Solve the System of Equations 4x^2-y^2=4 x^2+y^2=1

$4 x^{2} - y^{2} = 4$ $x^{2} + y^{2} = 1$

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

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

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

$4 x^{2} - y^{2} = 4$

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

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

$4 x^{2} - y^{2} = 4$

Simplify $\pm \sqrt{1 - y^{2}}$ .

Rewrite $1$ as $1^{2}$ .

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

$4 x^{2} - y^{2} = 4$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = y$ .

$x = \pm \sqrt{(1 + y) (1 - y)}$

$4 x^{2} - y^{2} = 4$

$x = \pm \sqrt{(1 + y) (1 - y)}$

$4 x^{2} - y^{2} = 4$

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.

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

$4 x^{2} - y^{2} = 4$

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

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

$4 x^{2} - y^{2} = 4$

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

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

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

$4 x^{2} - y^{2} = 4$

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

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

$4 x^{2} - y^{2} = 4$

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

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

$4 x^{2} - y^{2} = 4$

Solve the system $x = \sqrt{(1 + y) (1 - y)}, 4 x^{2} - y^{2} = 4$ .

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

Replace all occurrences of $x$ in $4 x^{2} - y^{2} = 4$ with $\sqrt{(1 + y) (1 - y)}$ .

$4 {(\sqrt{(1 + y) (1 - y)})}^{2} - y^{2} = 4$

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

Simplify the left side.

Simplify $4 {(\sqrt{(1 + y) (1 - y)})}^{2} - y^{2}$ .

Simplify each term.

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

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

$4 {({((1 + y) (1 - y))}^{\frac{1}{2}})}^{2} - y^{2} = 4$

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

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

$4 {((1 + y) (1 - y))}^{\frac{1}{2} \cdot 2} - y^{2} = 4$

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

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

$4 {((1 + y) (1 - y))}^{\frac{2}{2}} - y^{2} = 4$

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

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {((1 + y) (1 - y))}^{\frac{2}{2}} - y^{2} = 4$

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

Rewrite the expression.

$4 ((1 + y) (1 - y)) - y^{2} = 4$

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

$4 ((1 + y) (1 - y)) - y^{2} = 4$

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

Simplify.

$4 ((1 + y) (1 - y)) - y^{2} = 4$

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

$4 ((1 + y) (1 - y)) - y^{2} = 4$

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

Expand $(1 + y) (1 - y)$ using the FOIL Method.

Apply the distributive property.

$4 (1 (1 - y) + y (1 - y)) - y^{2} = 4$

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

Apply the distributive property.

$4 (1 \cdot 1 + 1 (- y) + y (1 - y)) - y^{2} = 4$

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

Apply the distributive property.

$4 (1 \cdot 1 + 1 (- y) + y \cdot 1 + y (- y)) - y^{2} = 4$

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

$4 (1 \cdot 1 + 1 (- y) + y \cdot 1 + y (- y)) - y^{2} = 4$

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

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$4 (1 + 1 (- y) + y \cdot 1 + y (- y)) - y^{2} = 4$

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

Multiply $- y$ by $1$ .

$4 (1 - y + y \cdot 1 + y (- y)) - y^{2} = 4$

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

Multiply $y$ by $1$ .

$4 (1 - y + y + y (- y)) - y^{2} = 4$

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

Rewrite using the commutative property of multiplication.

$4 (1 - y + y - y \cdot y) - y^{2} = 4$

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

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$4 (1 - y + y - (y \cdot y)) - y^{2} = 4$

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

Multiply $y$ by $y$ .

$4 (1 - y + y - y^{2}) - y^{2} = 4$

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

$4 (1 - y + y - y^{2}) - y^{2} = 4$

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

$4 (1 - y + y - y^{2}) - y^{2} = 4$

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

Add $- y$ and $y$ .

$4 (1 + 0 - y^{2}) - y^{2} = 4$

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

Add $1$ and $0$ .

$4 (1 - y^{2}) - y^{2} = 4$

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

$4 (1 - y^{2}) - y^{2} = 4$

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

Apply the distributive property.

$4 \cdot 1 + 4 (- y^{2}) - y^{2} = 4$

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

Multiply $4$ by $1$ .

$4 + 4 (- y^{2}) - y^{2} = 4$

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

Multiply $- 1$ by $4$ .

