Solve by Substitution 2x^2-3y^2=-19 , 3x^2+2y^2=30

$2 x^{2} - 3 y^{2} = - 19$ , $3 x^{2} + 2 y^{2} = 30$

Step 1

Solve for $x$ in $2 x^{2} - 3 y^{2} = - 19$ .

Add $3 y^{2}$ to both sides of the equation.

$2 x^{2} = - 19 + 3 y^{2}$

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

Divide each term in $2 x^{2} = - 19 + 3 y^{2}$ by $2$ and simplify.

Divide each term in $2 x^{2} = - 19 + 3 y^{2}$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{- 19}{2} + \frac{3 y^{2}}{2}$

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{- 19}{2} + \frac{3 y^{2}}{2}$

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

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

$x^{2} = \frac{- 19}{2} + \frac{3 y^{2}}{2}$

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

$x^{2} = \frac{- 19}{2} + \frac{3 y^{2}}{2}$

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

$x^{2} = \frac{- 19}{2} + \frac{3 y^{2}}{2}$

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

Simplify the right side.

Move the negative in front of the fraction.

$x^{2} = - \frac{19}{2} + \frac{3 y^{2}}{2}$

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

$x^{2} = - \frac{19}{2} + \frac{3 y^{2}}{2}$

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

$x^{2} = - \frac{19}{2} + \frac{3 y^{2}}{2}$

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

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

$x = \pm \sqrt{- \frac{19}{2} + \frac{3 y^{2}}{2}}$

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

Simplify $\pm \sqrt{- \frac{19}{2} + \frac{3 y^{2}}{2}}$ .

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{- 19 + 3 y^{2}}{2}}$

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

Rewrite $\sqrt{\frac{- 19 + 3 y^{2}}{2}}$ as $\frac{\sqrt{- 19 + 3 y^{2}}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{- 19 + 3 y^{2}}}{\sqrt{2}}$

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

Multiply $\frac{\sqrt{- 19 + 3 y^{2}}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{- 19 + 3 y^{2}}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

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

Combine and simplify the denominator.

Multiply $\frac{\sqrt{- 19 + 3 y^{2}}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{- 19 + 3 y^{2}} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

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

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

$x = \pm \frac{\sqrt{- 19 + 3 y^{2}} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

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

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

$x = \pm \frac{\sqrt{- 19 + 3 y^{2}} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

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

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

$x = \pm \frac{\sqrt{- 19 + 3 y^{2}} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

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

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{- 19 + 3 y^{2}} \sqrt{2}}{{\sqrt{2}}^{2}}$

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

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

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

$x = \pm \frac{\sqrt{- 19 + 3 y^{2}} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

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

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

$x = \pm \frac{\sqrt{- 19 + 3 y^{2}} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

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

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

$x = \pm \frac{\sqrt{- 19 + 3 y^{2}} \sqrt{2}}{2^{\frac{2}{2}}}$

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

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{\sqrt{- 19 + 3 y^{2}} \sqrt{2}}{2^{\frac{2}{2}}}$

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

Rewrite the expression.

$x = \pm \frac{\sqrt{- 19 + 3 y^{2}} \sqrt{2}}{2}$

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

$x = \pm \frac{\sqrt{- 19 + 3 y^{2}} \sqrt{2}}{2}$

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

Evaluate the exponent.

$x = \pm \frac{\sqrt{- 19 + 3 y^{2}} \sqrt{2}}{2}$

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

$x = \pm \frac{\sqrt{- 19 + 3 y^{2}} \sqrt{2}}{2}$

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

$x = \pm \frac{\sqrt{- 19 + 3 y^{2}} \sqrt{2}}{2}$

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

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(- 19 + 3 y^{2}) \cdot 2}}{2}$

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

Reorder factors in $\pm \frac{\sqrt{(- 19 + 3 y^{2}) \cdot 2}}{2}$ .

$x = \pm \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$x = \pm \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

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.

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

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

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

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

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

Step 2

Solve the system $x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}, 3 x^{2} + 2 y^{2} = 30$ .

