Solve the System of Equations -2x^2-y^2=-200 -3x^2-y^2=-300

$- 2 x^{2} - y^{2} = - 200$ $- 3 x^{2} - y^{2} = - 300$

Solve for $x$ in $- 2 x^{2} - y^{2} = - 200$ .

Add $y^{2}$ to both sides of the equation.

$- 2 x^{2} = - 200 + y^{2}$

$- 3 x^{2} - y^{2} = - 300$

Divide each term in $- 2 x^{2} = - 200 + y^{2}$ by $- 2$ and simplify.

Divide each term in $- 2 x^{2} = - 200 + y^{2}$ by $- 2$ .

$\frac{- 2 x^{2}}{- 2} = \frac{- 200}{- 2} + \frac{y^{2}}{- 2}$

$- 3 x^{2} - y^{2} = - 300$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x^{2}}{- 2} = \frac{- 200}{- 2} + \frac{y^{2}}{- 2}$

$- 3 x^{2} - y^{2} = - 300$

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

$x^{2} = \frac{- 200}{- 2} + \frac{y^{2}}{- 2}$

$- 3 x^{2} - y^{2} = - 300$

$x^{2} = \frac{- 200}{- 2} + \frac{y^{2}}{- 2}$

$- 3 x^{2} - y^{2} = - 300$

$x^{2} = \frac{- 200}{- 2} + \frac{y^{2}}{- 2}$

$- 3 x^{2} - y^{2} = - 300$

Simplify the right side.

Simplify each term.

Divide $- 200$ by $- 2$ .

$x^{2} = 100 + \frac{y^{2}}{- 2}$

$- 3 x^{2} - y^{2} = - 300$

Move the negative in front of the fraction.

$x^{2} = 100 - \frac{y^{2}}{2}$

$- 3 x^{2} - y^{2} = - 300$

$x^{2} = 100 - \frac{y^{2}}{2}$

$- 3 x^{2} - y^{2} = - 300$

$x^{2} = 100 - \frac{y^{2}}{2}$

$- 3 x^{2} - y^{2} = - 300$

$x^{2} = 100 - \frac{y^{2}}{2}$

$- 3 x^{2} - y^{2} = - 300$

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

$x = \pm \sqrt{100 - \frac{y^{2}}{2}}$

$- 3 x^{2} - y^{2} = - 300$

Simplify $\pm \sqrt{100 - \frac{y^{2}}{2}}$ .

To write $100$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \pm \sqrt{100 \cdot \frac{2}{2} - \frac{y^{2}}{2}}$

$- 3 x^{2} - y^{2} = - 300$

Combine $100$ and $\frac{2}{2}$ .

$x = \pm \sqrt{\frac{100 \cdot 2}{2} - \frac{y^{2}}{2}}$

$- 3 x^{2} - y^{2} = - 300$

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{100 \cdot 2 - y^{2}}{2}}$

$- 3 x^{2} - y^{2} = - 300$

Multiply $100$ by $2$ .

$x = \pm \sqrt{\frac{200 - y^{2}}{2}}$

$- 3 x^{2} - y^{2} = - 300$

Rewrite $\sqrt{\frac{200 - y^{2}}{2}}$ as $\frac{\sqrt{200 - y^{2}}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{200 - y^{2}}}{\sqrt{2}}$

$- 3 x^{2} - y^{2} = - 300$

Multiply $\frac{\sqrt{200 - y^{2}}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{200 - y^{2}}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

$- 3 x^{2} - y^{2} = - 300$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{200 - y^{2}}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{200 - y^{2}} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

$- 3 x^{2} - y^{2} = - 300$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{200 - y^{2}} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

$- 3 x^{2} - y^{2} = - 300$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{200 - y^{2}} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

$- 3 x^{2} - y^{2} = - 300$

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

$x = \pm \frac{\sqrt{200 - y^{2}} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

$- 3 x^{2} - y^{2} = - 300$

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{200 - y^{2}} \sqrt{2}}{{\sqrt{2}}^{2}}$

$- 3 x^{2} - y^{2} = - 300$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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

$x = \pm \frac{\sqrt{200 - y^{2}} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

$- 3 x^{2} - y^{2} = - 300$

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

$x = \pm \frac{\sqrt{200 - y^{2}} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

$- 3 x^{2} - y^{2} = - 300$

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

$x = \pm \frac{\sqrt{200 - y^{2}} \sqrt{2}}{2^{\frac{2}{2}}}$

$- 3 x^{2} - y^{2} = - 300$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{\sqrt{200 - y^{2}} \sqrt{2}}{2^{\frac{2}{2}}}$

$- 3 x^{2} - y^{2} = - 300$

Rewrite the expression.

