Solve by Substitution y^2=x^2-64 , -3y=x+8

$y^{2} = x^{2} - 64$ , $- 3 y = x + 8$

Solve for $x$ in the first equation.

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Rewrite the equation as $x + 8 = - 3 y$ .

$x + 8 = - 3 y$

$y^{2} = x^{2} - 64$

Subtract $8$ from both sides of the equation.

$x = - 3 y - 8$

$y^{2} = x^{2} - 64$

$x = - 3 y - 8$

$y^{2} = x^{2} - 64$

Replace all occurrences of $x$ with $- 3 y - 8$ in each equation.

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Replace all occurrences of $x$ in $y^{2} = x^{2} - 64$ with $- 3 y - 8$ .

$y^{2} = {(- 3 y - 8)}^{2} - 64$

$x = - 3 y - 8$

Simplify ${(- 3 y - 8)}^{2} - 64$ .

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Simplify each term.

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Rewrite ${(- 3 y - 8)}^{2}$ as $(- 3 y - 8) (- 3 y - 8)$ .

$y^{2} = (- 3 y - 8) (- 3 y - 8) - 64$

$x = - 3 y - 8$

Expand $(- 3 y - 8) (- 3 y - 8)$ using the FOIL Method.

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Apply the distributive property.

$y^{2} = - 3 y (- 3 y - 8) - 8 (- 3 y - 8) - 64$

$x = - 3 y - 8$

Apply the distributive property.

$y^{2} = - 3 y (- 3 y) - 3 y \cdot - 8 - 8 (- 3 y - 8) - 64$

$x = - 3 y - 8$

Apply the distributive property.

$y^{2} = - 3 y (- 3 y) - 3 y \cdot - 8 - 8 (- 3 y) - 8 \cdot - 8 - 64$

$x = - 3 y - 8$

$y^{2} = - 3 y (- 3 y) - 3 y \cdot - 8 - 8 (- 3 y) - 8 \cdot - 8 - 64$

$x = - 3 y - 8$

Simplify and combine like terms.

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Simplify each term.

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Rewrite using the commutative property of multiplication.

$y^{2} = - 3 \cdot (- 3 y \cdot y) - 3 y \cdot - 8 - 8 (- 3 y) - 8 \cdot - 8 - 64$

$x = - 3 y - 8$

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

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Move $y$ .

$y^{2} = - 3 \cdot (- 3 (y \cdot y)) - 3 y \cdot - 8 - 8 (- 3 y) - 8 \cdot - 8 - 64$

$x = - 3 y - 8$

Multiply $y$ by $y$ .

$y^{2} = - 3 \cdot (- 3 y^{2}) - 3 y \cdot - 8 - 8 (- 3 y) - 8 \cdot - 8 - 64$

$x = - 3 y - 8$

$y^{2} = - 3 \cdot (- 3 y^{2}) - 3 y \cdot - 8 - 8 (- 3 y) - 8 \cdot - 8 - 64$

$x = - 3 y - 8$

Multiply $- 3$ by $- 3$ .

$y^{2} = 9 y^{2} - 3 y \cdot - 8 - 8 (- 3 y) - 8 \cdot - 8 - 64$

$x = - 3 y - 8$

Multiply $- 8$ by $- 3$ .

$y^{2} = 9 y^{2} + 24 y - 8 (- 3 y) - 8 \cdot - 8 - 64$

$x = - 3 y - 8$

Multiply $- 3$ by $- 8$ .

$y^{2} = 9 y^{2} + 24 y + 24 y - 8 \cdot - 8 - 64$

$x = - 3 y - 8$

Multiply $- 8$ by $- 8$ .

$y^{2} = 9 y^{2} + 24 y + 24 y + 64 - 64$

$x = - 3 y - 8$

$y^{2} = 9 y^{2} + 24 y + 24 y + 64 - 64$

$x = - 3 y - 8$

Add $24 y$ and $24 y$ .