Replace all occurrences of $x$ with $\frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$ in each equation.

Replace all occurrences of $x$ in $3 x^{2} + 2 y^{2} = 30$ with $\frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$ .

$3 {(\frac{\sqrt{2 (- 19 + 3 y^{2})}}{2})}^{2} + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify the left side.

Simplify $3 {(\frac{\sqrt{2 (- 19 + 3 y^{2})}}{2})}^{2} + 2 y^{2}$ .

Simplify each term.

Apply the product rule to $\frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$ .

$3 (\frac{{\sqrt{2 (- 19 + 3 y^{2})}}^{2}}{2^{2}}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify the numerator.

Rewrite ${\sqrt{2 (- 19 + 3 y^{2})}}^{2}$ as $2 (- 19 + 3 y^{2})$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (- 19 + 3 y^{2})}$ as ${(2 (- 19 + 3 y^{2}))}^{\frac{1}{2}}$ .

$3 (\frac{{({(2 (- 19 + 3 y^{2}))}^{\frac{1}{2}})}^{2}}{2^{2}}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$3 (\frac{{(2 (- 19 + 3 y^{2}))}^{\frac{1}{2} \cdot 2}}{2^{2}}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$3 (\frac{{(2 (- 19 + 3 y^{2}))}^{\frac{2}{2}}}{2^{2}}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 (\frac{{(2 (- 19 + 3 y^{2}))}^{\frac{2}{2}}}{2^{2}}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Rewrite the expression.

$3 (\frac{2 (- 19 + 3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$3 (\frac{2 (- 19 + 3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify.

$3 (\frac{2 (- 19 + 3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$3 (\frac{2 (- 19 + 3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Apply the distributive property.

$3 (\frac{2 \cdot - 19 + 2 (3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Multiply $2$ by $- 19$ .

$3 (\frac{- 38 + 2 (3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Multiply $3$ by $2$ .

$3 (\frac{- 38 + 6 y^{2}}{2^{2}}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Factor $2$ out of $- 38 + 6 y^{2}$ .

Factor $2$ out of $- 38$ .

$3 (\frac{2 (- 19) + 6 y^{2}}{2^{2}}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Factor $2$ out of $6 y^{2}$ .

$3 (\frac{2 (- 19) + 2 (3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Factor $2$ out of $2 (- 19) + 2 (3 y^{2})$ .

$3 (\frac{2 (- 19 + 3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$3 (\frac{2 (- 19 + 3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$3 (\frac{2 (- 19 + 3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$3 (\frac{2 (- 19 + 3 y^{2})}{4}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Cancel the common factors.

Factor $2$ out of $4$ .

$3 (\frac{2 (- 19 + 3 y^{2})}{2 \cdot 2}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Cancel the common factor.

$3 (\frac{2 (- 19 + 3 y^{2})}{2 \cdot 2}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Rewrite the expression.

$3 (\frac{- 19 + 3 y^{2}}{2}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$3 (\frac{- 19 + 3 y^{2}}{2}) + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Combine $3$ and $\frac{- 19 + 3 y^{2}}{2}$ .

$\frac{3 (- 19 + 3 y^{2})}{2} + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$\frac{3 (- 19 + 3 y^{2})}{2} + 2 y^{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$\frac{3 (- 19 + 3 y^{2})}{2} + 2 y^{2} \cdot \frac{2}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify terms.

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

$\frac{3 (- 19 + 3 y^{2})}{2} + \frac{2 y^{2} \cdot 2}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Combine the numerators over the common denominator.

$\frac{3 (- 19 + 3 y^{2}) + 2 y^{2} \cdot 2}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$\frac{3 (- 19 + 3 y^{2}) + 2 y^{2} \cdot 2}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify the numerator.

Apply the distributive property.

$\frac{3 \cdot - 19 + 3 (3 y^{2}) + 2 y^{2} \cdot 2}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Multiply $3$ by $- 19$ .