$x = \pm \frac{\sqrt{200 - y^{2}} \sqrt{2}}{2}$

$- 3 x^{2} - y^{2} = - 300$

$x = \pm \frac{\sqrt{200 - y^{2}} \sqrt{2}}{2}$

$- 3 x^{2} - y^{2} = - 300$

Evaluate the exponent.

$x = \pm \frac{\sqrt{200 - y^{2}} \sqrt{2}}{2}$

$- 3 x^{2} - y^{2} = - 300$

$x = \pm \frac{\sqrt{200 - y^{2}} \sqrt{2}}{2}$

$- 3 x^{2} - y^{2} = - 300$

$x = \pm \frac{\sqrt{200 - y^{2}} \sqrt{2}}{2}$

$- 3 x^{2} - y^{2} = - 300$

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(200 - y^{2}) \cdot 2}}{2}$

$- 3 x^{2} - y^{2} = - 300$

Reorder factors in $\pm \frac{\sqrt{(200 - y^{2}) \cdot 2}}{2}$ .

$x = \pm \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 x^{2} - y^{2} = - 300$

$x = \pm \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 x^{2} - y^{2} = - 300$

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

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

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 x^{2} - y^{2} = - 300$

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

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 x^{2} - y^{2} = - 300$

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

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 x^{2} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 x^{2} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 x^{2} - y^{2} = - 300$

Solve the system $x = \frac{\sqrt{2 (200 - y^{2})}}{2}, - 3 x^{2} - y^{2} = - 300$ .

Replace all occurrences of $x$ with $\frac{\sqrt{2 (200 - y^{2})}}{2}$ in each equation.

Replace all occurrences of $x$ in $- 3 x^{2} - y^{2} = - 300$ with $\frac{\sqrt{2 (200 - y^{2})}}{2}$ .

$- 3 {(\frac{\sqrt{2 (200 - y^{2})}}{2})}^{2} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify the left side.

Simplify $- 3 {(\frac{\sqrt{2 (200 - y^{2})}}{2})}^{2} - y^{2}$ .

Simplify each term.

Apply the product rule to $\frac{\sqrt{2 (200 - y^{2})}}{2}$ .

$- 3 \frac{{\sqrt{2 (200 - y^{2})}}^{2}}{2^{2}} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify the numerator.

Rewrite ${\sqrt{2 (200 - y^{2})}}^{2}$ as $2 (200 - y^{2})$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (200 - y^{2})}$ as ${(2 (200 - y^{2}))}^{\frac{1}{2}}$ .

$- 3 \frac{{({(2 (200 - y^{2}))}^{\frac{1}{2}})}^{2}}{2^{2}} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

$- 3 \frac{{(2 (200 - y^{2}))}^{\frac{1}{2} \cdot 2}}{2^{2}} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

$- 3 \frac{{(2 (200 - y^{2}))}^{\frac{2}{2}}}{2^{2}} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$- 3 \frac{{(2 (200 - y^{2}))}^{\frac{2}{2}}}{2^{2}} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Rewrite the expression.

$- 3 \frac{2 (200 - y^{2})}{2^{2}} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 \frac{2 (200 - y^{2})}{2^{2}} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify.

$- 3 \frac{2 (200 - y^{2})}{2^{2}} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 \frac{2 (200 - y^{2})}{2^{2}} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Apply the distributive property.

$- 3 \frac{2 \cdot 200 + 2 (- y^{2})}{2^{2}} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Multiply $2$ by $200$ .