$y^{2} = 9 y^{2} + 48 y + 64 - 64$

$x = - 3 y - 8$

$y^{2} = 9 y^{2} + 48 y + 64 - 64$

$x = - 3 y - 8$

$y^{2} = 9 y^{2} + 48 y + 64 - 64$

$x = - 3 y - 8$

Combine the opposite terms in $9 y^{2} + 48 y + 64 - 64$ .

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Subtract $64$ from $64$ .

$y^{2} = 9 y^{2} + 48 y + 0$

$x = - 3 y - 8$

Add $9 y^{2} + 48 y$ and $0$ .

$y^{2} = 9 y^{2} + 48 y$

$x = - 3 y - 8$

$y^{2} = 9 y^{2} + 48 y$

$x = - 3 y - 8$

$y^{2} = 9 y^{2} + 48 y$

$x = - 3 y - 8$

$y^{2} = 9 y^{2} + 48 y$

$x = - 3 y - 8$

Solve for $y$ in the first equation.

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Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$9 y^{2} + 48 y = y^{2}$

$x = - 3 y - 8$

Move all terms containing $y$ to the left side of the equation.

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

$9 y^{2} + 48 y - y^{2} = 0$

$x = - 3 y - 8$

Subtract $y^{2}$ from $9 y^{2}$ .

$8 y^{2} + 48 y = 0$

$x = - 3 y - 8$

$8 y^{2} + 48 y = 0$

$x = - 3 y - 8$

Factor $8 y$ out of $8 y^{2} + 48 y$ .

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Factor $8 y$ out of $8 y^{2}$ .

$8 y (y) + 48 y = 0$

$x = - 3 y - 8$

Factor $8 y$ out of $48 y$ .

$8 y (y) + 8 y (6) = 0$

$x = - 3 y - 8$

Factor $8 y$ out of $8 y (y) + 8 y (6)$ .

$8 y (y + 6) = 0$

$x = - 3 y - 8$

$8 y (y + 6) = 0$

$x = - 3 y - 8$

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

$y = 0$

$y + 6 = 0$

$x = - 3 y - 8$

Set $y$ equal to $0$ .

$y = 0$

$x = - 3 y - 8$

Set $y + 6$ equal to $0$ and solve for $y$ .

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Set $y + 6$ equal to $0$ .

$y + 6 = 0$

$x = - 3 y - 8$

Subtract $6$ from both sides of the equation.

$y = - 6$

$x = - 3 y - 8$

$y = - 6$

$x = - 3 y - 8$

The final solution is all the values that make $8 y (y + 6) = 0$ true.

$y = 0, - 6$

$x = - 3 y - 8$

$y = 0, - 6$

$x = - 3 y - 8$

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

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Replace all occurrences of $y$ in $x = - 3 y - 8$ with $0$ .

$x = - 3 \cdot 0 - 8$

$y = 0$

Simplify $- 3 \cdot 0 - 8$ .

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Multiply $- 3$ by $0$ .

$x = 0 - 8$

$y = 0$

Subtract $8$ from $0$ .

$x = - 8$

$y = 0$

$x = - 8$

$y = 0$

$x = - 8$

$y = 0$

Replace all occurrences of $y$ with $- 6$ in each equation.

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Replace all occurrences of $y$ in $x = - 3 y - 8$ with $- 6$ .

$x = - 3 \cdot - 6 - 8$

$y = - 6$

Simplify $- 3 \cdot - 6 - 8$ .

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Multiply $- 3$ by $- 6$ .

$x = 18 - 8$

$y = - 6$

Subtract $8$ from $18$ .

$x = 10$

$y = - 6$

$x = 10$

$y = - 6$

$x = 10$

$y = - 6$

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

$(- 8, 0)$

$(10, - 6)$

The result can be shown in multiple forms.

Point Form:

$(- 8, 0), (10, - 6)$

Equation Form:

$x = - 8, y = 0$

$x = 10, y = - 6$

Solve by Substitution y^2=x^2-64 , -3y=x+8

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