$\frac{- 57 + 3 (3 y^{2}) + 2 y^{2} \cdot 2}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Multiply $3$ by $3$ .

$\frac{- 57 + 9 y^{2} + 2 y^{2} \cdot 2}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Multiply $2$ by $2$ .

$\frac{- 57 + 9 y^{2} + 4 y^{2}}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Add $9 y^{2}$ and $4 y^{2}$ .

$\frac{- 57 + 13 y^{2}}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$\frac{- 57 + 13 y^{2}}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify with factoring out.

Rewrite $- 57$ as $- 1 (57)$ .

$\frac{- 1 \cdot 57 + 13 y^{2}}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Factor $- 1$ out of $13 y^{2}$ .

$\frac{- 1 \cdot 57 - (- 13 y^{2})}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Factor $- 1$ out of $- 1 (57) - (- 13 y^{2})$ .

$\frac{- 1 (57 - 13 y^{2})}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Move the negative in front of the fraction.

$- \frac{57 - 13 y^{2}}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$- \frac{57 - 13 y^{2}}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$- \frac{57 - 13 y^{2}}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$- \frac{57 - 13 y^{2}}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$- \frac{57 - 13 y^{2}}{2} = 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Solve for $y$ in $- \frac{57 - 13 y^{2}}{2} = 30$ .

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

$- 2 (- \frac{57 - 13 y^{2}}{2}) = - 2 \cdot 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 2 (- \frac{57 - 13 y^{2}}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{57 - 13 y^{2}}{2}$ into the numerator.

$- 2 \frac{- (57 - 13 y^{2})}{2} = - 2 \cdot 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Factor $2$ out of $- 2$ .

$2 (- 1) (\frac{- (57 - 13 y^{2})}{2}) = - 2 \cdot 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Cancel the common factor.

$2 \cdot (- 1 \frac{- (57 - 13 y^{2})}{2}) = - 2 \cdot 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Rewrite the expression.

$57 - 13 y^{2} = - 2 \cdot 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$57 - 13 y^{2} = - 2 \cdot 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 (57 - 13 y^{2}) = - 2 \cdot 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Multiply $57 - 13 y^{2}$ by $1$ .

$57 - 13 y^{2} = - 2 \cdot 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$57 - 13 y^{2} = - 2 \cdot 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$57 - 13 y^{2} = - 2 \cdot 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$57 - 13 y^{2} = - 2 \cdot 30$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify the right side.

Multiply $- 2$ by $30$ .

$57 - 13 y^{2} = - 60$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$57 - 13 y^{2} = - 60$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$57 - 13 y^{2} = - 60$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

Subtract $57$ from both sides of the equation.

$- 13 y^{2} = - 60 - 57$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Subtract $57$ from $- 60$ .

$- 13 y^{2} = - 117$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$- 13 y^{2} = - 117$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Divide each term in $- 13 y^{2} = - 117$ by $- 13$ and simplify.

Divide each term in $- 13 y^{2} = - 117$ by $- 13$ .

$\frac{- 13 y^{2}}{- 13} = \frac{- 117}{- 13}$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify the left side.

Cancel the common factor of $- 13$ .

Cancel the common factor.

$\frac{- 13 y^{2}}{- 13} = \frac{- 117}{- 13}$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$y^{2} = \frac{- 117}{- 13}$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$y^{2} = \frac{- 117}{- 13}$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$y^{2} = \frac{- 117}{- 13}$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify the right side.

Divide $- 117$ by $- 13$ .

$y^{2} = 9$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$y^{2} = 9$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$y^{2} = 9$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$y = \pm \sqrt{9}$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify $\pm \sqrt{9}$ .

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

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

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$y = \pm 3$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$y = \pm 3$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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.

$y = 3$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$y = - 3$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$y = 3, - 3$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$y = 3, - 3$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$y = 3, - 3$

$x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Replace all occurrences of $y$ with $3$ in each equation.

Replace all occurrences of $y$ in $x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$ with $3$ .

