Solve the System of Equations x^2-y^2=16 x^2+y^2=20

$x^{2} - y^{2} = 16$ $x^{2} + y^{2} = 20$

Step 1

Solve for $x$ in $x^{2} - y^{2} = 16$ .

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

$x^{2} = 16 + y^{2}$

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

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

$x = \pm \sqrt{16 + y^{2}}$

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

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.

$x = \sqrt{16 + y^{2}}$

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

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

$x = - \sqrt{16 + y^{2}}$

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

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

$x = \sqrt{16 + y^{2}}$

$x = - \sqrt{16 + y^{2}}$

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

$x = \sqrt{16 + y^{2}}$

$x = - \sqrt{16 + y^{2}}$

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

$x = \sqrt{16 + y^{2}}$

$x = - \sqrt{16 + y^{2}}$

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

Step 2

Solve the system $x = \sqrt{16 + y^{2}}, x^{2} + y^{2} = 20$ .

Replace all occurrences of $x$ with $\sqrt{16 + y^{2}}$ in each equation.

Replace all occurrences of $x$ in $x^{2} + y^{2} = 20$ with $\sqrt{16 + y^{2}}$ .

${(\sqrt{16 + y^{2}})}^{2} + y^{2} = 20$

$x = \sqrt{16 + y^{2}}$

Simplify the left side.

Simplify ${(\sqrt{16 + y^{2}})}^{2} + y^{2}$ .

Rewrite ${\sqrt{16 + y^{2}}}^{2}$ as $16 + y^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{16 + y^{2}}$ as ${(16 + y^{2})}^{\frac{1}{2}}$ .

${({(16 + y^{2})}^{\frac{1}{2}})}^{2} + y^{2} = 20$

$x = \sqrt{16 + y^{2}}$

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

${(16 + y^{2})}^{\frac{1}{2} \cdot 2} + y^{2} = 20$

$x = \sqrt{16 + y^{2}}$

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

${(16 + y^{2})}^{\frac{2}{2}} + y^{2} = 20$

$x = \sqrt{16 + y^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(16 + y^{2})}^{\frac{2}{2}} + y^{2} = 20$

$x = \sqrt{16 + y^{2}}$

Rewrite the expression.

$(16 + y^{2}) + y^{2} = 20$

$x = \sqrt{16 + y^{2}}$

$(16 + y^{2}) + y^{2} = 20$

$x = \sqrt{16 + y^{2}}$

Simplify.

$16 + y^{2} + y^{2} = 20$

$x = \sqrt{16 + y^{2}}$

$16 + y^{2} + y^{2} = 20$

$x = \sqrt{16 + y^{2}}$

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

$16 + 2 y^{2} = 20$

$x = \sqrt{16 + y^{2}}$

$16 + 2 y^{2} = 20$

$x = \sqrt{16 + y^{2}}$

$16 + 2 y^{2} = 20$

$x = \sqrt{16 + y^{2}}$

$16 + 2 y^{2} = 20$

$x = \sqrt{16 + y^{2}}$

Solve for $y$ in $16 + 2 y^{2} = 20$ .

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

Subtract $16$ from both sides of the equation.

$2 y^{2} = 20 - 16$

$x = \sqrt{16 + y^{2}}$

Subtract $16$ from $20$ .

$2 y^{2} = 4$

$x = \sqrt{16 + y^{2}}$

$2 y^{2} = 4$

$x = \sqrt{16 + y^{2}}$

Divide each term in $2 y^{2} = 4$ by $2$ and simplify.

Divide each term in $2 y^{2} = 4$ by $2$ .

$\frac{2 y^{2}}{2} = \frac{4}{2}$

$x = \sqrt{16 + y^{2}}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y^{2}}{2} = \frac{4}{2}$

$x = \sqrt{16 + y^{2}}$

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

$y^{2} = \frac{4}{2}$

$x = \sqrt{16 + y^{2}}$

$y^{2} = \frac{4}{2}$

$x = \sqrt{16 + y^{2}}$

$y^{2} = \frac{4}{2}$

$x = \sqrt{16 + y^{2}}$

Simplify the right side.

Divide $4$ by $2$ .

$y^{2} = 2$

$x = \sqrt{16 + y^{2}}$

$y^{2} = 2$

$x = \sqrt{16 + y^{2}}$

$y^{2} = 2$

$x = \sqrt{16 + y^{2}}$

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

$y = \pm \sqrt{2}$

$x = \sqrt{16 + y^{2}}$

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.

$y = \sqrt{2}$

$x = \sqrt{16 + y^{2}}$

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

$y = - \sqrt{2}$

$x = \sqrt{16 + y^{2}}$

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

$y = \sqrt{2}, - \sqrt{2}$

$x = \sqrt{16 + y^{2}}$

$y = \sqrt{2}, - \sqrt{2}$

$x = \sqrt{16 + y^{2}}$

$y = \sqrt{2}, - \sqrt{2}$

$x = \sqrt{16 + y^{2}}$

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

Replace all occurrences of $y$ in $x = \sqrt{16 + y^{2}}$ with $\sqrt{2}$ .

