Algebra Examples

$x^{2} + y^{2} = 1$ , $x^{2} - y^{2} = 1$

Step 1

Solve for $x$ in $x^{2} - y^{2} = 1$ .

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Step 1.1

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

$x^{2} = 1 + y^{2}$

$x^{2} + y^{2} = 1$

Step 1.2

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

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

$x^{2} + y^{2} = 1$

Step 1.3

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

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Step 1.3.1

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

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

$x^{2} + y^{2} = 1$

Step 1.3.2

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

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

$x^{2} + y^{2} = 1$

Step 1.3.3

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

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

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

$x^{2} + y^{2} = 1$

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

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

$x^{2} + y^{2} = 1$

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

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

$x^{2} + y^{2} = 1$

Step 2

Solve the system $x = \sqrt{1 + y^{2}}, x^{2} + y^{2} = 1$ .

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Step 2.1

Replace all occurrences of $x$ with $\sqrt{1 + y^{2}}$ in each equation.

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Step 2.1.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 1$ with $\sqrt{1 + y^{2}}$ .

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

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

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify ${(\sqrt{1 + y^{2}})}^{2} + y^{2}$ .

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Step 2.1.2.1.1

Rewrite ${\sqrt{1 + y^{2}}}^{2}$ as $1 + y^{2}$ .

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Step 2.1.2.1.1.1

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

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

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

Step 2.1.2.1.1.2

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

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

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

Step 2.1.2.1.1.3

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

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

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

Step 2.1.2.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.4.1

Cancel the common factor.

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

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

Step 2.1.2.1.1.4.2

Rewrite the expression.

$(1 + y^{2}) + y^{2} = 1$

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

$(1 + y^{2}) + y^{2} = 1$

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

Step 2.1.2.1.1.5

Simplify.

$1 + y^{2} + y^{2} = 1$

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

$1 + y^{2} + y^{2} = 1$

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

Step 2.1.2.1.2

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

$1 + 2 y^{2} = 1$

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

$1 + 2 y^{2} = 1$

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

$1 + 2 y^{2} = 1$

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

$1 + 2 y^{2} = 1$

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

Step 2.2

Solve for $y$ in $1 + 2 y^{2} = 1$ .

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Step 2.2.1

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

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Step 2.2.1.1

Subtract $1$ from both sides of the equation.

$2 y^{2} = 1 - 1$

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

Step 2.2.1.2

Subtract $1$ from $1$ .

$2 y^{2} = 0$

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

$2 y^{2} = 0$

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

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

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

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

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

Step 2.2.2.2.1.2

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

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

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

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

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

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

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

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $0$ by $2$ .

$y^{2} = 0$

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

$y^{2} = 0$

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

$y^{2} = 0$

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

Step 2.2.3

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

$y = \pm \sqrt{0}$

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

Step 2.2.4

Simplify $\pm \sqrt{0}$ .

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Step 2.2.4.1

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

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

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

Step 2.2.4.2

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

$y = \pm 0$

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

Step 2.2.4.3

Plus or minus $0$ is $0$ .

$y = 0$

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

$y = 0$

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

$y = 0$

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

Step 2.3

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

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Step 2.3.1

Replace all occurrences of $y$ in $x = \sqrt{1 + y^{2}}$ with $0$ .

$x = \sqrt{1 + {(0)}^{2}}$

$y = 0$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{1 + {(0)}^{2}}$ .

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Step 2.3.2.1.1

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

$x = \sqrt{1 + 0}$

$y = 0$

Step 2.3.2.1.2

Add $1$ and $0$ .

$x = \sqrt{1}$

$y = 0$

Step 2.3.2.1.3

Any root of $1$ is $1$ .

$x = 1$

$y = 0$

$x = 1$

$y = 0$

$x = 1$

$y = 0$

$x = 1$

$y = 0$

$x = 1$

$y = 0$

Step 3

Solve the system $x = - \sqrt{1 + y^{2}}, x^{2} + y^{2} = 1$ .

