Solve by Substitution 81x^2+9y^2=729 , y=x^2-9

$81 x^{2} + 9 y^{2} = 729$ , $y = x^{2} - 9$

Step 1

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

Replace all occurrences of $y$ in $81 x^{2} + 9 y^{2} = 729$ with $x^{2} - 9$ .

$81 x^{2} + 9 {(x^{2} - 9)}^{2} = 729$

$y = x^{2} - 9$

Simplify the left side.

Simplify $81 x^{2} + 9 {(x^{2} - 9)}^{2}$ .

Simplify each term.

Rewrite ${(x^{2} - 9)}^{2}$ as $(x^{2} - 9) (x^{2} - 9)$ .

$81 x^{2} + 9 ((x^{2} - 9) (x^{2} - 9)) = 729$

$y = x^{2} - 9$

Expand $(x^{2} - 9) (x^{2} - 9)$ using the FOIL Method.

Apply the distributive property.

$81 x^{2} + 9 (x^{2} (x^{2} - 9) - 9 (x^{2} - 9)) = 729$

$y = x^{2} - 9$

Apply the distributive property.

$81 x^{2} + 9 (x^{2} x^{2} + x^{2} \cdot - 9 - 9 (x^{2} - 9)) = 729$

$y = x^{2} - 9$

Apply the distributive property.

$81 x^{2} + 9 (x^{2} x^{2} + x^{2} \cdot - 9 - 9 x^{2} - 9 \cdot - 9) = 729$

$y = x^{2} - 9$

$81 x^{2} + 9 (x^{2} x^{2} + x^{2} \cdot - 9 - 9 x^{2} - 9 \cdot - 9) = 729$

$y = x^{2} - 9$

Simplify and combine like terms.

Simplify each term.

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

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

$81 x^{2} + 9 (x^{2 + 2} + x^{2} \cdot - 9 - 9 x^{2} - 9 \cdot - 9) = 729$

$y = x^{2} - 9$

Add $2$ and $2$ .

$81 x^{2} + 9 (x^{4} + x^{2} \cdot - 9 - 9 x^{2} - 9 \cdot - 9) = 729$

$y = x^{2} - 9$

$81 x^{2} + 9 (x^{4} + x^{2} \cdot - 9 - 9 x^{2} - 9 \cdot - 9) = 729$

$y = x^{2} - 9$

Move $- 9$ to the left of $x^{2}$ .

$81 x^{2} + 9 (x^{4} - 9 \cdot x^{2} - 9 x^{2} - 9 \cdot - 9) = 729$

$y = x^{2} - 9$

Multiply $- 9$ by $- 9$ .

$81 x^{2} + 9 (x^{4} - 9 x^{2} - 9 x^{2} + 81) = 729$

$y = x^{2} - 9$

$81 x^{2} + 9 (x^{4} - 9 x^{2} - 9 x^{2} + 81) = 729$

$y = x^{2} - 9$

Subtract $9 x^{2}$ from $- 9 x^{2}$ .

$81 x^{2} + 9 (x^{4} - 18 x^{2} + 81) = 729$

$y = x^{2} - 9$

$81 x^{2} + 9 (x^{4} - 18 x^{2} + 81) = 729$

$y = x^{2} - 9$

Apply the distributive property.

$81 x^{2} + 9 x^{4} + 9 (- 18 x^{2}) + 9 \cdot 81 = 729$

$y = x^{2} - 9$

Simplify.

Multiply $- 18$ by $9$ .

$81 x^{2} + 9 x^{4} - 162 x^{2} + 9 \cdot 81 = 729$

$y = x^{2} - 9$

Multiply $9$ by $81$ .

$81 x^{2} + 9 x^{4} - 162 x^{2} + 729 = 729$

$y = x^{2} - 9$

$81 x^{2} + 9 x^{4} - 162 x^{2} + 729 = 729$

$y = x^{2} - 9$

$81 x^{2} + 9 x^{4} - 162 x^{2} + 729 = 729$

$y = x^{2} - 9$

Subtract $162 x^{2}$ from $81 x^{2}$ .

$9 x^{4} - 81 x^{2} + 729 = 729$

$y = x^{2} - 9$

$9 x^{4} - 81 x^{2} + 729 = 729$

$y = x^{2} - 9$

$9 x^{4} - 81 x^{2} + 729 = 729$

$y = x^{2} - 9$

$9 x^{4} - 81 x^{2} + 729 = 729$

$y = x^{2} - 9$

Step 2

Solve for $x$ in $9 x^{4} - 81 x^{2} + 729 = 729$ .

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

$9 u^{2} - 81 u + 729 = 729$

$u = x^{2}$

$y = x^{2} - 9$

Subtract $729$ from both sides of the equation.

$9 u^{2} - 81 u + 729 - 729 = 0$

$y = x^{2} - 9$

Combine the opposite terms in $9 u^{2} - 81 u + 729 - 729$ .

Subtract $729$ from $729$ .

$9 u^{2} - 81 u + 0 = 0$

$y = x^{2} - 9$

Add $9 u^{2} - 81 u$ and $0$ .

$9 u^{2} - 81 u = 0$

$y = x^{2} - 9$

$9 u^{2} - 81 u = 0$

$y = x^{2} - 9$

Factor $9 u$ out of $9 u^{2} - 81 u$ .