$- 3 \frac{400 + 2 (- y^{2})}{2^{2}} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Multiply $- 1$ by $2$ .

$- 3 \frac{400 - 2 y^{2}}{2^{2}} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Factor $2$ out of $400 - 2 y^{2}$ .

Factor $2$ out of $400$ .

$- 3 \frac{2 (200) - 2 y^{2}}{2^{2}} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

$- 3 \frac{2 (200) + 2 (- y^{2})}{2^{2}} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Factor $2$ out of $2 (200) + 2 (- y^{2})$ .

$- 3 \frac{2 (200 - y^{2})}{2^{2}} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 \frac{2 (200 - y^{2})}{2^{2}} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 \frac{2 (200 - y^{2})}{2^{2}} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

$- 3 \frac{2 (200 - y^{2})}{4} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Cancel the common factors.

Factor $2$ out of $4$ .

$- 3 \frac{2 (200 - y^{2})}{2 \cdot 2} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Cancel the common factor.

$- 3 \frac{2 (200 - y^{2})}{2 \cdot 2} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Rewrite the expression.

$- 3 \frac{200 - y^{2}}{2} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 \frac{200 - y^{2}}{2} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Combine $- 3$ and $\frac{200 - y^{2}}{2}$ .

$\frac{- 3 (200 - y^{2})}{2} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Move the negative in front of the fraction.

$- \frac{3 (200 - y^{2})}{2} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- \frac{3 (200 - y^{2})}{2} - y^{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

$- \frac{3 (200 - y^{2})}{2} - y^{2} \cdot \frac{2}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify terms.

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

$- \frac{3 (200 - y^{2})}{2} + \frac{- y^{2} \cdot 2}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Combine the numerators over the common denominator.

$\frac{- 3 (200 - y^{2}) - y^{2} \cdot 2}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$\frac{- 3 (200 - y^{2}) - y^{2} \cdot 2}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify the numerator.

Apply the distributive property.

$\frac{- 3 \cdot 200 - 3 (- y^{2}) - y^{2} \cdot 2}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Multiply $- 3$ by $200$ .

$\frac{- 600 - 3 (- y^{2}) - y^{2} \cdot 2}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Multiply $- 1$ by $- 3$ .

$\frac{- 600 + 3 y^{2} - y^{2} \cdot 2}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Multiply $2$ by $- 1$ .

$\frac{- 600 + 3 y^{2} - 2 y^{2}}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Subtract $2 y^{2}$ from $3 y^{2}$ .

$\frac{- 600 + y^{2}}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$\frac{- 600 + y^{2}}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify with factoring out.

Rewrite $- 600$ as $- 1 (600)$ .

$\frac{- 1 \cdot 600 + y^{2}}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Factor $- 1$ out of $y^{2}$ .

$\frac{- 1 \cdot 600 - 1 (- y^{2})}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Factor $- 1$ out of $- 1 (600) - 1 (- y^{2})$ .

$\frac{- 1 (600 - y^{2})}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Move the negative in front of the fraction.

$- \frac{600 - y^{2}}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- \frac{600 - y^{2}}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- \frac{600 - y^{2}}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- \frac{600 - y^{2}}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- \frac{600 - y^{2}}{2} = - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Solve for $y$ in $- \frac{600 - y^{2}}{2} = - 300$ .

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{600 - y^{2}}{2}) = - 2 \cdot - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 2 (- \frac{600 - y^{2}}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{600 - y^{2}}{2}$ into the numerator.

$- 2 \frac{- (600 - y^{2})}{2} = - 2 \cdot - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Factor $2$ out of $- 2$ .

$2 (- 1) (\frac{- (600 - y^{2})}{2}) = - 2 \cdot - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Cancel the common factor.

$2 \cdot (- 1 \frac{- (600 - y^{2})}{2}) = - 2 \cdot - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Rewrite the expression.

$600 - y^{2} = - 2 \cdot - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$600 - y^{2} = - 2 \cdot - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 (600 - y^{2}) = - 2 \cdot - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Multiply $600 - y^{2}$ by $1$ .

