Finite Math Examples

$7 x^{2} - 3 y^{2} = 0$ , $5 x^{2} + 7 y^{2} = 0$

Step 1

Solve for $x$ in $5 x^{2} + 7 y^{2} = 0$ .

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Step 1.1

Subtract $7 y^{2}$ from both sides of the equation.

$5 x^{2} = - 7 y^{2}$

$7 x^{2} - 3 y^{2} = 0$

Step 1.2

Divide each term in $5 x^{2} = - 7 y^{2}$ by $5$ and simplify.

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Step 1.2.1

Divide each term in $5 x^{2} = - 7 y^{2}$ by $5$ .

$\frac{5 x^{2}}{5} = \frac{- 7 y^{2}}{5}$

$7 x^{2} - 3 y^{2} = 0$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{5 x^{2}}{5} = \frac{- 7 y^{2}}{5}$

$7 x^{2} - 3 y^{2} = 0$

Step 1.2.2.1.2

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

$x^{2} = \frac{- 7 y^{2}}{5}$

$7 x^{2} - 3 y^{2} = 0$

$x^{2} = \frac{- 7 y^{2}}{5}$

$7 x^{2} - 3 y^{2} = 0$

$x^{2} = \frac{- 7 y^{2}}{5}$

$7 x^{2} - 3 y^{2} = 0$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{7 y^{2}}{5}$

$7 x^{2} - 3 y^{2} = 0$

$x^{2} = - \frac{7 y^{2}}{5}$

$7 x^{2} - 3 y^{2} = 0$

$x^{2} = - \frac{7 y^{2}}{5}$

$7 x^{2} - 3 y^{2} = 0$

Step 1.3

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

$x = \pm \sqrt{- \frac{7 y^{2}}{5}}$

$7 x^{2} - 3 y^{2} = 0$

Step 1.4

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

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Step 1.4.1

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

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$7 x^{2} - 3 y^{2} = 0$

Step 1.4.2

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

$x = - \sqrt{- \frac{7 y^{2}}{5}}$

$7 x^{2} - 3 y^{2} = 0$

Step 1.4.3

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

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$x = - \sqrt{- \frac{7 y^{2}}{5}}$

$7 x^{2} - 3 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$x = - \sqrt{- \frac{7 y^{2}}{5}}$

$7 x^{2} - 3 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$x = - \sqrt{- \frac{7 y^{2}}{5}}$

$7 x^{2} - 3 y^{2} = 0$

Step 2

Solve the system $x = \sqrt{- \frac{7 y^{2}}{5}}, 7 x^{2} - 3 y^{2} = 0$ .

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Step 2.1

Replace all occurrences of $x$ with $\sqrt{- \frac{7 y^{2}}{5}}$ in each equation.

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Step 2.1.1

Replace all occurrences of $x$ in $7 x^{2} - 3 y^{2} = 0$ with $\sqrt{- \frac{7 y^{2}}{5}}$ .

$7 {(\sqrt{- \frac{7 y^{2}}{5}})}^{2} - 3 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $7 {(\sqrt{- \frac{7 y^{2}}{5}})}^{2} - 3 y^{2}$ .

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Step 2.1.2.1.1

Simplify each term.

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Step 2.1.2.1.1.1

Rewrite ${\sqrt{- \frac{7 y^{2}}{5}}}^{2}$ as $- \frac{7 y^{2}}{5}$ .

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Step 2.1.2.1.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- \frac{7 y^{2}}{5}}$ as ${(- \frac{7 y^{2}}{5})}^{\frac{1}{2}}$ .

$7 {({(- \frac{7 y^{2}}{5})}^{\frac{1}{2}})}^{2} - 3 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.1.1.2

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

$7 {(- \frac{7 y^{2}}{5})}^{\frac{1}{2} \cdot 2} - 3 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.1.1.3

Combine $\frac{1}{2}$ and $2$ .

$7 {(- \frac{7 y^{2}}{5})}^{\frac{2}{2}} - 3 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.1.4.1

Cancel the common factor.

$7 {(- \frac{7 y^{2}}{5})}^{\frac{2}{2}} - 3 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.1.1.4.2

Rewrite the expression.

$7 (- \frac{7 y^{2}}{5}) - 3 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$7 (- \frac{7 y^{2}}{5}) - 3 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.1.1.5

Simplify.

$7 (- \frac{7 y^{2}}{5}) - 3 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$7 (- \frac{7 y^{2}}{5}) - 3 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.1.2

Multiply $7 (- \frac{7 y^{2}}{5})$ .

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Step 2.1.2.1.1.2.1

Multiply $- 1$ by $7$ .

$- 7 \frac{7 y^{2}}{5} - 3 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.1.2.2

Combine $- 7$ and $\frac{7 y^{2}}{5}$ .

$\frac{- 7 (7 y^{2})}{5} - 3 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.1.2.3

Multiply $7$ by $- 7$ .

$\frac{- 49 y^{2}}{5} - 3 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$\frac{- 49 y^{2}}{5} - 3 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.1.3

Move the negative in front of the fraction.

