel the common factor.

$\frac{y^{2}}{2} + \frac{y}{2} \cdot (2 \cdot 3) - y^{2} + 5 = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.1.3.3

Rewrite the expression.

$\frac{y^{2}}{2} + y \cdot 3 - y^{2} + 5 = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.1.4

Move $3$ to the left of $y$ .

$\frac{y^{2}}{2} + 3 y - y^{2} + 5 = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.2

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

$3 y + \frac{y^{2}}{2} - y^{2} \cdot \frac{2}{2} + 5 = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.3

Combine $- y^{2}$ and $\frac{2}{2}$ .

$3 y + \frac{y^{2}}{2} + \frac{- y^{2} \cdot 2}{2} + 5 = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.4

Combine the numerators over the common denominator.

$3 y + \frac{y^{2} - y^{2} \cdot 2}{2} + 5 = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.5

Find the common denominator.

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Step 1.5.1

Write $3 y$ as a fraction with denominator $1$ .

$\frac{3 y}{1} + \frac{y^{2} - y^{2} \cdot 2}{2} + 5 = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.5.2

Multiply $\frac{3 y}{1}$ by $\frac{2}{2}$ .

$\frac{3 y}{1} \cdot \frac{2}{2} + \frac{y^{2} - y^{2} \cdot 2}{2} + 5 = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.5.3

Multiply $\frac{3 y}{1}$ by $\frac{2}{2}$ .

$\frac{3 y \cdot 2}{2} + \frac{y^{2} - y^{2} \cdot 2}{2} + 5 = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.5.4

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

$\frac{3 y \cdot 2}{2} + \frac{y^{2} - y^{2} \cdot 2}{2} + \frac{5}{1} = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.5.5

Multiply $\frac{5}{1}$ by $\frac{2}{2}$ .

$\frac{3 y \cdot 2}{2} + \frac{y^{2} - y^{2} \cdot 2}{2} + \frac{5}{1} \cdot \frac{2}{2} = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.5.6

Multiply $\frac{5}{1}$ by $\frac{2}{2}$ .

$\frac{3 y \cdot 2}{2} + \frac{y^{2} - y^{2} \cdot 2}{2} + \frac{5 \cdot 2}{2} = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.6

Combine the numerators over the common denominator.

$\frac{3 y \cdot 2 + y^{2} - y^{2} \cdot 2 + 5 \cdot 2}{2} = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.7

Simplify each term.

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Step 1.7.1

Multiply $2$ by $3$ .

$\frac{6 y + y^{2} - y^{2} \cdot 2 + 5 \cdot 2}{2} = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.7.2

Multiply $2$ by $- 1$ .

$\frac{6 y + y^{2} - 2 y^{2} + 5 \cdot 2}{2} = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.7.3

Multiply $5$ by $2$ .

$\frac{6 y + y^{2} - 2 y^{2} + 10}{2} = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.8

Simplify terms.

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Step 1.8.1

Subtract $2 y^{2}$ from $y^{2}$ .

$\frac{6 y - y^{2} + 10}{2} = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.8.2

Reorder terms.

$\frac{- y^{2} + 6 y + 10}{2} = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.8.3

Factor $- 1$ out of $- y^{2}$ .

$\frac{- (y^{2}) + 6 y + 10}{2} = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.8.4

Factor $- 1$ out of $6 y$ .

$\frac{- (y^{2}) - (- 6 y) + 10}{2} = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.8.5

Factor $- 1$ out of $- (y^{2}) - (- 6 y)$ .

$\frac{- (y^{2} - 6 y) + 10}{2} = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.8.6

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{- (y^{2} - 6 y) - 1 (- 10)}{2} = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.8.7

Factor $- 1$ out of $- (y^{2} - 6 y) - 1 (- 10)$ .

$\frac{- (y^{2} - 6 y - 10)}{2} = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.8.8

Simplify the expression.

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Step 1.8.8.1

Rewrite $- (y^{2} - 6 y - 10)$ as $- 1 (y^{2} - 6 y - 10)$ .

$\frac{- 1 (y^{2} - 6 y - 10)}{2} = 7 - y \cdot (3 + \frac{y}{2})$

Step 1.8.8.2

Move the negative in front of the fraction.