$600 - y^{2} = - 2 \cdot - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$600 - y^{2} = - 2 \cdot - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$600 - y^{2} = - 2 \cdot - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$600 - y^{2} = - 2 \cdot - 300$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify the right side.

Multiply $- 2$ by $- 300$ .

$600 - y^{2} = 600$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$600 - y^{2} = 600$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$600 - y^{2} = 600$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Move all terms not containing $y$ to the right side of the equation.

Subtract $600$ from both sides of the equation.

$- y^{2} = 600 - 600$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Subtract $600$ from $600$ .

$- y^{2} = 0$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- y^{2} = 0$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Divide each term in $- y^{2} = 0$ by $- 1$ and simplify.

Divide each term in $- y^{2} = 0$ by $- 1$ .

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

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify the left side.

Dividing two negative values results in a positive value.

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

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

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

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify the right side.

Divide $0$ by $- 1$ .

$y^{2} = 0$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$y^{2} = 0$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$y^{2} = 0$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

$y = \pm \sqrt{0}$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify $\pm \sqrt{0}$ .

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

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

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

$y = \pm 0$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

Plus or minus $0$ is $0$ .

$y = 0$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$y = 0$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

$y = 0$

$x = \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

Replace all occurrences of $y$ in $x = \frac{\sqrt{2 (200 - y^{2})}}{2}$ with $0$ .

$x = \frac{\sqrt{2 (200 - {(0)}^{2})}}{2}$

$y = 0$

Simplify the right side.

Simplify $\frac{\sqrt{2 (200 - {(0)}^{2})}}{2}$ .

Simplify the numerator.

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

$x = \frac{\sqrt{2 (200 - 0)}}{2}$

$y = 0$

Multiply $- 1$ by $0$ .

$x = \frac{\sqrt{2 (200 + 0)}}{2}$

$y = 0$

Add $200$ and $0$ .

$x = \frac{\sqrt{2 \cdot 200}}{2}$

$y = 0$

Multiply $2$ by $200$ .

$x = \frac{\sqrt{400}}{2}$

$y = 0$

Rewrite $400$ as $20^{2}$ .

$x = \frac{\sqrt{20^{2}}}{2}$

$y = 0$

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

$x = \frac{20}{2}$

$y = 0$

$x = \frac{20}{2}$

$y = 0$

Divide $20$ by $2$ .

$x = 10$

$y = 0$

$x = 10$

$y = 0$

$x = 10$

$y = 0$

$x = 10$

$y = 0$

$x = 10$

$y = 0$

Solve the system $x = - \frac{\sqrt{2 (200 - y^{2})}}{2}, - 3 x^{2} - y^{2} = - 300$ .

Replace all occurrences of $x$ with $- \frac{\sqrt{2 (200 - y^{2})}}{2}$ in each equation.

Replace all occurrences of $x$ in $- 3 x^{2} - y^{2} = - 300$ with $- \frac{\sqrt{2 (200 - y^{2})}}{2}$ .

$- 3 {(- \frac{\sqrt{2 (200 - y^{2})}}{2})}^{2} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify the left side.

Simplify $- 3 {(- \frac{\sqrt{2 (200 - y^{2})}}{2})}^{2} - y^{2}$ .

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{\sqrt{2 (200 - y^{2})}}{2}$ .

$- 3 ({(- 1)}^{2} {(\frac{\sqrt{2 (200 - y^{2})}}{2})}^{2}) - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Apply the product rule to $\frac{\sqrt{2 (200 - y^{2})}}{2}$ .

$- 3 ({(- 1)}^{2} (\frac{{\sqrt{2 (200 - y^{2})}}^{2}}{2^{2}})) - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 ({(- 1)}^{2} (\frac{{\sqrt{2 (200 - y^{2})}}^{2}}{2^{2}})) - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Raise $- 1$ to the power of $2$ .

$- 3 (1 (\frac{{\sqrt{2 (200 - y^{2})}}^{2}}{2^{2}})) - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Multiply $\frac{{\sqrt{2 (200 - y^{2})}}^{2}}{2^{2}}$ by $1$ .

$- 3 \frac{{\sqrt{2 (200 - y^{2})}}^{2}}{2^{2}} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify the numerator.