$- \frac{49 y^{2}}{5} - 3 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$- \frac{49 y^{2}}{5} - 3 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.2

To write $- 3 y^{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{49 y^{2}}{5} - 3 y^{2} \cdot \frac{5}{5} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.3

Simplify terms.

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Step 2.1.2.1.3.1

Combine $- 3 y^{2}$ and $\frac{5}{5}$ .

$- \frac{49 y^{2}}{5} + \frac{- 3 y^{2} \cdot 5}{5} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.3.2

Combine the numerators over the common denominator.

$\frac{- 49 y^{2} - 3 y^{2} \cdot 5}{5} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$\frac{- 49 y^{2} - 3 y^{2} \cdot 5}{5} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.4

Simplify the numerator.

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Step 2.1.2.1.4.1

Factor $y^{2}$ out of $- 49 y^{2} - 3 y^{2} \cdot 5$ .

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Step 2.1.2.1.4.1.1

Factor $y^{2}$ out of $- 49 y^{2}$ .

$\frac{y^{2} \cdot - 49 - 3 y^{2} \cdot 5}{5} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.4.1.2

Factor $y^{2}$ out of $- 3 y^{2} \cdot 5$ .

$\frac{y^{2} \cdot - 49 + y^{2} (- 3 \cdot 5)}{5} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.4.1.3

Factor $y^{2}$ out of $y^{2} \cdot - 49 + y^{2} (- 3 \cdot 5)$ .

$\frac{y^{2} (- 49 - 3 \cdot 5)}{5} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$\frac{y^{2} (- 49 - 3 \cdot 5)}{5} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.4.2

Multiply $- 3$ by $5$ .

$\frac{y^{2} (- 49 - 15)}{5} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.4.3

Subtract $15$ from $- 49$ .

$\frac{y^{2} \cdot - 64}{5} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$\frac{y^{2} \cdot - 64}{5} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.5

Simplify the expression.

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Step 2.1.2.1.5.1

Move $- 64$ to the left of $y^{2}$ .

$\frac{- 64 \cdot y^{2}}{5} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.1.2.1.5.2

Move the negative in front of the fraction.

$- \frac{64 y^{2}}{5} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$- \frac{64 y^{2}}{5} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$- \frac{64 y^{2}}{5} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$- \frac{64 y^{2}}{5} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$- \frac{64 y^{2}}{5} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.2

Solve for $y$ in $- \frac{64 y^{2}}{5} = 0$ .

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Step 2.2.1

Set the numerator equal to zero.

$64 y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.2.2

Solve the equation for $y$ .

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Step 2.2.2.1

Divide each term in $64 y^{2} = 0$ by $64$ and simplify.

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Step 2.2.2.1.1

Divide each term in $64 y^{2} = 0$ by $64$ .

$\frac{64 y^{2}}{64} = \frac{0}{64}$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.2.2.1.2

Simplify the left side.

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Step 2.2.2.1.2.1

Cancel the common factor of $64$ .

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Step 2.2.2.1.2.1.1

Cancel the common factor.

$\frac{64 y^{2}}{64} = \frac{0}{64}$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.2.2.1.2.1.2

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

$y^{2} = \frac{0}{64}$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$y^{2} = \frac{0}{64}$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$y^{2} = \frac{0}{64}$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.2.2.1.3

Simplify the right side.

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Step 2.2.2.1.3.1

Divide $0$ by $64$ .

$y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$y^{2} = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.2.2.2

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

$y = \pm \sqrt{0}$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.2.2.3

Simplify $\pm \sqrt{0}$ .

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Step 2.2.2.3.1

Rewrite $0$ as $0^{2}$ .

$y = \pm \sqrt{0^{2}}$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.2.2.3.2

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

$y = \pm 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.2.2.3.3

Plus or minus $0$ is $0$ .

$y = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$y = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$y = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

$y = 0$

$x = \sqrt{- \frac{7 y^{2}}{5}}$

Step 2.3

Replace all occurrences of $y$ with $0$ in each equation.

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Step 2.3.1

Replace all occurrences of $y$ in $x = \sqrt{- \frac{7 y^{2}}{5}}$ with $0$ .

$x = \sqrt{- \frac{7 {(0)}^{2}}{5}}$

$y = 0$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{- \frac{7 {(0)}^{2}}{5}}$ .

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Step 2.3.2.1.1

Raising $0$ to any positive power yields $0$ .

$x = \sqrt{- \frac{7 \cdot 0}{5}}$

$y = 0$

Step 2.3.2.1.2

Multiply $7$ by $0$ .

$x = \sqrt{- \frac{0}{5}}$

$y = 0$

Step 2.3.2.1.3

Divide $0$ by $5$ .

$x = \sqrt{- 0}$

$y = 0$

Step 2.3.2.1.4

Multiply $- 1$ by $0$ .

$x = \sqrt{0}$

$y = 0$

Step 2.3.2.1.5

Rewrite $0$ as $0^{2}$ .

$x = \sqrt{0^{2}}$

$y = 0$

Step 2.3.2.1.6

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

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

Step 3

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(0, 0)$

Step 4

The result can be shown in multiple forms.

Point Form:

$(0, 0)$

Equation Form:

$x = 0, y = 0$

Step 5