Factor $9 u$ out of $9 u^{2}$ .

$9 u (u) - 81 u = 0$

$y = x^{2} - 9$

Factor $9 u$ out of $- 81 u$ .

$9 u (u) + 9 u (- 9) = 0$

$y = x^{2} - 9$

Factor $9 u$ out of $9 u (u) + 9 u (- 9)$ .

$9 u (u - 9) = 0$

$y = x^{2} - 9$

$9 u (u - 9) = 0$

$y = x^{2} - 9$

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

$u = 0$

$u - 9 = 0$

$y = x^{2} - 9$

Set $u$ equal to $0$ .

$u = 0$

$y = x^{2} - 9$

Set $u - 9$ equal to $0$ and solve for $u$ .

Set $u - 9$ equal to $0$ .

$u - 9 = 0$

$y = x^{2} - 9$

Add $9$ to both sides of the equation.

$u = 9$

$y = x^{2} - 9$

$u = 9$

$y = x^{2} - 9$

The final solution is all the values that make $9 u (u - 9) = 0$ true.

$u = 0, 9$

$y = x^{2} - 9$

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

$x^{2} = 0$

$x^{2} = 9$

$y = x^{2} - 9$

Solve the first equation for $x$ .

$x^{2} = 0$

$y = x^{2} - 9$

Solve the equation for $x$ .

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

$x = \pm \sqrt{0}$

$y = x^{2} - 9$

Simplify $\pm \sqrt{0}$ .

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

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

$y = x^{2} - 9$

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

$x = \pm 0$

$y = x^{2} - 9$

Plus or minus $0$ is $0$ .

$x = 0$

$y = x^{2} - 9$

$x = 0$

$y = x^{2} - 9$

$x = 0$

$y = x^{2} - 9$

Solve the second equation for $x$ .

$x^{2} = 9$

$y = x^{2} - 9$

Solve the equation for $x$ .

Remove parentheses.

$x^{2} = 9$

$y = x^{2} - 9$

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

$x = \pm \sqrt{9}$

$y = x^{2} - 9$

Simplify $\pm \sqrt{9}$ .

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

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

$y = x^{2} - 9$

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

$x = \pm 3$

$y = x^{2} - 9$

$x = \pm 3$

$y = x^{2} - 9$

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.

$x = 3$

$y = x^{2} - 9$

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

$x = - 3$

$y = x^{2} - 9$

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

$x = 3, - 3$

$y = x^{2} - 9$

$x = 3, - 3$

$y = x^{2} - 9$

$x = 3, - 3$

$y = x^{2} - 9$

The solution to $9 x^{4} - 81 x^{2} + 729 = 729$ is $x = 0, 3, - 3$ .

$x = 0, 3, - 3$

$y = x^{2} - 9$

$x = 0, 3, - 3$

$y = x^{2} - 9$

Step 3

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

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

$y = {(0)}^{2} - 9$

$x = 0$

Simplify the right side.

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

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

$y = 0 - 9$

$x = 0$

Subtract $9$ from $0$ .

$y = - 9$

$x = 0$

$y = - 9$

$x = 0$

$y = - 9$

$x = 0$

$y = - 9$

$x = 0$

Step 4

Replace all occurrences of $x$ with $3$ in each equation.

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

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

$x = 3$

Simplify the right side.

Simplify ${(3)}^{2} - 9$ .

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

$y = 9 - 9$

$x = 3$

Subtract $9$ from $9$ .

$y = 0$

$x = 3$

$y = 0$

$x = 3$

$y = 0$

$x = 3$

$y = 0$

$x = 3$

Step 5

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

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

$y = {(0)}^{2} - 9$

$x = 0$

Simplify the right side.

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

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

$y = 0 - 9$

$x = 0$

Subtract $9$ from $0$ .

$y = - 9$

$x = 0$

$y = - 9$

$x = 0$

$y = - 9$

$x = 0$

$y = - 9$

$x = 0$

Step 6

Replace all occurrences of $x$ with $3$ in each equation.

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

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

$x = 3$

Simplify the right side.

Simplify ${(3)}^{2} - 9$ .

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

$y = 9 - 9$

$x = 3$

Subtract $9$ from $9$ .

$y = 0$

$x = 3$

$y = 0$

$x = 3$

$y = 0$

$x = 3$

$y = 0$

$x = 3$

Step 7

Replace all occurrences of $x$ with $- 3$ in each equation.

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

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

$x = - 3$

Simplify the right side.

Simplify ${(- 3)}^{2} - 9$ .

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

$y = 9 - 9$

$x = - 3$

Subtract $9$ from $9$ .

$y = 0$

$x = - 3$

$y = 0$

$x = - 3$

$y = 0$

$x = - 3$

$y = 0$

$x = - 3$

Step 8

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

$(0, - 9)$

$(3, 0)$

$(- 3, 0)$

Step 9

The result can be shown in multiple forms.

Point Form:

$(0, - 9), (3, 0), (- 3, 0)$

Equation Form:

$x = 0, y = - 9$

$x = 3, y = 0$

$x = - 3, y = 0$

Step 10

Solve by Substitution 81x^2+9y^2=729 , y=x^2-9

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