$x = \sqrt{16 + {(\sqrt{2})}^{2}}$

$y = \sqrt{2}$

Simplify the right side.

Simplify $\sqrt{16 + {(\sqrt{2})}^{2}}$ .

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

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

$x = \sqrt{16 + {(2^{\frac{1}{2}})}^{2}}$

$y = \sqrt{2}$

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

$x = \sqrt{16 + 2^{\frac{1}{2} \cdot 2}}$

$y = \sqrt{2}$

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

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

$y = \sqrt{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

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

$y = \sqrt{2}$

Rewrite the expression.

$x = \sqrt{16 + 2}$

$y = \sqrt{2}$

$x = \sqrt{16 + 2}$

$y = \sqrt{2}$

Evaluate the exponent.

$x = \sqrt{16 + 2}$

$y = \sqrt{2}$

$x = \sqrt{16 + 2}$

$y = \sqrt{2}$

Add $16$ and $2$ .

$x = \sqrt{18}$

$y = \sqrt{2}$

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$x = \sqrt{9 (2)}$

$y = \sqrt{2}$

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

$x = \sqrt{3^{2} \cdot 2}$

$y = \sqrt{2}$

$x = \sqrt{3^{2} \cdot 2}$

$y = \sqrt{2}$

Pull terms out from under the radical.

$x = 3 \sqrt{2}$

$y = \sqrt{2}$

$x = 3 \sqrt{2}$

$y = \sqrt{2}$

$x = 3 \sqrt{2}$

$y = \sqrt{2}$

$x = 3 \sqrt{2}$

$y = \sqrt{2}$

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

Replace all occurrences of $y$ in $x = \sqrt{16 + y^{2}}$ with $- \sqrt{2}$ .

$x = \sqrt{16 + {(- \sqrt{2})}^{2}}$

$y = - \sqrt{2}$

Simplify the right side.

Simplify $\sqrt{16 + {(- \sqrt{2})}^{2}}$ .

Simplify the expression.

Apply the product rule to $- \sqrt{2}$ .

$x = \sqrt{16 + {(- 1)}^{2} {\sqrt{2}}^{2}}$

$y = - \sqrt{2}$

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

$x = \sqrt{16 + 1 {\sqrt{2}}^{2}}$

$y = - \sqrt{2}$

Multiply ${\sqrt{2}}^{2}$ by $1$ .

$x = \sqrt{16 + {\sqrt{2}}^{2}}$

$y = - \sqrt{2}$

$x = \sqrt{16 + {\sqrt{2}}^{2}}$

$y = - \sqrt{2}$

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

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

$x = \sqrt{16 + {(2^{\frac{1}{2}})}^{2}}$

$y = - \sqrt{2}$

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

$x = \sqrt{16 + 2^{\frac{1}{2} \cdot 2}}$

$y = - \sqrt{2}$

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

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

$y = - \sqrt{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

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

$y = - \sqrt{2}$

Rewrite the expression.

$x = \sqrt{16 + 2}$

$y = - \sqrt{2}$

$x = \sqrt{16 + 2}$

$y = - \sqrt{2}$

Evaluate the exponent.

$x = \sqrt{16 + 2}$

$y = - \sqrt{2}$

$x = \sqrt{16 + 2}$

$y = - \sqrt{2}$

Add $16$ and $2$ .

$x = \sqrt{18}$

$y = - \sqrt{2}$

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$x = \sqrt{9 (2)}$

$y = - \sqrt{2}$

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

$x = \sqrt{3^{2} \cdot 2}$

$y = - \sqrt{2}$

$x = \sqrt{3^{2} \cdot 2}$

$y = - \sqrt{2}$

Pull terms out from under the radical.

$x = 3 \sqrt{2}$

$y = - \sqrt{2}$

$x = 3 \sqrt{2}$

$y = - \sqrt{2}$

$x = 3 \sqrt{2}$

$y = - \sqrt{2}$

$x = 3 \sqrt{2}$

$y = - \sqrt{2}$

$x = 3 \sqrt{2}$

$y = - \sqrt{2}$

Step 3

Solve the system $x = - \sqrt{16 + y^{2}}, x^{2} + y^{2} = 20$ .

Replace all occurrences of $x$ with $- \sqrt{16 + y^{2}}$ in each equation.

Replace all occurrences of $x$ in $x^{2} + y^{2} = 20$ with $- \sqrt{16 + y^{2}}$ .

${(- \sqrt{16 + y^{2}})}^{2} + y^{2} = 20$

$x = - \sqrt{16 + y^{2}}$

Simplify the left side.

