Solve the System of Equations (x-5)^2+(y-3)^2=29 -x+y=-9

${(x - 5)}^{2} + {(y - 3)}^{2} = 29$ $- x + y = - 9$

Step 1

Add $x$ to both sides of the equation.

$y = - 9 + x$

${(x - 5)}^{2} + {(y - 3)}^{2} = 29$

Step 2

Replace all occurrences of $y$ with $- 9 + x$ in each equation.

Replace all occurrences of $y$ in ${(x - 5)}^{2} + {(y - 3)}^{2} = 29$ with $- 9 + x$ .

${(x - 5)}^{2} + {((- 9 + x) - 3)}^{2} = 29$

$y = - 9 + x$

Simplify the left side.

Simplify ${(x - 5)}^{2} + {((- 9 + x) - 3)}^{2}$ .

Simplify each term.

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$(x - 5) (x - 5) + {((- 9 + x) - 3)}^{2} = 29$

$y = - 9 + x$

Expand $(x - 5) (x - 5)$ using the FOIL Method.

Apply the distributive property.

$x (x - 5) - 5 (x - 5) + {((- 9 + x) - 3)}^{2} = 29$

$y = - 9 + x$

Apply the distributive property.

$x \cdot x + x \cdot - 5 - 5 (x - 5) + {((- 9 + x) - 3)}^{2} = 29$

$y = - 9 + x$

Apply the distributive property.

$x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 + {((- 9 + x) - 3)}^{2} = 29$

$y = - 9 + x$

$x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 + {((- 9 + x) - 3)}^{2} = 29$

$y = - 9 + x$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5 + {((- 9 + x) - 3)}^{2} = 29$

$y = - 9 + x$

Move $- 5$ to the left of $x$ .

$x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5 + {((- 9 + x) - 3)}^{2} = 29$

$y = - 9 + x$

Multiply $- 5$ by $- 5$ .

$x^{2} - 5 x - 5 x + 25 + {((- 9 + x) - 3)}^{2} = 29$

$y = - 9 + x$

$x^{2} - 5 x - 5 x + 25 + {((- 9 + x) - 3)}^{2} = 29$

$y = - 9 + x$

Subtract $5 x$ from $- 5 x$ .

$x^{2} - 10 x + 25 + {((- 9 + x) - 3)}^{2} = 29$

$y = - 9 + x$

$x^{2} - 10 x + 25 + {((- 9 + x) - 3)}^{2} = 29$

$y = - 9 + x$

Subtract $3$ from $- 9$ .

$x^{2} - 10 x + 25 + {(x - 12)}^{2} = 29$

$y = - 9 + x$

Rewrite ${(x - 12)}^{2}$ as $(x - 12) (x - 12)$ .

$x^{2} - 10 x + 25 + (x - 12) (x - 12) = 29$

$y = - 9 + x$

Expand $(x - 12) (x - 12)$ using the FOIL Method.

Apply the distributive property.

$x^{2} - 10 x + 25 + x (x - 12) - 12 (x - 12) = 29$

$y = - 9 + x$

Apply the distributive property.

$x^{2} - 10 x + 25 + x \cdot x + x \cdot - 12 - 12 (x - 12) = 29$

$y = - 9 + x$

Apply the distributive property.

$x^{2} - 10 x + 25 + x \cdot x + x \cdot - 12 - 12 x - 12 \cdot - 12 = 29$

$y = - 9 + x$

$x^{2} - 10 x + 25 + x \cdot x + x \cdot - 12 - 12 x - 12 \cdot - 12 = 29$

$y = - 9 + x$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} - 10 x + 25 + x^{2} + x \cdot - 12 - 12 x - 12 \cdot - 12 = 29$

$y = - 9 + x$

Move $- 12$ to the left of $x$ .

$x^{2} - 10 x + 25 + x^{2} - 12 \cdot x - 12 x - 12 \cdot - 12 = 29$

$y = - 9 + x$

Multiply $- 12$ by $- 12$ .

$x^{2} - 10 x + 25 + x^{2} - 12 x - 12 x + 144 = 29$

$y = - 9 + x$

$x^{2} - 10 x + 25 + x^{2} - 12 x - 12 x + 144 = 29$

$y = - 9 + x$

Subtract $12 x$ from $- 12 x$ .

$x^{2} - 10 x + 25 + x^{2} - 24 x + 144 = 29$

$y = - 9 + x$

$x^{2} - 10 x + 25 + x^{2} - 24 x + 144 = 29$

$y = - 9 + x$

$x^{2} - 10 x + 25 + x^{2} - 24 x + 144 = 29$

$y = - 9 + x$

Simplify by adding terms.