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Step 3.1

Replace all occurrences of $x$ with $- \sqrt{1 + y^{2}}$ in each equation.

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Step 3.1.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 1$ with $- \sqrt{1 + y^{2}}$ .

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

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify ${(- \sqrt{1 + y^{2}})}^{2} + y^{2}$ .

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Step 3.1.2.1.1

Simplify each term.

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Step 3.1.2.1.1.1

Apply the product rule to $- \sqrt{1 + y^{2}}$ .

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

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

Step 3.1.2.1.1.2

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

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

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

Step 3.1.2.1.1.3

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

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

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

Step 3.1.2.1.1.4

Rewrite ${\sqrt{1 + y^{2}}}^{2}$ as $1 + y^{2}$ .

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Step 3.1.2.1.1.4.1

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

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

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

Step 3.1.2.1.1.4.2

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

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

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

Step 3.1.2.1.1.4.3

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

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

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

Step 3.1.2.1.1.4.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.1.4.4.1

Cancel the common factor.

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

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

Step 3.1.2.1.1.4.4.2

Rewrite the expression.

$(1 + y^{2}) + y^{2} = 1$

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

$(1 + y^{2}) + y^{2} = 1$

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

Step 3.1.2.1.1.4.5

Simplify.

$1 + y^{2} + y^{2} = 1$

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

$1 + y^{2} + y^{2} = 1$

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

$1 + y^{2} + y^{2} = 1$

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

Step 3.1.2.1.2

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

$1 + 2 y^{2} = 1$

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

$1 + 2 y^{2} = 1$

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

$1 + 2 y^{2} = 1$

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

$1 + 2 y^{2} = 1$

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

Step 3.2

Solve for $y$ in $1 + 2 y^{2} = 1$ .

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Step 3.2.1

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

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Step 3.2.1.1

Subtract $1$ from both sides of the equation.

$2 y^{2} = 1 - 1$

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

Step 3.2.1.2

Subtract $1$ from $1$ .

$2 y^{2} = 0$

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

$2 y^{2} = 0$

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

Step 3.2.2

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

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Step 3.2.2.1

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

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

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

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

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

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

Step 3.2.2.2.1.2

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

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

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

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

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

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

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

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $0$ by $2$ .

$y^{2} = 0$

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

$y^{2} = 0$

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

$y^{2} = 0$

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

Step 3.2.3

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

$y = \pm \sqrt{0}$

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

Step 3.2.4

Simplify $\pm \sqrt{0}$ .

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Step 3.2.4.1

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

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

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

Step 3.2.4.2

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

$y = \pm 0$

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

Step 3.2.4.3

Plus or minus $0$ is $0$ .

$y = 0$

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

$y = 0$

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

$y = 0$

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

Step 3.3

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

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Step 3.3.1

Replace all occurrences of $y$ in $x = - \sqrt{1 + y^{2}}$ with $0$ .

$x = - \sqrt{1 + {(0)}^{2}}$

$y = 0$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \sqrt{1 + {(0)}^{2}}$ .

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Step 3.3.2.1.1

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

$x = - \sqrt{1 + 0}$

$y = 0$

Step 3.3.2.1.2

Add $1$ and $0$ .

$x = - \sqrt{1}$

$y = 0$

Step 3.3.2.1.3

Any root of $1$ is $1$ .

$x = - 1 \cdot 1$

$y = 0$

Step 3.3.2.1.4

Multiply $- 1$ by $1$ .

$x = - 1$

$y = 0$

$x = - 1$

$y = 0$

$x = - 1$

$y = 0$

$x = - 1$

$y = 0$

$x = - 1$

$y = 0$

Step 4

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

$(1, 0)$

$(- 1, 0)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(1, 0), (- 1, 0)$

Equation Form:

$x = 1, y = 0$

$x = - 1, y = 0$

Step 6