Solve the System of Equations x^2-4y^2=9 x+2y^2=3

$x^{2} - 4 y^{2} = 9$ $x + 2 y^{2} = 3$

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

$x = 3 - 2 y^{2}$

$x^{2} - 4 y^{2} = 9$

Replace all occurrences of $x$ with $3 - 2 y^{2}$ in each equation.

Replace all occurrences of $x$ in $x^{2} - 4 y^{2} = 9$ with $3 - 2 y^{2}$ .

${(3 - 2 y^{2})}^{2} - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

Simplify the left side.

Simplify ${(3 - 2 y^{2})}^{2} - 4 y^{2}$ .

Simplify each term.

Rewrite ${(3 - 2 y^{2})}^{2}$ as $(3 - 2 y^{2}) (3 - 2 y^{2})$ .

$(3 - 2 y^{2}) (3 - 2 y^{2}) - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

Expand $(3 - 2 y^{2}) (3 - 2 y^{2})$ using the FOIL Method.

Apply the distributive property.

$3 (3 - 2 y^{2}) - 2 y^{2} (3 - 2 y^{2}) - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

Apply the distributive property.

$3 \cdot 3 + 3 (- 2 y^{2}) - 2 y^{2} (3 - 2 y^{2}) - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

Apply the distributive property.

$3 \cdot 3 + 3 (- 2 y^{2}) - 2 y^{2} \cdot 3 - 2 y^{2} (- 2 y^{2}) - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

$3 \cdot 3 + 3 (- 2 y^{2}) - 2 y^{2} \cdot 3 - 2 y^{2} (- 2 y^{2}) - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$9 + 3 (- 2 y^{2}) - 2 y^{2} \cdot 3 - 2 y^{2} (- 2 y^{2}) - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

Multiply $- 2$ by $3$ .

$9 - 6 y^{2} - 2 y^{2} \cdot 3 - 2 y^{2} (- 2 y^{2}) - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

Multiply $3$ by $- 2$ .

$9 - 6 y^{2} - 6 y^{2} - 2 y^{2} (- 2 y^{2}) - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

Rewrite using the commutative property of multiplication.

$9 - 6 y^{2} - 6 y^{2} - 2 \cdot (- 2 y^{2} y^{2}) - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

$9 - 6 y^{2} - 6 y^{2} - 2 \cdot (- 2 (y^{2} y^{2})) - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$9 - 6 y^{2} - 6 y^{2} - 2 \cdot (- 2 y^{2 + 2}) - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

Add $2$ and $2$ .

$9 - 6 y^{2} - 6 y^{2} - 2 \cdot (- 2 y^{4}) - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

$9 - 6 y^{2} - 6 y^{2} - 2 \cdot (- 2 y^{4}) - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

Multiply $- 2$ by $- 2$ .

$9 - 6 y^{2} - 6 y^{2} + 4 y^{4} - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

$9 - 6 y^{2} - 6 y^{2} + 4 y^{4} - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

Subtract $6 y^{2}$ from $- 6 y^{2}$ .

$9 - 12 y^{2} + 4 y^{4} - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

$9 - 12 y^{2} + 4 y^{4} - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

$9 - 12 y^{2} + 4 y^{4} - 4 y^{2} = 9$

$x = 3 - 2 y^{2}$

Subtract $4 y^{2}$ from $- 12 y^{2}$ .

$9 + 4 y^{4} - 16 y^{2} = 9$

$x = 3 - 2 y^{2}$

$9 + 4 y^{4} - 16 y^{2} = 9$

$x = 3 - 2 y^{2}$

$9 + 4 y^{4} - 16 y^{2} = 9$

$x = 3 - 2 y^{2}$

$9 + 4 y^{4} - 16 y^{2} = 9$

$x = 3 - 2 y^{2}$

Solve for $y$ in $9 + 4 y^{4} - 16 y^{2} = 9$ .

Substitute $u = y^{2}$ into the equation. This will make the quadratic formula easy to use.

$4 u^{2} - 16 u + 9 = 9$

$u = y^{2}$

$x = 3 - 2 y^{2}$

Subtract $9$ from both sides of the equation.

