Solve the System of Equations (x-2)^2+(y+1)^2=89 x+y=4

${(x - 2)}^{2} + {(y + 1)}^{2} = 89$ $x + y = 4$

Subtract $y$ from both sides of the equation.

$x = 4 - y$

${(x - 2)}^{2} + {(y + 1)}^{2} = 89$

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

Replace all occurrences of $x$ in ${(x - 2)}^{2} + {(y + 1)}^{2} = 89$ with $4 - y$ .

${((4 - y) - 2)}^{2} + {(y + 1)}^{2} = 89$

$x = 4 - y$

Simplify the left side.

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

Simplify each term.

Subtract $2$ from $4$ .

${(- y + 2)}^{2} + {(y + 1)}^{2} = 89$

$x = 4 - y$

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

$(- y + 2) (- y + 2) + {(y + 1)}^{2} = 89$

$x = 4 - y$

Expand $(- y + 2) (- y + 2)$ using the FOIL Method.

Apply the distributive property.

$- y (- y + 2) + 2 (- y + 2) + {(y + 1)}^{2} = 89$

$x = 4 - y$

Apply the distributive property.

$- y (- y) - y \cdot 2 + 2 (- y + 2) + {(y + 1)}^{2} = 89$

$x = 4 - y$

Apply the distributive property.

$- y (- y) - y \cdot 2 + 2 (- y) + 2 \cdot 2 + {(y + 1)}^{2} = 89$

$x = 4 - y$

$- y (- y) - y \cdot 2 + 2 (- y) + 2 \cdot 2 + {(y + 1)}^{2} = 89$

$x = 4 - y$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$- 1 \cdot (- 1 y \cdot y) - y \cdot 2 + 2 (- y) + 2 \cdot 2 + {(y + 1)}^{2} = 89$

$x = 4 - y$

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

Move $y$ .

$- 1 \cdot (- 1 (y \cdot y)) - y \cdot 2 + 2 (- y) + 2 \cdot 2 + {(y + 1)}^{2} = 89$

$x = 4 - y$

Multiply $y$ by $y$ .

$- 1 \cdot (- 1 y^{2}) - y \cdot 2 + 2 (- y) + 2 \cdot 2 + {(y + 1)}^{2} = 89$

$x = 4 - y$

$- 1 \cdot (- 1 y^{2}) - y \cdot 2 + 2 (- y) + 2 \cdot 2 + {(y + 1)}^{2} = 89$

$x = 4 - y$

Multiply $- 1$ by $- 1$ .

$1 y^{2} - y \cdot 2 + 2 (- y) + 2 \cdot 2 + {(y + 1)}^{2} = 89$

$x = 4 - y$

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

$y^{2} - y \cdot 2 + 2 (- y) + 2 \cdot 2 + {(y + 1)}^{2} = 89$

$x = 4 - y$

Multiply $2$ by $- 1$ .

$y^{2} - 2 y + 2 (- y) + 2 \cdot 2 + {(y + 1)}^{2} = 89$

$x = 4 - y$

Multiply $- 1$ by $2$ .

$y^{2} - 2 y - 2 y + 2 \cdot 2 + {(y + 1)}^{2} = 89$

$x = 4 - y$

Multiply $2$ by $2$ .

$y^{2} - 2 y - 2 y + 4 + {(y + 1)}^{2} = 89$

$x = 4 - y$

$y^{2} - 2 y - 2 y + 4 + {(y + 1)}^{2} = 89$

$x = 4 - y$

Subtract $2 y$ from $- 2 y$ .

$y^{2} - 4 y + 4 + {(y + 1)}^{2} = 89$

$x = 4 - y$

$y^{2} - 4 y + 4 + {(y + 1)}^{2} = 89$

$x = 4 - y$

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

$y^{2} - 4 y + 4 + (y + 1) (y + 1) = 89$

$x = 4 - y$

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

Apply the distributive property.

$y^{2} - 4 y + 4 + y (y + 1) + 1 (y + 1) = 89$

$x = 4 - y$

Apply the distributive property.

$y^{2} - 4 y + 4 + y \cdot y + y \cdot 1 + 1 (y + 1) = 89$

$x = 4 - y$

Apply the distributive property.

$y^{2} - 4 y + 4 + y \cdot y + y \cdot 1 + 1 y + 1 \cdot 1 = 89$

$x = 4 - y$

$y^{2} - 4 y + 4 + y \cdot y + y \cdot 1 + 1 y + 1 \cdot 1 = 89$

$x = 4 - y$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$y^{2} - 4 y + 4 + y^{2} + y \cdot 1 + 1 y + 1 \cdot 1 = 89$

$x = 4 - y$

Multiply $y$ by $1$ .

$y^{2} - 4 y + 4 + y^{2} + y + 1 y + 1 \cdot 1 = 89$

$x = 4 - y$

Multiply $y$ by $1$ .

$y^{2} - 4 y + 4 + y^{2} + y + y + 1 \cdot 1 = 89$

$x = 4 - y$

Multiply $1$ by $1$ .