$4 - 4 y^{2} - y^{2} = 4$

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

$4 - 4 y^{2} - y^{2} = 4$

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

Subtract $y^{2}$ from $- 4 y^{2}$ .

$4 - 5 y^{2} = 4$

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

$4 - 5 y^{2} = 4$

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

$4 - 5 y^{2} = 4$

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

$4 - 5 y^{2} = 4$

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

Solve for $y$ in $4 - 5 y^{2} = 4$ .

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

Subtract $4$ from both sides of the equation.

$- 5 y^{2} = 4 - 4$

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

Subtract $4$ from $4$ .

$- 5 y^{2} = 0$

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

$- 5 y^{2} = 0$

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

Divide each term in $- 5 y^{2} = 0$ by $- 5$ and simplify.

Divide each term in $- 5 y^{2} = 0$ by $- 5$ .

$\frac{- 5 y^{2}}{- 5} = \frac{0}{- 5}$

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

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 y^{2}}{- 5} = \frac{0}{- 5}$

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

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

$y^{2} = \frac{0}{- 5}$

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

$y^{2} = \frac{0}{- 5}$

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

$y^{2} = \frac{0}{- 5}$

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

Simplify the right side.

Divide $0$ by $- 5$ .

$y^{2} = 0$

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

$y^{2} = 0$

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

$y^{2} = 0$

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

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

$y = \pm \sqrt{0}$

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

Simplify $\pm \sqrt{0}$ .

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

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

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

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

$y = \pm 0$

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

Plus or minus $0$ is $0$ .

$y = 0$

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

$y = 0$

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

$y = 0$

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

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

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

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

$y = 0$

Simplify $x = \sqrt{(1 + 0) (1 - (0))}$ .

Simplify the left side.

Remove parentheses.

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

$y = 0$

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

$y = 0$

Simplify the right side.

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

Add $1$ and $0$ .

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

$y = 0$

Multiply $1 - (0)$ by $1$ .

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

$y = 0$

Multiply $- 1$ by $0$ .

$x = \sqrt{1 + 0}$

$y = 0$

Add $1$ and $0$ .

$x = \sqrt{1}$

$y = 0$

Any root of $1$ is $1$ .

$x = 1$

$y = 0$

$x = 1$

$y = 0$

$x = 1$

$y = 0$

$x = 1$

$y = 0$

$x = 1$

$y = 0$

$x = 1$

$y = 0$

Solve the system $x = - \sqrt{(1 + y) (1 - y)}, 4 x^{2} - y^{2} = 4$ .

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

Replace all occurrences of $x$ in $4 x^{2} - y^{2} = 4$ with $- \sqrt{(1 + y) (1 - y)}$ .

$4 {(- \sqrt{(1 + y) (1 - y)})}^{2} - y^{2} = 4$

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

Simplify the left side.

Simplify $4 {(- \sqrt{(1 + y) (1 - y)})}^{2} - y^{2}$ .

Simplify each term.

Apply the product rule to $- \sqrt{(1 + y) (1 - y)}$ .

$4 ({(- 1)}^{2} {\sqrt{(1 + y) (1 - y)}}^{2}) - y^{2} = 4$

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

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

$4 (1 {\sqrt{(1 + y) (1 - y)}}^{2}) - y^{2} = 4$

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

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

$4 {\sqrt{(1 + y) (1 - y)}}^{2} - y^{2} = 4$

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

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

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

$4 {({((1 + y) (1 - y))}^{\frac{1}{2}})}^{2} - y^{2} = 4$

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

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

$4 {((1 + y) (1 - y))}^{\frac{1}{2} \cdot 2} - y^{2} = 4$

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

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

$4 {((1 + y) (1 - y))}^{\frac{2}{2}} - y^{2} = 4$

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

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {((1 + y) (1 - y))}^{\frac{2}{2}} - y^{2} = 4$

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

Rewrite the expression.

$4 ((1 + y) (1 - y)) - y^{2} = 4$

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

$4 ((1 + y) (1 - y)) - y^{2} = 4$

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

Simplify.

$4 ((1 + y) (1 - y)) - y^{2} = 4$

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

$4 ((1 + y) (1 - y)) - y^{2} = 4$

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

Expand $(1 + y) (1 - y)$ using the FOIL Method.

Apply the distributive property.

$4 (1 (1 - y) + y (1 - y)) - y^{2} = 4$

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

Apply the distributive property.

$4 (1 \cdot 1 + 1 (- y) + y (1 - y)) - y^{2} = 4$

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

Apply the distributive property.