$4 u^{2} - 16 u + 9 - 9 = 0$

$x = 3 - 2 y^{2}$

Combine the opposite terms in $4 u^{2} - 16 u + 9 - 9$ .

Subtract $9$ from $9$ .

$4 u^{2} - 16 u + 0 = 0$

$x = 3 - 2 y^{2}$

Add $4 u^{2} - 16 u$ and $0$ .

$4 u^{2} - 16 u = 0$

$x = 3 - 2 y^{2}$

$4 u^{2} - 16 u = 0$

$x = 3 - 2 y^{2}$

Factor $4 u$ out of $4 u^{2} - 16 u$ .

Factor $4 u$ out of $4 u^{2}$ .

$4 u (u) - 16 u = 0$

$x = 3 - 2 y^{2}$

Factor $4 u$ out of $- 16 u$ .

$4 u (u) + 4 u (- 4) = 0$

$x = 3 - 2 y^{2}$

Factor $4 u$ out of $4 u (u) + 4 u (- 4)$ .

$4 u (u - 4) = 0$

$x = 3 - 2 y^{2}$

$4 u (u - 4) = 0$

$x = 3 - 2 y^{2}$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u = 0$

$u - 4 = 0$

$x = 3 - 2 y^{2}$

Set $u$ equal to $0$ .

$u = 0$

$x = 3 - 2 y^{2}$

Set $u - 4$ equal to $0$ and solve for $u$ .

Set $u - 4$ equal to $0$ .

$u - 4 = 0$

$x = 3 - 2 y^{2}$

Add $4$ to both sides of the equation.

$u = 4$

$x = 3 - 2 y^{2}$

$u = 4$

$x = 3 - 2 y^{2}$

The final solution is all the values that make $4 u (u - 4) = 0$ true.

$u = 0, 4$

$x = 3 - 2 y^{2}$

Substitute the real value of $u = y^{2}$ back into the solved equation.

$y^{2} = 0$

$y^{2} = 4$

$x = 3 - 2 y^{2}$

Solve the first equation for $y$ .

$y^{2} = 0$

$x = 3 - 2 y^{2}$

Solve the equation for $y$ .

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

$y = \pm \sqrt{0}$

$x = 3 - 2 y^{2}$

Simplify $\pm \sqrt{0}$ .

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

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

$x = 3 - 2 y^{2}$

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

$y = \pm 0$

$x = 3 - 2 y^{2}$

Plus or minus $0$ is $0$ .

$y = 0$

$x = 3 - 2 y^{2}$

$y = 0$

$x = 3 - 2 y^{2}$

$y = 0$

$x = 3 - 2 y^{2}$

Solve the second equation for $y$ .

$y^{2} = 4$

$x = 3 - 2 y^{2}$

Solve the equation for $y$ .

Remove parentheses.

$y^{2} = 4$

$x = 3 - 2 y^{2}$

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

$y = \pm \sqrt{4}$

$x = 3 - 2 y^{2}$

Simplify $\pm \sqrt{4}$ .

Rewrite $4$ as $2^{2}$ .

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

$x = 3 - 2 y^{2}$

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

$y = \pm 2$

$x = 3 - 2 y^{2}$

$y = \pm 2$

$x = 3 - 2 y^{2}$

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.

$y = 2$

$x = 3 - 2 y^{2}$

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

$y = - 2$

$x = 3 - 2 y^{2}$

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

$y = 2, - 2$

$x = 3 - 2 y^{2}$

$y = 2, - 2$

$x = 3 - 2 y^{2}$

$y = 2, - 2$

$x = 3 - 2 y^{2}$

The solution to $4 y^{4} - 16 y^{2} + 9 = 9$ is $y = 0, 2, - 2$ .

$y = 0, 2, - 2$

$x = 3 - 2 y^{2}$

$y = 0, 2, - 2$

$x = 3 - 2 y^{2}$

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

Replace all occurrences of $y$ in $x = 3 - 2 y^{2}$ with $0$ .

$x = 3 - 2 {(0)}^{2}$

$y = 0$

Simplify the right side.

Simplify $3 - 2 {(0)}^{2}$ .

Simplify each term.