Rewrite ${\sqrt{2 (200 - y^{2})}}^{2}$ as $2 (200 - y^{2})$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (200 - y^{2})}$ as ${(2 (200 - y^{2}))}^{\frac{1}{2}}$ .

$- 3 \frac{{({(2 (200 - y^{2}))}^{\frac{1}{2}})}^{2}}{2^{2}} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

$- 3 \frac{{(2 (200 - y^{2}))}^{\frac{1}{2} \cdot 2}}{2^{2}} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

$- 3 \frac{{(2 (200 - y^{2}))}^{\frac{2}{2}}}{2^{2}} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$- 3 \frac{{(2 (200 - y^{2}))}^{\frac{2}{2}}}{2^{2}} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Rewrite the expression.

$- 3 \frac{2 (200 - y^{2})}{2^{2}} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 \frac{2 (200 - y^{2})}{2^{2}} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify.

$- 3 \frac{2 (200 - y^{2})}{2^{2}} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 \frac{2 (200 - y^{2})}{2^{2}} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Apply the distributive property.

$- 3 \frac{2 \cdot 200 + 2 (- y^{2})}{2^{2}} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Multiply $2$ by $200$ .

$- 3 \frac{400 + 2 (- y^{2})}{2^{2}} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Multiply $- 1$ by $2$ .

$- 3 \frac{400 - 2 y^{2}}{2^{2}} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Factor $2$ out of $400 - 2 y^{2}$ .

Factor $2$ out of $400$ .

$- 3 \frac{2 (200) - 2 y^{2}}{2^{2}} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

$- 3 \frac{2 (200) + 2 (- y^{2})}{2^{2}} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Factor $2$ out of $2 (200) + 2 (- y^{2})$ .

$- 3 \frac{2 (200 - y^{2})}{2^{2}} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 \frac{2 (200 - y^{2})}{2^{2}} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 \frac{2 (200 - y^{2})}{2^{2}} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

$- 3 \frac{2 (200 - y^{2})}{4} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Cancel the common factors.

Factor $2$ out of $4$ .

$- 3 \frac{2 (200 - y^{2})}{2 \cdot 2} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Cancel the common factor.

$- 3 \frac{2 (200 - y^{2})}{2 \cdot 2} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Rewrite the expression.

$- 3 \frac{200 - y^{2}}{2} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- 3 \frac{200 - y^{2}}{2} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Combine $- 3$ and $\frac{200 - y^{2}}{2}$ .

$\frac{- 3 (200 - y^{2})}{2} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Move the negative in front of the fraction.

$- \frac{3 (200 - y^{2})}{2} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- \frac{3 (200 - y^{2})}{2} - y^{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

$- \frac{3 (200 - y^{2})}{2} - y^{2} \cdot \frac{2}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify terms.

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

$- \frac{3 (200 - y^{2})}{2} + \frac{- y^{2} \cdot 2}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Combine the numerators over the common denominator.

$\frac{- 3 (200 - y^{2}) - y^{2} \cdot 2}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$\frac{- 3 (200 - y^{2}) - y^{2} \cdot 2}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify the numerator.

Apply the distributive property.

$\frac{- 3 \cdot 200 - 3 (- y^{2}) - y^{2} \cdot 2}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Multiply $- 3$ by $200$ .

$\frac{- 600 - 3 (- y^{2}) - y^{2} \cdot 2}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Multiply $- 1$ by $- 3$ .

$\frac{- 600 + 3 y^{2} - y^{2} \cdot 2}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Multiply $2$ by $- 1$ .

$\frac{- 600 + 3 y^{2} - 2 y^{2}}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Subtract $2 y^{2}$ from $3 y^{2}$ .

$\frac{- 600 + y^{2}}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$\frac{- 600 + y^{2}}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify with factoring out.

Rewrite $- 600$ as $- 1 (600)$ .

$\frac{- 1 \cdot 600 + y^{2}}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Factor $- 1$ out of $y^{2}$ .

$\frac{- 1 \cdot 600 - 1 (- y^{2})}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Factor $- 1$ out of $- 1 (600) - 1 (- y^{2})$ .