Simplify ${(- \sqrt{16 + y^{2}})}^{2} + y^{2}$ .

Simplify each term.

Apply the product rule to $- \sqrt{16 + y^{2}}$ .

${(- 1)}^{2} {\sqrt{16 + y^{2}}}^{2} + y^{2} = 20$

$x = - \sqrt{16 + y^{2}}$

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

$1 {\sqrt{16 + y^{2}}}^{2} + y^{2} = 20$

$x = - \sqrt{16 + y^{2}}$

Multiply ${\sqrt{16 + y^{2}}}^{2}$ by $1$ .

${\sqrt{16 + y^{2}}}^{2} + y^{2} = 20$

$x = - \sqrt{16 + y^{2}}$

Rewrite ${\sqrt{16 + y^{2}}}^{2}$ as $16 + y^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{16 + y^{2}}$ as ${(16 + y^{2})}^{\frac{1}{2}}$ .

${({(16 + y^{2})}^{\frac{1}{2}})}^{2} + y^{2} = 20$

$x = - \sqrt{16 + y^{2}}$

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

${(16 + y^{2})}^{\frac{1}{2} \cdot 2} + y^{2} = 20$

$x = - \sqrt{16 + y^{2}}$

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

${(16 + y^{2})}^{\frac{2}{2}} + y^{2} = 20$

$x = - \sqrt{16 + y^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(16 + y^{2})}^{\frac{2}{2}} + y^{2} = 20$

$x = - \sqrt{16 + y^{2}}$

Rewrite the expression.

$(16 + y^{2}) + y^{2} = 20$

$x = - \sqrt{16 + y^{2}}$

$(16 + y^{2}) + y^{2} = 20$

$x = - \sqrt{16 + y^{2}}$

Simplify.

$16 + y^{2} + y^{2} = 20$

$x = - \sqrt{16 + y^{2}}$

$16 + y^{2} + y^{2} = 20$

$x = - \sqrt{16 + y^{2}}$

$16 + y^{2} + y^{2} = 20$

$x = - \sqrt{16 + y^{2}}$

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

$16 + 2 y^{2} = 20$

$x = - \sqrt{16 + y^{2}}$

$16 + 2 y^{2} = 20$

$x = - \sqrt{16 + y^{2}}$

$16 + 2 y^{2} = 20$

$x = - \sqrt{16 + y^{2}}$

$16 + 2 y^{2} = 20$

$x = - \sqrt{16 + y^{2}}$

Solve for $y$ in $16 + 2 y^{2} = 20$ .

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

Subtract $16$ from both sides of the equation.

$2 y^{2} = 20 - 16$

$x = - \sqrt{16 + y^{2}}$

Subtract $16$ from $20$ .

$2 y^{2} = 4$

$x = - \sqrt{16 + y^{2}}$

$2 y^{2} = 4$

$x = - \sqrt{16 + y^{2}}$

Divide each term in $2 y^{2} = 4$ by $2$ and simplify.

Divide each term in $2 y^{2} = 4$ by $2$ .

$\frac{2 y^{2}}{2} = \frac{4}{2}$

$x = - \sqrt{16 + y^{2}}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y^{2}}{2} = \frac{4}{2}$

$x = - \sqrt{16 + y^{2}}$

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

$y^{2} = \frac{4}{2}$

$x = - \sqrt{16 + y^{2}}$

$y^{2} = \frac{4}{2}$

$x = - \sqrt{16 + y^{2}}$

$y^{2} = \frac{4}{2}$

$x = - \sqrt{16 + y^{2}}$

Simplify the right side.

Divide $4$ by $2$ .

$y^{2} = 2$

$x = - \sqrt{16 + y^{2}}$

$y^{2} = 2$

$x = - \sqrt{16 + y^{2}}$

$y^{2} = 2$

$x = - \sqrt{16 + y^{2}}$

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

$y = \pm \sqrt{2}$

$x = - \sqrt{16 + y^{2}}$

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.

$y = \sqrt{2}$

$x = - \sqrt{16 + y^{2}}$

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

$y = - \sqrt{2}$

$x = - \sqrt{16 + y^{2}}$

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

$y = \sqrt{2}, - \sqrt{2}$

$x = - \sqrt{16 + y^{2}}$

$y = \sqrt{2}, - \sqrt{2}$

$x = - \sqrt{16 + y^{2}}$

$y = \sqrt{2}, - \sqrt{2}$

$x = - \sqrt{16 + y^{2}}$

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

Replace all occurrences of $y$ in $x = - \sqrt{16 + y^{2}}$ with $\sqrt{2}$ .

$x = - \sqrt{16 + {(\sqrt{2})}^{2}}$

$y = \sqrt{2}$

Simplify the right side.

Simplify $- \sqrt{16 + {(\sqrt{2})}^{2}}$ .