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

$x = 3 - 2 \cdot 0$

$y = 0$

Multiply $- 2$ by $0$ .

$x = 3 + 0$

$y = 0$

$x = 3 + 0$

$y = 0$

Add $3$ and $0$ .

$x = 3$

$y = 0$

$x = 3$

$y = 0$

$x = 3$

$y = 0$

$x = 3$

$y = 0$

Replace all occurrences of $y$ with $2$ in each equation.

Replace all occurrences of $y$ in $x = 3 - 2 y^{2}$ with $2$ .

$x = 3 - 2 {(2)}^{2}$

$y = 2$

Simplify the right side.

Simplify $3 - 2 {(2)}^{2}$ .

Simplify each term.

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

$x = 3 - 2 \cdot 4$

$y = 2$

Multiply $- 2$ by $4$ .

$x = 3 - 8$

$y = 2$

$x = 3 - 8$

$y = 2$

Subtract $8$ from $3$ .

$x = - 5$

$y = 2$

$x = - 5$

$y = 2$

$x = - 5$

$y = 2$

$x = - 5$

$y = 2$

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

Replace all occurrences of $y$ in $x = 3 - 2 y^{2}$ with $0$ .

$x = 3 - 2 {(0)}^{2}$

$y = 0$

Simplify the right side.

Simplify $3 - 2 {(0)}^{2}$ .

Simplify each term.

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

$x = 3 - 2 \cdot 0$

$y = 0$

Multiply $- 2$ by $0$ .

$x = 3 + 0$

$y = 0$

$x = 3 + 0$

$y = 0$

Add $3$ and $0$ .

$x = 3$

$y = 0$

$x = 3$

$y = 0$

$x = 3$

$y = 0$

$x = 3$

$y = 0$

Replace all occurrences of $y$ with $2$ in each equation.

Replace all occurrences of $y$ in $x = 3 - 2 y^{2}$ with $2$ .

$x = 3 - 2 {(2)}^{2}$

$y = 2$

Simplify the right side.

Simplify $3 - 2 {(2)}^{2}$ .

Simplify each term.

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

$x = 3 - 2 \cdot 4$

$y = 2$

Multiply $- 2$ by $4$ .

$x = 3 - 8$

$y = 2$

$x = 3 - 8$

$y = 2$

Subtract $8$ from $3$ .

$x = - 5$

$y = 2$

$x = - 5$

$y = 2$

$x = - 5$

$y = 2$

$x = - 5$

$y = 2$

Replace all occurrences of $y$ with $- 2$ in each equation.

Replace all occurrences of $y$ in $x = 3 - 2 y^{2}$ with $- 2$ .

$x = 3 - 2 {(- 2)}^{2}$

$y = - 2$

Simplify the right side.

Simplify $3 - 2 {(- 2)}^{2}$ .

Simplify each term.

Multiply $- 2$ by ${(- 2)}^{2}$ by adding the exponents.

Multiply $- 2$ by ${(- 2)}^{2}$ .

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

$x = 3 + (- 2) {(- 2)}^{2}$

$y = - 2$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = 3 + {(- 2)}^{1 + 2}$

$y = - 2$

$x = 3 + {(- 2)}^{1 + 2}$

$y = - 2$

Add $1$ and $2$ .

$x = 3 + {(- 2)}^{3}$

$y = - 2$

$x = 3 + {(- 2)}^{3}$

$y = - 2$

Raise $- 2$ to the power of $3$ .

$x = 3 - 8$

$y = - 2$

$x = 3 - 8$

$y = - 2$

Subtract $8$ from $3$ .

$x = - 5$

$y = - 2$

$x = - 5$

$y = - 2$

$x = - 5$

$y = - 2$

$x = - 5$

$y = - 2$

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

$(3, 0)$

$(- 5, 2)$

$(- 5, - 2)$

The result can be shown in multiple forms.

Point Form:

$(3, 0), (- 5, 2), (- 5, - 2)$

Equation Form:

$x = 3, y = 0$

$x = - 5, y = 2$

$x = - 5, y = - 2$

Solve the System of Equations x^2-4y^2=9 x+2y^2=3

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