Algebra Examples

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

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

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

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

Step 1

Solve for

x

$x$ in

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

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

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Step 1.1

Add

y^{2}

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

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

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

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

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

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}}

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

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

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

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

$\pm$ to find the first solution.

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

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

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

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

Step 1.3.2

Next, use the negative value of the

\pm

$\pm$ to find the second solution.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Step 2

Solve the system

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

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

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Step 2.1

Replace all occurrences of

x

$x$ with

\sqrt{1 + y^{2}}

$\sqrt{1 + y^{2}}$ in each equation.

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Step 2.1.1

Replace all occurrences of

x

$x$ in

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

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

\sqrt{1 + y^{2}}

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

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

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

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

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

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}

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

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Step 2.1.2.1.1

Rewrite

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

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

1 + y^{2}

$1 + y^{2}$ .

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Step 2.1.2.1.1.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{1 + y^{2}}

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

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

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

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

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

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

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

Step 2.1.2.1.1.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

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

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

Step 2.1.2.1.1.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

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

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

Step 2.1.2.1.1.4

Cancel the common factor of

2

$2$ .

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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