$x = \frac{\sqrt{2 (- 19 + 3 {(3)}^{2})}}{2}$

$y = 3$

Simplify the right side.

Simplify $\frac{\sqrt{2 (- 19 + 3 {(3)}^{2})}}{2}$ .

Simplify the numerator.

Multiply $3$ by $3^{2}$ by adding the exponents.

Multiply $3$ by $3^{2}$ .

Raise $3$ to the power of $1$ .

$x = \frac{\sqrt{2 (- 19 + 3 \cdot 3^{2})}}{2}$

$y = 3$

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

$x = \frac{\sqrt{2 (- 19 + 3^{1 + 2})}}{2}$

$y = 3$

$x = \frac{\sqrt{2 (- 19 + 3^{1 + 2})}}{2}$

$y = 3$

Add $1$ and $2$ .

$x = \frac{\sqrt{2 (- 19 + 3^{3})}}{2}$

$y = 3$

$x = \frac{\sqrt{2 (- 19 + 3^{3})}}{2}$

$y = 3$

Raise $3$ to the power of $3$ .

$x = \frac{\sqrt{2 (- 19 + 27)}}{2}$

$y = 3$

Add $- 19$ and $27$ .

$x = \frac{\sqrt{2 \cdot 8}}{2}$

$y = 3$

Multiply $2$ by $8$ .

$x = \frac{\sqrt{16}}{2}$

$y = 3$

Rewrite $16$ as $4^{2}$ .

$x = \frac{\sqrt{4^{2}}}{2}$

$y = 3$

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

$x = \frac{4}{2}$

$y = 3$

$x = \frac{4}{2}$

$y = 3$

Divide $4$ by $2$ .

$x = 2$

$y = 3$

$x = 2$

$y = 3$

$x = 2$

$y = 3$

$x = 2$

$y = 3$

Replace all occurrences of $y$ with $- 3$ in each equation.

Replace all occurrences of $y$ in $x = \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$ with $- 3$ .

$x = \frac{\sqrt{2 (- 19 + 3 {(- 3)}^{2})}}{2}$

$y = - 3$

Simplify the right side.

Simplify $\frac{\sqrt{2 (- 19 + 3 {(- 3)}^{2})}}{2}$ .

Simplify the numerator.

Raise $- 3$ to the power of $2$ .

$x = \frac{\sqrt{2 (- 19 + 3 \cdot 9)}}{2}$

$y = - 3$

Multiply $3$ by $9$ .

$x = \frac{\sqrt{2 (- 19 + 27)}}{2}$

$y = - 3$

Add $- 19$ and $27$ .

$x = \frac{\sqrt{2 \cdot 8}}{2}$

$y = - 3$

Multiply $2$ by $8$ .

$x = \frac{\sqrt{16}}{2}$

$y = - 3$

Rewrite $16$ as $4^{2}$ .

$x = \frac{\sqrt{4^{2}}}{2}$

$y = - 3$

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

$x = \frac{4}{2}$

$y = - 3$

$x = \frac{4}{2}$

$y = - 3$

Divide $4$ by $2$ .

$x = 2$

$y = - 3$

$x = 2$

$y = - 3$

$x = 2$

$y = - 3$

$x = 2$

$y = - 3$

$x = 2$

$y = - 3$

Step 3

Solve the system $x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}, 3 x^{2} + 2 y^{2} = 30$ .

Replace all occurrences of $x$ with $- \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$ in each equation.

Replace all occurrences of $x$ in $3 x^{2} + 2 y^{2} = 30$ with $- \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$ .

$3 {(- \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2})}^{2} + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify the left side.

Simplify $3 {(- \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2})}^{2} + 2 y^{2}$ .

Simplify each term.

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

Apply the product rule to $- \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$ .

$3 ({(- 1)}^{2} {(\frac{\sqrt{2 (- 19 + 3 y^{2})}}{2})}^{2}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Apply the product rule to $\frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$ .