$y^{2} - 4 y + 4 + y^{2} + y + y + 1 = 89$

$x = 4 - y$

$y^{2} - 4 y + 4 + y^{2} + y + y + 1 = 89$

$x = 4 - y$

Add $y$ and $y$ .

$y^{2} - 4 y + 4 + y^{2} + 2 y + 1 = 89$

$x = 4 - y$

$y^{2} - 4 y + 4 + y^{2} + 2 y + 1 = 89$

$x = 4 - y$

$y^{2} - 4 y + 4 + y^{2} + 2 y + 1 = 89$

$x = 4 - y$

Simplify by adding terms.

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

$2 y^{2} - 4 y + 4 + 2 y + 1 = 89$

$x = 4 - y$

Add $- 4 y$ and $2 y$ .

$2 y^{2} - 2 y + 4 + 1 = 89$

$x = 4 - y$

Add $4$ and $1$ .

$2 y^{2} - 2 y + 5 = 89$

$x = 4 - y$

$2 y^{2} - 2 y + 5 = 89$

$x = 4 - y$

$2 y^{2} - 2 y + 5 = 89$

$x = 4 - y$

$2 y^{2} - 2 y + 5 = 89$

$x = 4 - y$

$2 y^{2} - 2 y + 5 = 89$

$x = 4 - y$

Solve for $y$ in $2 y^{2} - 2 y + 5 = 89$ .

Subtract $89$ from both sides of the equation.

$2 y^{2} - 2 y + 5 - 89 = 0$

$x = 4 - y$

Subtract $89$ from $5$ .

$2 y^{2} - 2 y - 84 = 0$

$x = 4 - y$

Factor the left side of the equation.

Factor $2$ out of $2 y^{2} - 2 y - 84$ .

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

$2 (y^{2}) - 2 y - 84 = 0$

$x = 4 - y$

Factor $2$ out of $- 2 y$ .

$2 (y^{2}) + 2 (- y) - 84 = 0$

$x = 4 - y$

Factor $2$ out of $- 84$ .

$2 (y^{2}) + 2 (- y) + 2 (- 42) = 0$

$x = 4 - y$

Factor $2$ out of $2 (y^{2}) + 2 (- y)$ .

$2 (y^{2} - y) + 2 (- 42) = 0$

$x = 4 - y$

Factor $2$ out of $2 (y^{2} - y) + 2 (- 42)$ .

$2 (y^{2} - y - 42) = 0$

$x = 4 - y$

$2 (y^{2} - y - 42) = 0$

$x = 4 - y$

Factor.

Factor $y^{2} - y - 42$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 42$ and whose sum is $- 1$ .

$- 7, 6$

$x = 4 - y$

Write the factored form using these integers.

$2 ((y - 7) (y + 6)) = 0$

$x = 4 - y$

$2 ((y - 7) (y + 6)) = 0$

$x = 4 - y$

Remove unnecessary parentheses.

$2 (y - 7) (y + 6) = 0$

$x = 4 - y$

$2 (y - 7) (y + 6) = 0$

$x = 4 - y$

$2 (y - 7) (y + 6) = 0$

$x = 4 - y$

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

$y - 7 = 0$

$y + 6 = 0$

$x = 4 - y$

Set $y - 7$ equal to $0$ and solve for $y$ .

Set $y - 7$ equal to $0$ .

$y - 7 = 0$

$x = 4 - y$

Add $7$ to both sides of the equation.

$y = 7$

$x = 4 - y$

$y = 7$

$x = 4 - y$

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

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

$y + 6 = 0$

$x = 4 - y$

Subtract $6$ from both sides of the equation.

$y = - 6$

$x = 4 - y$

$y = - 6$

$x = 4 - y$

The final solution is all the values that make $2 (y - 7) (y + 6) = 0$ true.

$y = 7, - 6$

$x = 4 - y$

$y = 7, - 6$

$x = 4 - y$

Replace all occurrences of $y$ with $7$ in each equation.

Replace all occurrences of $y$ in $x = 4 - y$ with $7$ .

$x = 4 - (7)$

$y = 7$

Simplify the right side.

Simplify $4 - (7)$ .

Multiply $- 1$ by $7$ .

$x = 4 - 7$

$y = 7$

Subtract $7$ from $4$ .

$x = - 3$

$y = 7$

$x = - 3$

$y = 7$

$x = - 3$

$y = 7$

$x = - 3$

$y = 7$

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

Replace all occurrences of $y$ in $x = 4 - y$ with $- 6$ .

$x = 4 - (- 6)$

$y = - 6$

Simplify the right side.

Simplify $4 - (- 6)$ .

Multiply $- 1$ by $- 6$ .

$x = 4 + 6$

$y = - 6$

Add $4$ and $6$ .

$x = 10$

$y = - 6$

$x = 10$

$y = - 6$

$x = 10$

$y = - 6$

$x = 10$

$y = - 6$

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

$(- 3, 7)$

$(10, - 6)$

The result can be shown in multiple forms.

Point Form:

$(- 3, 7), (10, - 6)$

Equation Form:

$x = - 3, y = 7$

$x = 10, y = - 6$

Solve the System of Equations (x-2)^2+(y+1)^2=89 x+y=4

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