Solve the System of Equations (x+4)^2+y^2=64 -x+y=-4

${(x + 4)}^{2} + y^{2} = 64$ $- x + y = - 4$

Step 1

Add $x$ to both sides of the equation.

$y = - 4 + x$

${(x + 4)}^{2} + y^{2} = 64$

Step 2

Replace all occurrences of $y$ with $- 4 + x$ in each equation.

Replace all occurrences of $y$ in ${(x + 4)}^{2} + y^{2} = 64$ with $- 4 + x$ .

${(x + 4)}^{2} + {(- 4 + x)}^{2} = 64$

$y = - 4 + x$

Simplify the left side.

Simplify ${(x + 4)}^{2} + {(- 4 + x)}^{2}$ .

Simplify each term.

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$(x + 4) (x + 4) + {(- 4 + x)}^{2} = 64$

$y = - 4 + x$

Expand $(x + 4) (x + 4)$ using the FOIL Method.

Apply the distributive property.

$x (x + 4) + 4 (x + 4) + {(- 4 + x)}^{2} = 64$

$y = - 4 + x$

Apply the distributive property.

$x \cdot x + x \cdot 4 + 4 (x + 4) + {(- 4 + x)}^{2} = 64$

$y = - 4 + x$

Apply the distributive property.

$x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4 + {(- 4 + x)}^{2} = 64$

$y = - 4 + x$

$x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4 + {(- 4 + x)}^{2} = 64$

$y = - 4 + x$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot 4 + 4 x + 4 \cdot 4 + {(- 4 + x)}^{2} = 64$

$y = - 4 + x$

Move $4$ to the left of $x$ .

$x^{2} + 4 \cdot x + 4 x + 4 \cdot 4 + {(- 4 + x)}^{2} = 64$

$y = - 4 + x$

Multiply $4$ by $4$ .

$x^{2} + 4 x + 4 x + 16 + {(- 4 + x)}^{2} = 64$

$y = - 4 + x$

$x^{2} + 4 x + 4 x + 16 + {(- 4 + x)}^{2} = 64$

$y = - 4 + x$

Add $4 x$ and $4 x$ .

$x^{2} + 8 x + 16 + {(- 4 + x)}^{2} = 64$

$y = - 4 + x$

$x^{2} + 8 x + 16 + {(- 4 + x)}^{2} = 64$

$y = - 4 + x$

Rewrite ${(- 4 + x)}^{2}$ as $(- 4 + x) (- 4 + x)$ .

$x^{2} + 8 x + 16 + (- 4 + x) (- 4 + x) = 64$

$y = - 4 + x$

Expand $(- 4 + x) (- 4 + x)$ using the FOIL Method.

Apply the distributive property.

$x^{2} + 8 x + 16 - 4 (- 4 + x) + x (- 4 + x) = 64$

$y = - 4 + x$

Apply the distributive property.

$x^{2} + 8 x + 16 - 4 \cdot - 4 - 4 x + x (- 4 + x) = 64$

$y = - 4 + x$

Apply the distributive property.

$x^{2} + 8 x + 16 - 4 \cdot - 4 - 4 x + x \cdot - 4 + x \cdot x = 64$

$y = - 4 + x$

$x^{2} + 8 x + 16 - 4 \cdot - 4 - 4 x + x \cdot - 4 + x \cdot x = 64$

$y = - 4 + x$

Simplify and combine like terms.

Simplify each term.

Multiply $- 4$ by $- 4$ .

$x^{2} + 8 x + 16 + 16 - 4 x + x \cdot - 4 + x \cdot x = 64$

$y = - 4 + x$

Move $- 4$ to the left of $x$ .

$x^{2} + 8 x + 16 + 16 - 4 x - 4 \cdot x + x \cdot x = 64$

$y = - 4 + x$

Multiply $x$ by $x$ .

$x^{2} + 8 x + 16 + 16 - 4 x - 4 x + x^{2} = 64$

$y = - 4 + x$

$x^{2} + 8 x + 16 + 16 - 4 x - 4 x + x^{2} = 64$

$y = - 4 + x$

Subtract $4 x$ from $- 4 x$ .