$3 ({(- 1)}^{2} (\frac{{\sqrt{2 (- 19 + 3 y^{2})}}^{2}}{2^{2}})) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$3 ({(- 1)}^{2} (\frac{{\sqrt{2 (- 19 + 3 y^{2})}}^{2}}{2^{2}})) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$3 (1 (\frac{{\sqrt{2 (- 19 + 3 y^{2})}}^{2}}{2^{2}})) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Multiply $\frac{{\sqrt{2 (- 19 + 3 y^{2})}}^{2}}{2^{2}}$ by $1$ .

$3 (\frac{{\sqrt{2 (- 19 + 3 y^{2})}}^{2}}{2^{2}}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify the numerator.

Rewrite ${\sqrt{2 (- 19 + 3 y^{2})}}^{2}$ as $2 (- 19 + 3 y^{2})$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (- 19 + 3 y^{2})}$ as ${(2 (- 19 + 3 y^{2}))}^{\frac{1}{2}}$ .

$3 (\frac{{({(2 (- 19 + 3 y^{2}))}^{\frac{1}{2}})}^{2}}{2^{2}}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$3 (\frac{{(2 (- 19 + 3 y^{2}))}^{\frac{1}{2} \cdot 2}}{2^{2}}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$3 (\frac{{(2 (- 19 + 3 y^{2}))}^{\frac{2}{2}}}{2^{2}}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 (\frac{{(2 (- 19 + 3 y^{2}))}^{\frac{2}{2}}}{2^{2}}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Rewrite the expression.

$3 (\frac{2 (- 19 + 3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$3 (\frac{2 (- 19 + 3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify.

$3 (\frac{2 (- 19 + 3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$3 (\frac{2 (- 19 + 3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Apply the distributive property.

$3 (\frac{2 \cdot - 19 + 2 (3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Multiply $2$ by $- 19$ .

$3 (\frac{- 38 + 2 (3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Multiply $3$ by $2$ .

$3 (\frac{- 38 + 6 y^{2}}{2^{2}}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Factor $2$ out of $- 38 + 6 y^{2}$ .

Factor $2$ out of $- 38$ .

$3 (\frac{2 (- 19) + 6 y^{2}}{2^{2}}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Factor $2$ out of $6 y^{2}$ .

$3 (\frac{2 (- 19) + 2 (3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Factor $2$ out of $2 (- 19) + 2 (3 y^{2})$ .

$3 (\frac{2 (- 19 + 3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$3 (\frac{2 (- 19 + 3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$3 (\frac{2 (- 19 + 3 y^{2})}{2^{2}}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$3 (\frac{2 (- 19 + 3 y^{2})}{4}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Cancel the common factors.

Factor $2$ out of $4$ .

$3 (\frac{2 (- 19 + 3 y^{2})}{2 \cdot 2}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Cancel the common factor.

$3 (\frac{2 (- 19 + 3 y^{2})}{2 \cdot 2}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Rewrite the expression.

$3 (\frac{- 19 + 3 y^{2}}{2}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$3 (\frac{- 19 + 3 y^{2}}{2}) + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Combine $3$ and $\frac{- 19 + 3 y^{2}}{2}$ .

$\frac{3 (- 19 + 3 y^{2})}{2} + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$\frac{3 (- 19 + 3 y^{2})}{2} + 2 y^{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$\frac{3 (- 19 + 3 y^{2})}{2} + 2 y^{2} \cdot \frac{2}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify terms.

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

$\frac{3 (- 19 + 3 y^{2})}{2} + \frac{2 y^{2} \cdot 2}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Combine the numerators over the common denominator.

$\frac{3 (- 19 + 3 y^{2}) + 2 y^{2} \cdot 2}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$\frac{3 (- 19 + 3 y^{2}) + 2 y^{2} \cdot 2}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify the numerator.

Apply the distributive property.

$\frac{3 \cdot - 19 + 3 (3 y^{2}) + 2 y^{2} \cdot 2}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Multiply $3$ by $- 19$ .

$\frac{- 57 + 3 (3 y^{2}) + 2 y^{2} \cdot 2}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Multiply $3$ by $3$ .