$- \frac{y^{2} - 6 y - 10}{2} = 7 - y \cdot (3 + \frac{y}{2})$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$- \frac{y^{2} - 6 y - 10}{2} = 7 - y \cdot 3 - y \frac{y}{2}$

Step 2.2

Multiply $3$ by $- 1$ .

$- \frac{y^{2} - 6 y - 10}{2} = 7 - 3 y - y \frac{y}{2}$

Step 2.3

Multiply $- y \frac{y}{2}$ .

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Step 2.3.1

Combine $\frac{y}{2}$ and $y$ .

$- \frac{y^{2} - 6 y - 10}{2} = 7 - 3 y - \frac{y \cdot y}{2}$

Step 2.3.2

Raise $y$ to the power of $1$ .

$- \frac{y^{2} - 6 y - 10}{2} = 7 - 3 y - \frac{y^{1} y}{2}$

Step 2.3.3

Raise $y$ to the power of $1$ .

$- \frac{y^{2} - 6 y - 10}{2} = 7 - 3 y - \frac{y^{1} y^{1}}{2}$

Step 2.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{y^{2} - 6 y - 10}{2} = 7 - 3 y - \frac{y^{1 + 1}}{2}$

Step 2.3.5

Add $1$ and $1$ .

$- \frac{y^{2} - 6 y - 10}{2} = 7 - 3 y - \frac{y^{2}}{2}$

Step 3

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

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Step 3.1

Add $3 y$ to both sides of the equation.

$- \frac{y^{2} - 6 y - 10}{2} + 3 y = 7 - \frac{y^{2}}{2}$

Step 3.2

Add $\frac{y^{2}}{2}$ to both sides of the equation.

$- \frac{y^{2} - 6 y - 10}{2} + 3 y + \frac{y^{2}}{2} = 7$

Step 3.3

Combine the numerators over the common denominator.

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

Step 3.4

Simplify each term.

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Step 3.4.1

Apply the distributive property.

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

Step 3.4.2

Simplify.

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Step 3.4.2.1

Multiply $- 6$ by $- 1$ .

$3 y + \frac{- y^{2} + 6 y - - 10 + y^{2}}{2} = 7$

Step 3.4.2.2

Multiply $- 1$ by $- 10$ .

$3 y + \frac{- y^{2} + 6 y + 10 + y^{2}}{2} = 7$

Step 3.5

Combine the opposite terms in $- y^{2} + 6 y + 10 + y^{2}$ .

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Step 3.5.1

Add $- y^{2}$ and $y^{2}$ .

$3 y + \frac{6 y + 10 + 0}{2} = 7$

Step 3.5.2

Add $6 y + 10$ and $0$ .

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

Step 3.6

Cancel the common factor of $6 y + 10$ and $2$ .

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Step 3.6.1

Factor $2$ out of $6 y$ .

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

Step 3.6.2

Factor $2$ out of $10$ .

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

Step 3.6.3

Factor $2$ out of $2 (3 y) + 2 (5)$ .

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

Step 3.6.4

Cancel the common factors.

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Step 3.6.4.1

Factor $2$ out of $2$ .

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

Step 3.6.4.2

Cancel the common factor.

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

Step 3.6.4.3

Rewrite the expression.

$3 y + \frac{3 y + 5}{1} = 7$

Step 3.6.4.4

Divide $3 y + 5$ by $1$ .

$3 y + 3 y + 5 = 7$

Step 3.7

Add $3 y$ and $3 y$ .

$6 y + 5 = 7$

Step 4

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

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Step 4.1

Subtract $5$ from both sides of the equation.

$6 y = 7 - 5$

Step 4.2

Subtract $5$ from $7$ .

$6 y = 2$

Step 5

Divide each term in $6 y = 2$ by $6$ and simplify.

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Step 5.1

Divide each term in $6 y = 2$ by $6$ .

$\frac{6 y}{6} = \frac{2}{6}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $6$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{2}{6}$

Step 5.2.1.2

Divide $y$ by $1$ .

$y = \frac{2}{6}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $2$ and $6$ .

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Step 5.3.1.1

Factor $2$ out of $2$ .

$y = \frac{2 (1)}{6}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $2$ out of $6$ .

$y = \frac{2 \cdot 1}{2 \cdot 3}$

Step 5.3.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot 1}{2 \cdot 3}$

Step 5.3.1.2.3

Rewrite the expression.

$y = \frac{1}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = \frac{1}{3}$

Decimal Form:

$y = 0.\overline{3}$