Finite Math Examples

$3 x^{2} + 3 y^{2} = 30$ , $9 x^{2} + 9 y^{2} = 60$

Step 1

Solve for $x$ in $3 x^{2} + 3 y^{2} = 30$ .

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Step 1.1

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

$3 x^{2} = 30 - 3 y^{2}$

$9 x^{2} + 9 y^{2} = 60$

Step 1.2

Divide each term in $3 x^{2} = 30 - 3 y^{2}$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x^{2} = 30 - 3 y^{2}$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{30}{3} + \frac{- 3 y^{2}}{3}$

$9 x^{2} + 9 y^{2} = 60$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{30}{3} + \frac{- 3 y^{2}}{3}$

$9 x^{2} + 9 y^{2} = 60$

Step 1.2.2.1.2

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

$x^{2} = \frac{30}{3} + \frac{- 3 y^{2}}{3}$

$9 x^{2} + 9 y^{2} = 60$

$x^{2} = \frac{30}{3} + \frac{- 3 y^{2}}{3}$

$9 x^{2} + 9 y^{2} = 60$

$x^{2} = \frac{30}{3} + \frac{- 3 y^{2}}{3}$

$9 x^{2} + 9 y^{2} = 60$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $30$ by $3$ .

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

$9 x^{2} + 9 y^{2} = 60$

Step 1.2.3.1.2

Cancel the common factor of $- 3$ and $3$ .

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Step 1.2.3.1.2.1

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

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

$9 x^{2} + 9 y^{2} = 60$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $3$ out of $3$ .

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

$9 x^{2} + 9 y^{2} = 60$

Step 1.2.3.1.2.2.2

Cancel the common factor.

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

$9 x^{2} + 9 y^{2} = 60$

Step 1.2.3.1.2.2.3

Rewrite the expression.

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

$9 x^{2} + 9 y^{2} = 60$

Step 1.2.3.1.2.2.4

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

$x^{2} = 10 - y^{2}$

$9 x^{2} + 9 y^{2} = 60$

$x^{2} = 10 - y^{2}$

$9 x^{2} + 9 y^{2} = 60$

$x^{2} = 10 - y^{2}$

$9 x^{2} + 9 y^{2} = 60$

$x^{2} = 10 - y^{2}$

$9 x^{2} + 9 y^{2} = 60$

$x^{2} = 10 - y^{2}$

$9 x^{2} + 9 y^{2} = 60$

$x^{2} = 10 - y^{2}$

$9 x^{2} + 9 y^{2} = 60$

Step 1.3

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

$x = \pm \sqrt{10 - y^{2}}$

$9 x^{2} + 9 y^{2} = 60$

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.

$x = \sqrt{10 - y^{2}}$

$9 x^{2} + 9 y^{2} = 60$

Step 1.4.2

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

$x = - \sqrt{10 - y^{2}}$

$9 x^{2} + 9 y^{2} = 60$

Step 1.4.3

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

$x = \sqrt{10 - y^{2}}$

$x = - \sqrt{10 - y^{2}}$

$9 x^{2} + 9 y^{2} = 60$

$x = \sqrt{10 - y^{2}}$

$x = - \sqrt{10 - y^{2}}$

$9 x^{2} + 9 y^{2} = 60$

$x = \sqrt{10 - y^{2}}$

$x = - \sqrt{10 - y^{2}}$

$9 x^{2} + 9 y^{2} = 60$

Step 2

Solve the system $x = \sqrt{10 - y^{2}}, 9 x^{2} + 9 y^{2} = 60$ .

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Step 2.1

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

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Step 2.1.1

Replace all occurrences of $x$ in $9 x^{2} + 9 y^{2} = 60$ with $\sqrt{10 - y^{2}}$ .

$9 {(\sqrt{10 - y^{2}})}^{2} + 9 y^{2} = 60$

$x = \sqrt{10 - y^{2}}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $9 {(\sqrt{10 - y^{2}})}^{2} + 9 y^{2}$ .

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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 ${\sqrt{10 - y^{2}}}^{2}$ as $10 - y^{2}$ .

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Step 2.1.2.1.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 - y^{2}}$ as ${(10 - y^{2})}^{\frac{1}{2}}$ .

$9 {({(10 - y^{2})}^{\frac{1}{2}})}^{2} + 9 y^{2} = 60$

$x = \sqrt{10 - y^{2}}$

Step 2.1.2.1.1.1.2

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

$9 {(10 - y^{2})}^{\frac{1}{2} \cdot 2} + 9 y^{2} = 60$

$x = \sqrt{10 - y^{2}}$

Step 2.1.2.1.1.1.3

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

$9 {(10 - y^{2})}^{\frac{2}{2}} + 9 y^{2} = 60$

$x = \sqrt{10 - y^{2}}$

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.

$9 {(10 - y^{2})}^{\frac{2}{2}} + 9 y^{2} = 60$

$x = \sqrt{10 - y^{2}}$

Step 2.1.2.1.1.1.4.2

Rewrite the expression.

$9 (10 - y^{2}) + 9 y^{2} = 60$

$x = \sqrt{10 - y^{2}}$

$9 (10 - y^{2}) + 9 y^{2} = 60$

$x = \sqrt{10 - y^{2}}$

Step 2.1.2.1.1.1.5

Simplify.

$9 (10 - y^{2}) + 9 y^{2} = 60$

$x = \sqrt{10 - y^{2}}$

$9 (10 - y^{2}) + 9 y^{2} = 60$

$x = \sqrt{10 - y^{2}}$

Step 2.1.2.1.1.2

Apply the distributive property.

$9 \cdot 10 + 9 (- y^{2}) + 9 y^{2} = 60$

$x = \sqrt{10 - y^{2}}$

Step 2.1.2.1.1.3

Multiply $9$ by $10$ .

$90 + 9 (- y^{2}) + 9 y^{2} = 60$

$x = \sqrt{10 - y^{2}}$

Step 2.1.2.1.1.4

Multiply $- 1$ by $9$ .

$90 - 9 y^{2} + 9 y^{2} = 60$

$x = \sqrt{10 - y^{2}}$

$90 - 9 y^{2} + 9 y^{2} = 60$

$x = \sqrt{10 - y^{2}}$

Step 2.1.2.1.2

Combine the opposite terms in $90 - 9 y^{2} + 9 y^{2}$ .

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Step 2.1.2.1.2.1

Add $- 9 y^{2}$ and $9 y^{2}$ .

$90 + 0 = 60$

$x = \sqrt{10 - y^{2}}$

Step 2.1.2.1.2.2

Add $90$ and $0$ .

$90 = 60$

$x = \sqrt{10 - y^{2}}$

$90 = 60$

$x = \sqrt{10 - y^{2}}$

$90 = 60$

$x = \sqrt{10 - y^{2}}$

$90 = 60$

$x = \sqrt{10 - y^{2}}$

$90 = 60$

$x = \sqrt{10 - y^{2}}$

Step 2.2

Since $90 = 60$ is not true, there is no solution.

No solution

Step 3