Finite Math Examples

$20 x^{2} - 2 y = 0$ , $30 y^{2} - 2 x = 0$

Step 1

Solve for $y$ in $20 x^{2} - 2 y = 0$ .

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Step 1.1

Subtract $20 x^{2}$ from both sides of the equation.

$- 2 y = - 20 x^{2}$

$30 y^{2} - 2 x = 0$

Step 1.2

Divide each term in $- 2 y = - 20 x^{2}$ by $- 2$ and simplify.

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Step 1.2.1

Divide each term in $- 2 y = - 20 x^{2}$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{- 20 x^{2}}{- 2}$

$30 y^{2} - 2 x = 0$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{- 20 x^{2}}{- 2}$

$30 y^{2} - 2 x = 0$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 20 x^{2}}{- 2}$

$30 y^{2} - 2 x = 0$

$y = \frac{- 20 x^{2}}{- 2}$

$30 y^{2} - 2 x = 0$

$y = \frac{- 20 x^{2}}{- 2}$

$30 y^{2} - 2 x = 0$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $- 20$ and $- 2$ .

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Step 1.2.3.1.1

Factor $- 2$ out of $- 20 x^{2}$ .

$y = \frac{- 2 (10 x^{2})}{- 2}$

$30 y^{2} - 2 x = 0$

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Factor $- 2$ out of $- 2$ .

$y = \frac{- 2 (10 x^{2})}{- 2 \cdot 1}$

$30 y^{2} - 2 x = 0$

Step 1.2.3.1.2.2

Cancel the common factor.

$y = \frac{- 2 (10 x^{2})}{- 2 \cdot 1}$

$30 y^{2} - 2 x = 0$

Step 1.2.3.1.2.3

Rewrite the expression.

$y = \frac{10 x^{2}}{1}$

$30 y^{2} - 2 x = 0$

Step 1.2.3.1.2.4

Divide $10 x^{2}$ by $1$ .

$y = 10 x^{2}$

$30 y^{2} - 2 x = 0$

$y = 10 x^{2}$

$30 y^{2} - 2 x = 0$

$y = 10 x^{2}$

$30 y^{2} - 2 x = 0$

$y = 10 x^{2}$

$30 y^{2} - 2 x = 0$

$y = 10 x^{2}$

$30 y^{2} - 2 x = 0$

$y = 10 x^{2}$

$30 y^{2} - 2 x = 0$

Step 2

Replace all occurrences of $y$ with $10 x^{2}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $30 y^{2} - 2 x = 0$ with $10 x^{2}$ .

$30 {(10 x^{2})}^{2} - 2 x = 0$

$y = 10 x^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the product rule to $10 x^{2}$ .

$30 (10^{2} {(x^{2})}^{2}) - 2 x = 0$

$y = 10 x^{2}$

Step 2.2.1.2

Raise $10$ to the power of $2$ .

$30 (100 {(x^{2})}^{2}) - 2 x = 0$

$y = 10 x^{2}$

Step 2.2.1.3

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 2.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$30 (100 x^{2 \cdot 2}) - 2 x = 0$

$y = 10 x^{2}$

Step 2.2.1.3.2

Multiply $2$ by $2$ .

$30 (100 x^{4}) - 2 x = 0$

$y = 10 x^{2}$

$30 (100 x^{4}) - 2 x = 0$

$y = 10 x^{2}$

Step 2.2.1.4

Multiply $100$ by $30$ .

$3000 x^{4} - 2 x = 0$

$y = 10 x^{2}$

$3000 x^{4} - 2 x = 0$

$y = 10 x^{2}$

$3000 x^{4} - 2 x = 0$

$y = 10 x^{2}$

$3000 x^{4} - 2 x = 0$

$y = 10 x^{2}$

Step 3

Solve for $x$ in $3000 x^{4} - 2 x = 0$ .

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Step 3.1

Factor $2 x$ out of $3000 x^{4} - 2 x$ .

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Step 3.1.1

Factor $2 x$ out of $3000 x^{4}$ .

$2 x (1500 x^{3}) - 2 x = 0$

$y = 10 x^{2}$

Step 3.1.2

Factor $2 x$ out of $- 2 x$ .

$2 x (1500 x^{3}) + 2 x (- 1) = 0$

$y = 10 x^{2}$

Step 3.1.3

Factor $2 x$ out of $2 x (1500 x^{3}) + 2 x (- 1)$ .

$2 x (1500 x^{3} - 1) = 0$

$y = 10 x^{2}$

$2 x (1500 x^{3} - 1) = 0$

$y = 10 x^{2}$

Step 3.2

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

$x = 0$

$1500 x^{3} - 1 = 0$

$y = 10 x^{2}$

Step 3.3

Set $x$ equal to $0$ .

$x = 0$

$y = 10 x^{2}$

Step 3.4

Set $1500 x^{3} - 1$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $1500 x^{3} - 1$ equal to $0$ .

$1500 x^{3} - 1 = 0$

$y = 10 x^{2}$

Step 3.4.2

Solve $1500 x^{3} - 1 = 0$ for $x$ .

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Step 3.4.2.1

Add $1$ to both sides of the equation.

$1500 x^{3} = 1$

$y = 10 x^{2}$

Step 3.4.2.2

Divide each term in $1500 x^{3} = 1$ by $1500$ and simplify.

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Step 3.4.2.2.1

Divide each term in $1500 x^{3} = 1$ by $1500$ .