$\frac{- 57 + 9 y^{2} + 2 y^{2} \cdot 2}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Multiply $2$ by $2$ .

$\frac{- 57 + 9 y^{2} + 4 y^{2}}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Add $9 y^{2}$ and $4 y^{2}$ .

$\frac{- 57 + 13 y^{2}}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$\frac{- 57 + 13 y^{2}}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify with factoring out.

Rewrite $- 57$ as $- 1 (57)$ .

$\frac{- 1 \cdot 57 + 13 y^{2}}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Factor $- 1$ out of $13 y^{2}$ .

$\frac{- 1 \cdot 57 - (- 13 y^{2})}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Factor $- 1$ out of $- 1 (57) - (- 13 y^{2})$ .

$\frac{- 1 (57 - 13 y^{2})}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Move the negative in front of the fraction.

$- \frac{57 - 13 y^{2}}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$- \frac{57 - 13 y^{2}}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$- \frac{57 - 13 y^{2}}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$- \frac{57 - 13 y^{2}}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$- \frac{57 - 13 y^{2}}{2} = 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Solve for $y$ in $- \frac{57 - 13 y^{2}}{2} = 30$ .

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

$- 2 (- \frac{57 - 13 y^{2}}{2}) = - 2 \cdot 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 2 (- \frac{57 - 13 y^{2}}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{57 - 13 y^{2}}{2}$ into the numerator.

$- 2 \frac{- (57 - 13 y^{2})}{2} = - 2 \cdot 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Factor $2$ out of $- 2$ .

$2 (- 1) (\frac{- (57 - 13 y^{2})}{2}) = - 2 \cdot 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Cancel the common factor.

$2 \cdot (- 1 \frac{- (57 - 13 y^{2})}{2}) = - 2 \cdot 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Rewrite the expression.

$57 - 13 y^{2} = - 2 \cdot 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$57 - 13 y^{2} = - 2 \cdot 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 (57 - 13 y^{2}) = - 2 \cdot 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Multiply $57 - 13 y^{2}$ by $1$ .

$57 - 13 y^{2} = - 2 \cdot 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$57 - 13 y^{2} = - 2 \cdot 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$57 - 13 y^{2} = - 2 \cdot 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$57 - 13 y^{2} = - 2 \cdot 30$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify the right side.

Multiply $- 2$ by $30$ .

$57 - 13 y^{2} = - 60$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$57 - 13 y^{2} = - 60$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$57 - 13 y^{2} = - 60$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

Subtract $57$ from both sides of the equation.

$- 13 y^{2} = - 60 - 57$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Subtract $57$ from $- 60$ .

$- 13 y^{2} = - 117$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$- 13 y^{2} = - 117$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Divide each term in $- 13 y^{2} = - 117$ by $- 13$ and simplify.

Divide each term in $- 13 y^{2} = - 117$ by $- 13$ .

$\frac{- 13 y^{2}}{- 13} = \frac{- 117}{- 13}$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify the left side.

Cancel the common factor of $- 13$ .

Cancel the common factor.

$\frac{- 13 y^{2}}{- 13} = \frac{- 117}{- 13}$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$y^{2} = \frac{- 117}{- 13}$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$y^{2} = \frac{- 117}{- 13}$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$y^{2} = \frac{- 117}{- 13}$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify the right side.

Divide $- 117$ by $- 13$ .

$y^{2} = 9$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$y^{2} = 9$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$y^{2} = 9$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$y = \pm \sqrt{9}$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Simplify $\pm \sqrt{9}$ .

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

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

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$y = \pm 3$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$y = \pm 3$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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.

$y = 3$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$y = - 3$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

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

$y = 3, - 3$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$y = 3, - 3$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

$y = 3, - 3$

$x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$

Replace all occurrences of $y$ with $3$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$ with $3$ .

$x = - \frac{\sqrt{2 (- 19 + 3 {(3)}^{2})}}{2}$

$y = 3$

Simplify the right side.

