Solve by Substitution x^2+y^2=9 , x^2-y^2=9

$x^{2} + y^{2} = 9$ , $x^{2} - y^{2} = 9$

Step 1

Solve for $x$ in $x^{2} - y^{2} = 9$ .

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

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

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

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

$x = \pm \sqrt{9 + y^{2}}$

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

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.

$x = \sqrt{9 + y^{2}}$

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

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

$x = - \sqrt{9 + y^{2}}$

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

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

$x = \sqrt{9 + y^{2}}$

$x = - \sqrt{9 + y^{2}}$

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

$x = \sqrt{9 + y^{2}}$

$x = - \sqrt{9 + y^{2}}$

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

$x = \sqrt{9 + y^{2}}$

$x = - \sqrt{9 + y^{2}}$

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

Step 2

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

Replace all occurrences of $x$ with $\sqrt{9 + y^{2}}$ in each equation.

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

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

$x = \sqrt{9 + y^{2}}$

Simplify the left side.

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

Rewrite ${\sqrt{9 + y^{2}}}^{2}$ as $9 + y^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 + y^{2}}$ as ${(9 + y^{2})}^{\frac{1}{2}}$ .

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

$x = \sqrt{9 + y^{2}}$

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

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

$x = \sqrt{9 + y^{2}}$

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

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

$x = \sqrt{9 + y^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

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

$x = \sqrt{9 + y^{2}}$

Rewrite the expression.

$(9 + y^{2}) + y^{2} = 9$

$x = \sqrt{9 + y^{2}}$

$(9 + y^{2}) + y^{2} = 9$

$x = \sqrt{9 + y^{2}}$

Simplify.

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

$x = \sqrt{9 + y^{2}}$

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

$x = \sqrt{9 + y^{2}}$

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

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

$x = \sqrt{9 + y^{2}}$

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

$x = \sqrt{9 + y^{2}}$

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

$x = \sqrt{9 + y^{2}}$

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

$x = \sqrt{9 + y^{2}}$

Solve for $y$ in $9 + 2 y^{2} = 9$ .

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

Subtract $9$ from both sides of the equation.

$2 y^{2} = 9 - 9$

$x = \sqrt{9 + y^{2}}$

Subtract $9$ from $9$ .

$2 y^{2} = 0$

$x = \sqrt{9 + y^{2}}$

$2 y^{2} = 0$

$x = \sqrt{9 + y^{2}}$

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

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

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

$x = \sqrt{9 + y^{2}}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

$x = \sqrt{9 + y^{2}}$

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

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

$x = \sqrt{9 + y^{2}}$

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

$x = \sqrt{9 + y^{2}}$

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

$x = \sqrt{9 + y^{2}}$

Simplify the right side.

Divide $0$ by $2$ .

$y^{2} = 0$

$x = \sqrt{9 + y^{2}}$

$y^{2} = 0$

$x = \sqrt{9 + y^{2}}$

$y^{2} = 0$

$x = \sqrt{9 + y^{2}}$

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

$y = \pm \sqrt{0}$

$x = \sqrt{9 + y^{2}}$

Simplify $\pm \sqrt{0}$ .

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

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

$x = \sqrt{9 + y^{2}}$

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

$y = \pm 0$

$x = \sqrt{9 + y^{2}}$

Plus or minus $0$ is $0$ .

$y = 0$

$x = \sqrt{9 + y^{2}}$

$y = 0$

$x = \sqrt{9 + y^{2}}$

$y = 0$

$x = \sqrt{9 + y^{2}}$

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

Replace all occurrences of $y$ in $x = \sqrt{9 + y^{2}}$ with $0$ .

$x = \sqrt{9 + {(0)}^{2}}$

$y = 0$

Simplify the right side.

Simplify $\sqrt{9 + {(0)}^{2}}$ .

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

$x = \sqrt{9 + 0}$

$y = 0$

Add $9$ and $0$ .

$x = \sqrt{9}$

$y = 0$

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

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

$y = 0$

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

$x = 3$

$y = 0$

$x = 3$

$y = 0$

$x = 3$

$y = 0$

$x = 3$

$y = 0$

$x = 3$

$y = 0$

Step 3

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

Replace all occurrences of $x$ with $- \sqrt{9 + y^{2}}$ in each equation.

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

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

$x = - \sqrt{9 + y^{2}}$

Simplify the left side.