$\frac{1500 x^{3}}{1500} = \frac{1}{1500}$

$y = 10 x^{2}$

Step 3.4.2.2.2

Simplify the left side.

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Step 3.4.2.2.2.1

Cancel the common factor of $1500$ .

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Step 3.4.2.2.2.1.1

Cancel the common factor.

$\frac{1500 x^{3}}{1500} = \frac{1}{1500}$

$y = 10 x^{2}$

Step 3.4.2.2.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{1}{1500}$

$y = 10 x^{2}$

$x^{3} = \frac{1}{1500}$

$y = 10 x^{2}$

$x^{3} = \frac{1}{1500}$

$y = 10 x^{2}$

$x^{3} = \frac{1}{1500}$

$y = 10 x^{2}$

Step 3.4.2.3

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

$x = \sqrt[3]{\frac{1}{1500}}$

$y = 10 x^{2}$

Step 3.4.2.4

Simplify $\sqrt[3]{\frac{1}{1500}}$ .

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Step 3.4.2.4.1

Rewrite $\sqrt[3]{\frac{1}{1500}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{1500}}$ .

$x = \frac{\sqrt[3]{1}}{\sqrt[3]{1500}}$

$y = 10 x^{2}$

Step 3.4.2.4.2

Any root of $1$ is $1$ .

$x = \frac{1}{\sqrt[3]{1500}}$

$y = 10 x^{2}$

Step 3.4.2.4.3

Simplify the denominator.

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Step 3.4.2.4.3.1

Rewrite $1500$ as $5^{3} \cdot 12$ .

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Step 3.4.2.4.3.1.1

Factor $125$ out of $1500$ .

$x = \frac{1}{\sqrt[3]{125 (12)}}$

$y = 10 x^{2}$

Step 3.4.2.4.3.1.2

Rewrite $125$ as $5^{3}$ .

$x = \frac{1}{\sqrt[3]{5^{3} \cdot 12}}$

$y = 10 x^{2}$

$x = \frac{1}{\sqrt[3]{5^{3} \cdot 12}}$

$y = 10 x^{2}$

Step 3.4.2.4.3.2

Pull terms out from under the radical.

$x = \frac{1}{5 \sqrt[3]{12}}$

$y = 10 x^{2}$

$x = \frac{1}{5 \sqrt[3]{12}}$

$y = 10 x^{2}$

Step 3.4.2.4.4

Multiply $\frac{1}{5 \sqrt[3]{12}}$ by $\frac{{\sqrt[3]{12}}^{2}}{{\sqrt[3]{12}}^{2}}$ .

$x = \frac{1}{5 \sqrt[3]{12}} \cdot \frac{{\sqrt[3]{12}}^{2}}{{\sqrt[3]{12}}^{2}}$

$y = 10 x^{2}$

Step 3.4.2.4.5

Combine and simplify the denominator.

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Step 3.4.2.4.5.1

Multiply $\frac{1}{5 \sqrt[3]{12}}$ by $\frac{{\sqrt[3]{12}}^{2}}{{\sqrt[3]{12}}^{2}}$ .

$x = \frac{{\sqrt[3]{12}}^{2}}{5 \sqrt[3]{12} {\sqrt[3]{12}}^{2}}$

$y = 10 x^{2}$

Step 3.4.2.4.5.2

Move $\sqrt[3]{12}$ .

$x = \frac{{\sqrt[3]{12}}^{2}}{5 (\sqrt[3]{12} {\sqrt[3]{12}}^{2})}$

$y = 10 x^{2}$

Step 3.4.2.4.5.3

Raise $\sqrt[3]{12}$ to the power of $1$ .

$x = \frac{{\sqrt[3]{12}}^{2}}{5 (\sqrt[3]{12} {\sqrt[3]{12}}^{2})}$

$y = 10 x^{2}$

Step 3.4.2.4.5.4

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

$x = \frac{{\sqrt[3]{12}}^{2}}{5 {\sqrt[3]{12}}^{1 + 2}}$

$y = 10 x^{2}$

Step 3.4.2.4.5.5

Add $1$ and $2$ .

$x = \frac{{\sqrt[3]{12}}^{2}}{5 {\sqrt[3]{12}}^{3}}$

$y = 10 x^{2}$

Step 3.4.2.4.5.6

Rewrite ${\sqrt[3]{12}}^{3}$ as $12$ .

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Step 3.4.2.4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{12}$ as $12^{\frac{1}{3}}$ .

$x = \frac{{\sqrt[3]{12}}^{2}}{5 {(12^{\frac{1}{3}})}^{3}}$

$y = 10 x^{2}$

Step 3.4.2.4.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{{\sqrt[3]{12}}^{2}}{5 \cdot 12^{\frac{1}{3} \cdot 3}}$

$y = 10 x^{2}$

Step 3.4.2.4.5.6.3

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

$x = \frac{{\sqrt[3]{12}}^{2}}{5 \cdot 12^{\frac{3}{3}}}$

$y = 10 x^{2}$

Step 3.4.2.4.5.6.4

Cancel the common factor of $3$ .

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Step 3.4.2.4.5.6.4.1

Cancel the common factor.

$x = \frac{{\sqrt[3]{12}}^{2}}{5 \cdot 12^{\frac{3}{3}}}$

$y = 10 x^{2}$

Step 3.4.2.4.5.6.4.2

Rewrite the expression.