Algebra Examples

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

Step 1

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

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Step 1.1

Rewrite.

$0 + 0 + \frac{1}{3} y = \frac{2}{3} y^{2}$

Step 1.2

Simplify by adding zeros.

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

Step 1.3

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

$\frac{y}{3} = \frac{2}{3} y^{2}$

Step 2

Combine $\frac{2}{3}$ and $y^{2}$ .

$\frac{y}{3} = \frac{2 y^{2}}{3}$

Step 3

Subtract $\frac{2 y^{2}}{3}$ from both sides of the equation.

$\frac{y}{3} - \frac{2 y^{2}}{3} = 0$

Step 4

Multiply each term in $\frac{y}{3} - \frac{2 y^{2}}{3} = 0$ by $3$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{y}{3} - \frac{2 y^{2}}{3} = 0$ by $3$ .

$\frac{y}{3} \cdot 3 - \frac{2 y^{2}}{3} \cdot 3 = 0 \cdot 3$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Cancel the common factor of $3$ .

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Step 4.2.1.1.1

Cancel the common factor.

$\frac{y}{3} \cdot 3 - \frac{2 y^{2}}{3} \cdot 3 = 0 \cdot 3$

Step 4.2.1.1.2

Rewrite the expression.

$y - \frac{2 y^{2}}{3} \cdot 3 = 0 \cdot 3$

Step 4.2.1.2

Cancel the common factor of $3$ .

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Step 4.2.1.2.1

Move the leading negative in $- \frac{2 y^{2}}{3}$ into the numerator.

$y + \frac{- 2 y^{2}}{3} \cdot 3 = 0 \cdot 3$

Step 4.2.1.2.2

Cancel the common factor.

$y + \frac{- 2 y^{2}}{3} \cdot 3 = 0 \cdot 3$

Step 4.2.1.2.3

Rewrite the expression.

$y - 2 y^{2} = 0 \cdot 3$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply $0$ by $3$ .

$y - 2 y^{2} = 0$

Step 5

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

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Step 5.1

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

$y \cdot 1 - 2 y^{2} = 0$

Step 5.2

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

$y \cdot 1 + y (- 2 y) = 0$

Step 5.3

Factor $y$ out of $y \cdot 1 + y (- 2 y)$ .

$y (1 - 2 y) = 0$

Step 6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y = 0$

$1 - 2 y = 0$

Step 7

Set $y$ equal to $0$ .

$y = 0$

Step 8

Set $1 - 2 y$ equal to $0$ and solve for $y$ .

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Step 8.1

Set $1 - 2 y$ equal to $0$ .

$1 - 2 y = 0$

Step 8.2

Solve $1 - 2 y = 0$ for $y$ .

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Step 8.2.1

Subtract $1$ from both sides of the equation.

$- 2 y = - 1$

Step 8.2.2

Divide each term in $- 2 y = - 1$ by $- 2$ and simplify.

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Step 8.2.2.1

Divide each term in $- 2 y = - 1$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{- 1}{- 2}$

Step 8.2.2.2

Simplify the left side.

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Step 8.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 8.2.2.2.1.1

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{- 1}{- 2}$

Step 8.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 1}{- 2}$

Step 8.2.2.3

Simplify the right side.

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Step 8.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 9

The final solution is all the values that make $y (1 - 2 y) = 0$ true.

$y = 0, \frac{1}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$y = 0, \frac{1}{2}$

Decimal Form:

$y = 0, 0.5$