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

Simplify each term.

Apply the product rule to $- \sqrt{9 + y^{2}}$ .

${(- 1)}^{2} {\sqrt{9 + y^{2}}}^{2} + y^{2} = 9$

$x = - \sqrt{9 + y^{2}}$

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

$1 {\sqrt{9 + y^{2}}}^{2} + y^{2} = 9$

$x = - \sqrt{9 + y^{2}}$

Multiply ${\sqrt{9 + y^{2}}}^{2}$ by $1$ .

${\sqrt{9 + y^{2}}}^{2} + y^{2} = 9$

$x = - \sqrt{9 + y^{2}}$

Rewrite ${\sqrt{9 + y^{2}}}^{2}$ as $9 + y^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 + y^{2}}$ as ${(9 + y^{2})}^{\frac{1}{2}}$ .

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

$x = - \sqrt{9 + y^{2}}$

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

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

$x = - \sqrt{9 + y^{2}}$

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

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

$x = - \sqrt{9 + y^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

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

$x = - \sqrt{9 + y^{2}}$

Rewrite the expression.

$(9 + y^{2}) + y^{2} = 9$

$x = - \sqrt{9 + y^{2}}$

$(9 + y^{2}) + y^{2} = 9$

$x = - \sqrt{9 + y^{2}}$

Simplify.

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

$x = - \sqrt{9 + y^{2}}$

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

$x = - \sqrt{9 + y^{2}}$

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

$x = - \sqrt{9 + y^{2}}$

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

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

$x = - \sqrt{9 + y^{2}}$

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

$x = - \sqrt{9 + y^{2}}$

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

$x = - \sqrt{9 + y^{2}}$

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

$x = - \sqrt{9 + y^{2}}$

Solve for $y$ in $9 + 2 y^{2} = 9$ .

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

Subtract $9$ from both sides of the equation.

$2 y^{2} = 9 - 9$

$x = - \sqrt{9 + y^{2}}$

Subtract $9$ from $9$ .

$2 y^{2} = 0$

$x = - \sqrt{9 + y^{2}}$

$2 y^{2} = 0$

$x = - \sqrt{9 + y^{2}}$

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

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

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

$x = - \sqrt{9 + y^{2}}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

$x = - \sqrt{9 + y^{2}}$

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

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

$x = - \sqrt{9 + y^{2}}$

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

$x = - \sqrt{9 + y^{2}}$

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

$x = - \sqrt{9 + y^{2}}$

Simplify the right side.

Divide $0$ by $2$ .

$y^{2} = 0$

$x = - \sqrt{9 + y^{2}}$

$y^{2} = 0$

$x = - \sqrt{9 + y^{2}}$

$y^{2} = 0$

$x = - \sqrt{9 + y^{2}}$

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

$y = \pm \sqrt{0}$

$x = - \sqrt{9 + y^{2}}$

Simplify $\pm \sqrt{0}$ .

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

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

$x = - \sqrt{9 + y^{2}}$

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

$y = \pm 0$

$x = - \sqrt{9 + y^{2}}$

Plus or minus $0$ is $0$ .

$y = 0$

$x = - \sqrt{9 + y^{2}}$

$y = 0$

$x = - \sqrt{9 + y^{2}}$

$y = 0$

$x = - \sqrt{9 + y^{2}}$

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

Replace all occurrences of $y$ in $x = - \sqrt{9 + y^{2}}$ with $0$ .

$x = - \sqrt{9 + {(0)}^{2}}$

$y = 0$

Simplify the right side.

Simplify $- \sqrt{9 + {(0)}^{2}}$ .

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

$x = - \sqrt{9 + 0}$

$y = 0$

Add $9$ and $0$ .

$x = - \sqrt{9}$

$y = 0$

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

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

$y = 0$

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

$x = - 1 \cdot 3$

$y = 0$

Multiply $- 1$ by $3$ .

$x = - 3$

$y = 0$

$x = - 3$

$y = 0$

$x = - 3$

$y = 0$

$x = - 3$

$y = 0$

$x = - 3$

$y = 0$

Step 4

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

$(3, 0)$

$(- 3, 0)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(3, 0), (- 3, 0)$

Equation Form:

$x = 3, y = 0$

$x = - 3, y = 0$

Step 6

Solve by Substitution x^2+y^2=9 , x^2-y^2=9

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