$x^{2} + 8 x + 16 + 16 - 8 x + x^{2} = 64$

$y = - 4 + x$

$x^{2} + 8 x + 16 + 16 - 8 x + x^{2} = 64$

$y = - 4 + x$

$x^{2} + 8 x + 16 + 16 - 8 x + x^{2} = 64$

$y = - 4 + x$

Simplify by adding terms.

Combine the opposite terms in $x^{2} + 8 x + 16 + 16 - 8 x + x^{2}$ .

Subtract $8 x$ from $8 x$ .

$x^{2} + 0 + 16 + 16 + x^{2} = 64$

$y = - 4 + x$

Add $x^{2}$ and $0$ .

$x^{2} + 16 + 16 + x^{2} = 64$

$y = - 4 + x$

$x^{2} + 16 + 16 + x^{2} = 64$

$y = - 4 + x$

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} + 16 + 16 = 64$

$y = - 4 + x$

Add $16$ and $16$ .

$2 x^{2} + 32 = 64$

$y = - 4 + x$

$2 x^{2} + 32 = 64$

$y = - 4 + x$

$2 x^{2} + 32 = 64$

$y = - 4 + x$

$2 x^{2} + 32 = 64$

$y = - 4 + x$

$2 x^{2} + 32 = 64$

$y = - 4 + x$

Step 3

Solve for $x$ in $2 x^{2} + 32 = 64$ .

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

Subtract $32$ from both sides of the equation.

$2 x^{2} = 64 - 32$

$y = - 4 + x$

Subtract $32$ from $64$ .

$2 x^{2} = 32$

$y = - 4 + x$

$2 x^{2} = 32$

$y = - 4 + x$

Divide each term in $2 x^{2} = 32$ by $2$ and simplify.

Divide each term in $2 x^{2} = 32$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{32}{2}$

$y = - 4 + x$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{32}{2}$

$y = - 4 + x$

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

$x^{2} = \frac{32}{2}$

$y = - 4 + x$

$x^{2} = \frac{32}{2}$

$y = - 4 + x$

$x^{2} = \frac{32}{2}$

$y = - 4 + x$

Simplify the right side.

Divide $32$ by $2$ .

$x^{2} = 16$

$y = - 4 + x$

$x^{2} = 16$

$y = - 4 + x$

$x^{2} = 16$

$y = - 4 + x$

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

$x = \pm \sqrt{16}$

$y = - 4 + x$

Simplify $\pm \sqrt{16}$ .

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2}}$

$y = - 4 + x$

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

$x = \pm 4$

$y = - 4 + x$

$x = \pm 4$

$y = - 4 + x$

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.

$x = 4$

$y = - 4 + x$

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

$x = - 4$

$y = - 4 + x$

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

$x = 4, - 4$

$y = - 4 + x$

$x = 4, - 4$

$y = - 4 + x$

$x = 4, - 4$

$y = - 4 + x$

Step 4

Replace all occurrences of $x$ with $4$ in each equation.

Replace all occurrences of $x$ in $y = - 4 + x$ with $4$ .

$y = - 4 + 4$

$x = 4$

Simplify $y = - 4 + 4$ .

Simplify the left side.

Remove parentheses.

$y = - 4 + 4$

$x = 4$

$y = - 4 + 4$

$x = 4$

Simplify the right side.

Add $- 4$ and $4$ .

$y = 0$

$x = 4$

$y = 0$

$x = 4$

$y = 0$

$x = 4$

$y = 0$

$x = 4$

Step 5

Replace all occurrences of $x$ with $- 4$ in each equation.

Replace all occurrences of $x$ in $y = - 4 + x$ with $- 4$ .

$y = - 4 - 4$

$x = - 4$

Simplify $y = - 4 - 4$ .

Simplify the left side.

Remove parentheses.

$y = - 4 - 4$

$x = - 4$

$y = - 4 - 4$

$x = - 4$

Simplify the right side.

Subtract $4$ from $- 4$ .

$y = - 8$

$x = - 4$

$y = - 8$

$x = - 4$

$y = - 8$

$x = - 4$

$y = - 8$

$x = - 4$

Step 6

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

$(4, 0)$

$(- 4, - 8)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(4, 0), (- 4, - 8)$

Equation Form:

$x = 4, y = 0$

$x = - 4, y = - 8$

Step 8

Solve the System of Equations (x+4)^2+y^2=64 -x+y=-4

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