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

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

$x = - \sqrt{16 + {(2^{\frac{1}{2}})}^{2}}$

$y = \sqrt{2}$

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

$x = - \sqrt{16 + 2^{\frac{1}{2} \cdot 2}}$

$y = \sqrt{2}$

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

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

$y = \sqrt{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

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

$y = \sqrt{2}$

Rewrite the expression.

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

$y = \sqrt{2}$

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

$y = \sqrt{2}$

Evaluate the exponent.

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

$y = \sqrt{2}$

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

$y = \sqrt{2}$

Add $16$ and $2$ .

$x = - \sqrt{18}$

$y = \sqrt{2}$

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$x = - \sqrt{9 (2)}$

$y = \sqrt{2}$

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

$x = - \sqrt{3^{2} \cdot 2}$

$y = \sqrt{2}$

$x = - \sqrt{3^{2} \cdot 2}$

$y = \sqrt{2}$

Pull terms out from under the radical.

$x = - (3 \sqrt{2})$

$y = \sqrt{2}$

Multiply $3$ by $- 1$ .

$x = - 3 \sqrt{2}$

$y = \sqrt{2}$

$x = - 3 \sqrt{2}$

$y = \sqrt{2}$

$x = - 3 \sqrt{2}$

$y = \sqrt{2}$

$x = - 3 \sqrt{2}$

$y = \sqrt{2}$

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

Replace all occurrences of $y$ in $x = - \sqrt{16 + y^{2}}$ with $- \sqrt{2}$ .

$x = - \sqrt{16 + {(- \sqrt{2})}^{2}}$

$y = - \sqrt{2}$

Simplify the right side.

Simplify $- \sqrt{16 + {(- \sqrt{2})}^{2}}$ .

Simplify the expression.

Apply the product rule to $- \sqrt{2}$ .

$x = - \sqrt{16 + {(- 1)}^{2} {\sqrt{2}}^{2}}$

$y = - \sqrt{2}$

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

$x = - \sqrt{16 + 1 {\sqrt{2}}^{2}}$

$y = - \sqrt{2}$

Multiply ${\sqrt{2}}^{2}$ by $1$ .

$x = - \sqrt{16 + {\sqrt{2}}^{2}}$

$y = - \sqrt{2}$

$x = - \sqrt{16 + {\sqrt{2}}^{2}}$

$y = - \sqrt{2}$

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

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

$x = - \sqrt{16 + {(2^{\frac{1}{2}})}^{2}}$

$y = - \sqrt{2}$

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

$x = - \sqrt{16 + 2^{\frac{1}{2} \cdot 2}}$

$y = - \sqrt{2}$

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

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

$y = - \sqrt{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

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

$y = - \sqrt{2}$

Rewrite the expression.

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

$y = - \sqrt{2}$

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

$y = - \sqrt{2}$

Evaluate the exponent.

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

$y = - \sqrt{2}$

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

$y = - \sqrt{2}$

Add $16$ and $2$ .

$x = - \sqrt{18}$

$y = - \sqrt{2}$

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$x = - \sqrt{9 (2)}$

$y = - \sqrt{2}$

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

$x = - \sqrt{3^{2} \cdot 2}$

$y = - \sqrt{2}$

$x = - \sqrt{3^{2} \cdot 2}$

$y = - \sqrt{2}$

Pull terms out from under the radical.

$x = - (3 \sqrt{2})$

$y = - \sqrt{2}$

Multiply $3$ by $- 1$ .

$x = - 3 \sqrt{2}$

$y = - \sqrt{2}$

$x = - 3 \sqrt{2}$

$y = - \sqrt{2}$

$x = - 3 \sqrt{2}$

$y = - \sqrt{2}$

$x = - 3 \sqrt{2}$

$y = - \sqrt{2}$

$x = - 3 \sqrt{2}$

$y = - \sqrt{2}$

Step 4

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

$(3 \sqrt{2}, \sqrt{2})$

$(3 \sqrt{2}, - \sqrt{2})$

$(- 3 \sqrt{2}, \sqrt{2})$

$(- 3 \sqrt{2}, - \sqrt{2})$

Step 5

The result can be shown in multiple forms.

Point Form:

$(3 \sqrt{2}, \sqrt{2}), (3 \sqrt{2}, - \sqrt{2}), (- 3 \sqrt{2}, \sqrt{2}), (- 3 \sqrt{2}, - \sqrt{2})$

Equation Form:

$x = 3 \sqrt{2}, y = \sqrt{2}$

$x = 3 \sqrt{2}, y = - \sqrt{2}$

$x = - 3 \sqrt{2}, y = \sqrt{2}$

$x = - 3 \sqrt{2}, y = - \sqrt{2}$

Step 6

Solve the System of Equations x^2-y^2=16 x^2+y^2=20

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