Algebra Examples

$- \frac{3}{7} y^{2} + 2 \frac{1}{3} = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify $- \frac{3}{7} y^{2} + 2 \frac{1}{3}$ .

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Step 1.1.1

Convert $2 \frac{1}{3}$ to an improper fraction.

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Step 1.1.1.1

A mixed number is an addition of its whole and fractional parts.

$- \frac{3}{7} y^{2} + 2 + \frac{1}{3} = 0$

Step 1.1.1.2

Add $2$ and $\frac{1}{3}$ .

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Step 1.1.1.2.1

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

$- \frac{3}{7} y^{2} + 2 \cdot \frac{3}{3} + \frac{1}{3} = 0$

Step 1.1.1.2.2

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

$- \frac{3}{7} y^{2} + \frac{2 \cdot 3}{3} + \frac{1}{3} = 0$

Step 1.1.1.2.3

Combine the numerators over the common denominator.

$- \frac{3}{7} y^{2} + \frac{2 \cdot 3 + 1}{3} = 0$

Step 1.1.1.2.4

Simplify the numerator.

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Step 1.1.1.2.4.1

Multiply $2$ by $3$ .

$- \frac{3}{7} y^{2} + \frac{6 + 1}{3} = 0$

Step 1.1.1.2.4.2

Add $6$ and $1$ .

$- \frac{3}{7} y^{2} + \frac{7}{3} = 0$

Step 1.1.2

Simplify each term.

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Step 1.1.2.1

Combine $y^{2}$ and $\frac{3}{7}$ .

$- \frac{y^{2} \cdot 3}{7} + \frac{7}{3} = 0$

Step 1.1.2.2

Move $3$ to the left of $y^{2}$ .

$- \frac{3 y^{2}}{7} + \frac{7}{3} = 0$

Step 2

Subtract $\frac{7}{3}$ from both sides of the equation.

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

Step 3

Multiply both sides of the equation by $- \frac{7}{3}$ .

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

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $- \frac{7}{3} (- \frac{3 y^{2}}{7})$ .

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Step 4.1.1.1

Cancel the common factor of $7$ .

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Step 4.1.1.1.1

Move the leading negative in $- \frac{7}{3}$ into the numerator.

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

Step 4.1.1.1.2

Move the leading negative in $- \frac{3 y^{2}}{7}$ into the numerator.

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

Step 4.1.1.1.3

Factor $7$ out of $- 7$ .

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

Step 4.1.1.1.4

Cancel the common factor.

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

Step 4.1.1.1.5

Rewrite the expression.

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

Step 4.1.1.2

Cancel the common factor of $3$ .

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Step 4.1.1.2.1

Factor $3$ out of $- 3 y^{2}$ .

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

Step 4.1.1.2.2

Cancel the common factor.

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

Step 4.1.1.2.3

Rewrite the expression.

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

Step 4.1.1.3

Multiply.

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Step 4.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.1.3.2

Multiply $y^{2}$ by $1$ .

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

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $- \frac{7}{3} (- \frac{7}{3})$ .

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Step 4.2.1.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2.1.2

Multiply $\frac{7}{3}$ by $1$ .

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

Step 4.2.1.3

Multiply $\frac{7}{3}$ by $\frac{7}{3}$ .

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

Step 4.2.1.4

Multiply $7$ by $7$ .

$y^{2} = \frac{49}{3 \cdot 3}$

Step 4.2.1.5

Multiply $3$ by $3$ .

$y^{2} = \frac{49}{9}$

Step 5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{\frac{49}{9}}$

Step 6

Simplify $\pm \sqrt{\frac{49}{9}}$ .

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Step 6.1

Rewrite $\sqrt{\frac{49}{9}}$ as $\frac{\sqrt{49}}{\sqrt{9}}$ .

$y = \pm \frac{\sqrt{49}}{\sqrt{9}}$

Step 6.2

Simplify the numerator.

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Step 6.2.1

Rewrite $49$ as $7^{2}$ .

$y = \pm \frac{\sqrt{7^{2}}}{\sqrt{9}}$

Step 6.2.2

Pull terms out from under the radical, assuming positive real numbers.

$y = \pm \frac{7}{\sqrt{9}}$

Step 6.3

Simplify the denominator.

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Step 6.3.1

Rewrite $9$ as $3^{2}$ .

$y = \pm \frac{7}{\sqrt{3^{2}}}$

Step 6.3.2

Pull terms out from under the radical, assuming positive real numbers.

$y = \pm \frac{7}{3}$

Step 7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{7}{3}$

Step 7.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{7}{3}$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{7}{3}, - \frac{7}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$y = \frac{7}{3}, - \frac{7}{3}$

Decimal Form:

$y = 2.\overline{3}, - 2.\overline{3}$

Mixed Number Form:

$y = 2 \frac{1}{3}, - 2 \frac{1}{3}$