$4 (1 \cdot 1 + 1 (- y) + y \cdot 1 + y (- y)) - y^{2} = 4$

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

$4 (1 \cdot 1 + 1 (- y) + y \cdot 1 + y (- y)) - y^{2} = 4$

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

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$4 (1 + 1 (- y) + y \cdot 1 + y (- y)) - y^{2} = 4$

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

Multiply $- y$ by $1$ .

$4 (1 - y + y \cdot 1 + y (- y)) - y^{2} = 4$

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

Multiply $y$ by $1$ .

$4 (1 - y + y + y (- y)) - y^{2} = 4$

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

Rewrite using the commutative property of multiplication.

$4 (1 - y + y - y \cdot y) - y^{2} = 4$

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

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$4 (1 - y + y - (y \cdot y)) - y^{2} = 4$

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

Multiply $y$ by $y$ .

$4 (1 - y + y - y^{2}) - y^{2} = 4$

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

$4 (1 - y + y - y^{2}) - y^{2} = 4$

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

$4 (1 - y + y - y^{2}) - y^{2} = 4$

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

Add $- y$ and $y$ .

$4 (1 + 0 - y^{2}) - y^{2} = 4$

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

Add $1$ and $0$ .

$4 (1 - y^{2}) - y^{2} = 4$

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

$4 (1 - y^{2}) - y^{2} = 4$

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

Apply the distributive property.

$4 \cdot 1 + 4 (- y^{2}) - y^{2} = 4$

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

Multiply $4$ by $1$ .

$4 + 4 (- y^{2}) - y^{2} = 4$

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

Multiply $- 1$ by $4$ .

$4 - 4 y^{2} - y^{2} = 4$

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

$4 - 4 y^{2} - y^{2} = 4$

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

Subtract $y^{2}$ from $- 4 y^{2}$ .

$4 - 5 y^{2} = 4$

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

$4 - 5 y^{2} = 4$

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

$4 - 5 y^{2} = 4$

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

$4 - 5 y^{2} = 4$

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

Solve for $y$ in $4 - 5 y^{2} = 4$ .

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

Subtract $4$ from both sides of the equation.

$- 5 y^{2} = 4 - 4$

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

Subtract $4$ from $4$ .

$- 5 y^{2} = 0$

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

$- 5 y^{2} = 0$

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

Divide each term in $- 5 y^{2} = 0$ by $- 5$ and simplify.

Divide each term in $- 5 y^{2} = 0$ by $- 5$ .

$\frac{- 5 y^{2}}{- 5} = \frac{0}{- 5}$

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

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 y^{2}}{- 5} = \frac{0}{- 5}$

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

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

$y^{2} = \frac{0}{- 5}$

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

$y^{2} = \frac{0}{- 5}$

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

$y^{2} = \frac{0}{- 5}$

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

Simplify the right side.

Divide $0$ by $- 5$ .

$y^{2} = 0$

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

$y^{2} = 0$

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

$y^{2} = 0$

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

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

$y = \pm \sqrt{0}$

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

Simplify $\pm \sqrt{0}$ .

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

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

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

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

$y = \pm 0$

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

Plus or minus $0$ is $0$ .

$y = 0$

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

$y = 0$

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

$y = 0$

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

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

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

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

$y = 0$

Simplify $x = - \sqrt{(1 + 0) (1 - (0))}$ .

Simplify the left side.

Remove parentheses.

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

$y = 0$

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

$y = 0$

Simplify the right side.

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

Add $1$ and $0$ .

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

$y = 0$

Multiply $1 - (0)$ by $1$ .

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

$y = 0$

Multiply $- 1$ by $0$ .

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

$y = 0$

Add $1$ and $0$ .

$x = - \sqrt{1}$

$y = 0$

Any root of $1$ is $1$ .

$x = - 1 \cdot 1$

$y = 0$

Multiply $- 1$ by $1$ .

$x = - 1$

$y = 0$

$x = - 1$

$y = 0$

$x = - 1$

$y = 0$

$x = - 1$

$y = 0$

$x = - 1$

$y = 0$

$x = - 1$

$y = 0$

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

$(1, 0)$

$(- 1, 0)$

The result can be shown in multiple forms.

Point Form:

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

Equation Form:

$x = 1, y = 0$

$x = - 1, y = 0$

Solve the System of Equations 4x^2-y^2=4 x^2+y^2=1

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