Simplify $- \frac{\sqrt{2 (- 19 + 3 {(3)}^{2})}}{2}$ .

Simplify the numerator.

Multiply $3$ by $3^{2}$ by adding the exponents.

Multiply $3$ by $3^{2}$ .

Raise $3$ to the power of $1$ .

$x = - \frac{\sqrt{2 (- 19 + 3 \cdot 3^{2})}}{2}$

$y = 3$

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

$x = - \frac{\sqrt{2 (- 19 + 3^{1 + 2})}}{2}$

$y = 3$

$x = - \frac{\sqrt{2 (- 19 + 3^{1 + 2})}}{2}$

$y = 3$

Add $1$ and $2$ .

$x = - \frac{\sqrt{2 (- 19 + 3^{3})}}{2}$

$y = 3$

$x = - \frac{\sqrt{2 (- 19 + 3^{3})}}{2}$

$y = 3$

Raise $3$ to the power of $3$ .

$x = - \frac{\sqrt{2 (- 19 + 27)}}{2}$

$y = 3$

Add $- 19$ and $27$ .

$x = - \frac{\sqrt{2 \cdot 8}}{2}$

$y = 3$

Multiply $2$ by $8$ .

$x = - \frac{\sqrt{16}}{2}$

$y = 3$

Rewrite $16$ as $4^{2}$ .

$x = - \frac{\sqrt{4^{2}}}{2}$

$y = 3$

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

$x = - \frac{4}{2}$

$y = 3$

$x = - \frac{4}{2}$

$y = 3$

Simplify the expression.

Divide $4$ by $2$ .

$x = - 1 \cdot 2$

$y = 3$

Multiply $- 1$ by $2$ .

$x = - 2$

$y = 3$

$x = - 2$

$y = 3$

$x = - 2$

$y = 3$

$x = - 2$

$y = 3$

$x = - 2$

$y = 3$

Replace all occurrences of $y$ with $- 3$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{\sqrt{2 (- 19 + 3 y^{2})}}{2}$ with $- 3$ .

$x = - \frac{\sqrt{2 (- 19 + 3 {(- 3)}^{2})}}{2}$

$y = - 3$

Simplify the right side.

Simplify $- \frac{\sqrt{2 (- 19 + 3 {(- 3)}^{2})}}{2}$ .

Simplify the numerator.

Raise $- 3$ to the power of $2$ .

$x = - \frac{\sqrt{2 (- 19 + 3 \cdot 9)}}{2}$

$y = - 3$

Multiply $3$ by $9$ .

$x = - \frac{\sqrt{2 (- 19 + 27)}}{2}$

$y = - 3$

Add $- 19$ and $27$ .

$x = - \frac{\sqrt{2 \cdot 8}}{2}$

$y = - 3$

Multiply $2$ by $8$ .

$x = - \frac{\sqrt{16}}{2}$

$y = - 3$

Rewrite $16$ as $4^{2}$ .

$x = - \frac{\sqrt{4^{2}}}{2}$

$y = - 3$

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

$x = - \frac{4}{2}$

$y = - 3$

$x = - \frac{4}{2}$

$y = - 3$

Simplify the expression.

Divide $4$ by $2$ .

$x = - 1 \cdot 2$

$y = - 3$

Multiply $- 1$ by $2$ .

$x = - 2$

$y = - 3$

$x = - 2$

$y = - 3$

$x = - 2$

$y = - 3$

$x = - 2$

$y = - 3$

$x = - 2$

$y = - 3$

$x = - 2$

$y = - 3$

Step 4

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

$(2, 3)$

$(2, - 3)$

$(- 2, 3)$

$(- 2, - 3)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(2, 3), (2, - 3), (- 2, 3), (- 2, - 3)$

Equation Form:

$x = 2, y = 3$

$x = 2, y = - 3$

$x = - 2, y = 3$

$x = - 2, y = - 3$

Step 6

Solve by Substitution 2x^2-3y^2=-19 , 3x^2+2y^2=30

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