$\frac{- 1 (600 - y^{2})}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Move the negative in front of the fraction.

$- \frac{600 - y^{2}}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- \frac{600 - y^{2}}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- \frac{600 - y^{2}}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- \frac{600 - y^{2}}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- \frac{600 - y^{2}}{2} = - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Solve for $y$ in $- \frac{600 - y^{2}}{2} = - 300$ .

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{600 - y^{2}}{2}) = - 2 \cdot - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 2 (- \frac{600 - y^{2}}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{600 - y^{2}}{2}$ into the numerator.

$- 2 \frac{- (600 - y^{2})}{2} = - 2 \cdot - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Factor $2$ out of $- 2$ .

$2 (- 1) (\frac{- (600 - y^{2})}{2}) = - 2 \cdot - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Cancel the common factor.

$2 \cdot (- 1 \frac{- (600 - y^{2})}{2}) = - 2 \cdot - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Rewrite the expression.

$600 - y^{2} = - 2 \cdot - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$600 - y^{2} = - 2 \cdot - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 (600 - y^{2}) = - 2 \cdot - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Multiply $600 - y^{2}$ by $1$ .

$600 - y^{2} = - 2 \cdot - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$600 - y^{2} = - 2 \cdot - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$600 - y^{2} = - 2 \cdot - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$600 - y^{2} = - 2 \cdot - 300$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify the right side.

Multiply $- 2$ by $- 300$ .

$600 - y^{2} = 600$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$600 - y^{2} = 600$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$600 - y^{2} = 600$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Move all terms not containing $y$ to the right side of the equation.

Subtract $600$ from both sides of the equation.

$- y^{2} = 600 - 600$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Subtract $600$ from $600$ .

$- y^{2} = 0$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$- y^{2} = 0$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Divide each term in $- y^{2} = 0$ by $- 1$ and simplify.

Divide each term in $- y^{2} = 0$ by $- 1$ .

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

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify the left side.

Dividing two negative values results in a positive value.

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

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

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

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify the right side.

Divide $0$ by $- 1$ .

$y^{2} = 0$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$y^{2} = 0$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$y^{2} = 0$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

$y = \pm \sqrt{0}$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Simplify $\pm \sqrt{0}$ .

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

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

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

$y = \pm 0$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

Plus or minus $0$ is $0$ .

$y = 0$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$y = 0$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

$y = 0$

$x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$

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

Replace all occurrences of $y$ in $x = - \frac{\sqrt{2 (200 - y^{2})}}{2}$ with $0$ .

$x = - \frac{\sqrt{2 (200 - {(0)}^{2})}}{2}$

$y = 0$

Simplify the right side.

Simplify $- \frac{\sqrt{2 (200 - {(0)}^{2})}}{2}$ .

Simplify the numerator.

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

$x = - \frac{\sqrt{2 (200 - 0)}}{2}$

$y = 0$

Multiply $- 1$ by $0$ .

$x = - \frac{\sqrt{2 (200 + 0)}}{2}$

$y = 0$

Add $200$ and $0$ .

$x = - \frac{\sqrt{2 \cdot 200}}{2}$

$y = 0$

Multiply $2$ by $200$ .

$x = - \frac{\sqrt{400}}{2}$

$y = 0$

Rewrite $400$ as $20^{2}$ .

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

$y = 0$

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

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

$y = 0$

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

$y = 0$

Simplify the expression.

Divide $20$ by $2$ .

$x = - 1 \cdot 10$

$y = 0$

Multiply $- 1$ by $10$ .

$x = - 10$

$y = 0$

$x = - 10$

$y = 0$

$x = - 10$

$y = 0$

$x = - 10$

$y = 0$

$x = - 10$

$y = 0$

$x = - 10$

$y = 0$

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

$(10, 0)$

$(- 10, 0)$

The result can be shown in multiple forms.

Point Form:

$(10, 0), (- 10, 0)$

Equation Form:

$x = 10, y = 0$

$x = - 10, y = 0$

Solve the System of Equations -2x^2-y^2=-200 -3x^2-y^2=-300

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