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

$2 x^{2} - 10 x + 25 - 24 x + 144 = 29$

$y = - 9 + x$

Subtract $24 x$ from $- 10 x$ .

$2 x^{2} - 34 x + 25 + 144 = 29$

$y = - 9 + x$

Add $25$ and $144$ .

$2 x^{2} - 34 x + 169 = 29$

$y = - 9 + x$

$2 x^{2} - 34 x + 169 = 29$

$y = - 9 + x$

$2 x^{2} - 34 x + 169 = 29$

$y = - 9 + x$

$2 x^{2} - 34 x + 169 = 29$

$y = - 9 + x$

$2 x^{2} - 34 x + 169 = 29$

$y = - 9 + x$

Step 3

Solve for $x$ in $2 x^{2} - 34 x + 169 = 29$ .

Subtract $29$ from both sides of the equation.

$2 x^{2} - 34 x + 169 - 29 = 0$

$y = - 9 + x$

Subtract $29$ from $169$ .

$2 x^{2} - 34 x + 140 = 0$

$y = - 9 + x$

Factor the left side of the equation.

Factor $2$ out of $2 x^{2} - 34 x + 140$ .

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

$2 (x^{2}) - 34 x + 140 = 0$

$y = - 9 + x$

Factor $2$ out of $- 34 x$ .

$2 (x^{2}) + 2 (- 17 x) + 140 = 0$

$y = - 9 + x$

Factor $2$ out of $140$ .

$2 x^{2} + 2 (- 17 x) + 2 \cdot 70 = 0$

$y = - 9 + x$

Factor $2$ out of $2 x^{2} + 2 (- 17 x)$ .

$2 (x^{2} - 17 x) + 2 \cdot 70 = 0$

$y = - 9 + x$

Factor $2$ out of $2 (x^{2} - 17 x) + 2 \cdot 70$ .

$2 (x^{2} - 17 x + 70) = 0$

$y = - 9 + x$

$2 (x^{2} - 17 x + 70) = 0$

$y = - 9 + x$

Factor.

Factor $x^{2} - 17 x + 70$ 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 $70$ and whose sum is $- 17$ .

$- 10, - 7$

$y = - 9 + x$

Write the factored form using these integers.

$2 ((x - 10) (x - 7)) = 0$

$y = - 9 + x$

$2 ((x - 10) (x - 7)) = 0$

$y = - 9 + x$

Remove unnecessary parentheses.

$2 (x - 10) (x - 7) = 0$

$y = - 9 + x$

$2 (x - 10) (x - 7) = 0$

$y = - 9 + x$

$2 (x - 10) (x - 7) = 0$

$y = - 9 + x$

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

$x - 10 = 0$

$x - 7 = 0$

$y = - 9 + x$

Set $x - 10$ equal to $0$ and solve for $x$ .

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

$y = - 9 + x$

Add $10$ to both sides of the equation.

$x = 10$

$y = - 9 + x$

$x = 10$

$y = - 9 + x$

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

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

$x - 7 = 0$

$y = - 9 + x$

Add $7$ to both sides of the equation.

$x = 7$

$y = - 9 + x$

$x = 7$

$y = - 9 + x$

The final solution is all the values that make $2 (x - 10) (x - 7) = 0$ true.

$x = 10, 7$

$y = - 9 + x$

$x = 10, 7$

$y = - 9 + x$

Step 4

Replace all occurrences of $x$ with $10$ in each equation.

Replace all occurrences of $x$ in $y = - 9 + x$ with $10$ .

$y = - 9 + 10$

$x = 10$

Simplify $y = - 9 + 10$ .

Simplify the left side.

Remove parentheses.

$y = - 9 + 10$

$x = 10$

$y = - 9 + 10$

$x = 10$

Simplify the right side.

Add $- 9$ and $10$ .

$y = 1$

$x = 10$

$y = 1$

$x = 10$

$y = 1$

$x = 10$

$y = 1$

$x = 10$

Step 5

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

Replace all occurrences of $x$ in $y = - 9 + x$ with $7$ .

$y = - 9 + 7$

$x = 7$

Simplify $y = - 9 + 7$ .

Simplify the left side.

Remove parentheses.

$y = - 9 + 7$

$x = 7$

$y = - 9 + 7$

$x = 7$

Simplify the right side.

Add $- 9$ and $7$ .

$y = - 2$

$x = 7$

$y = - 2$

$x = 7$

$y = - 2$

$x = 7$

$y = - 2$

$x = 7$

Step 6

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

$(10, 1)$

$(7, - 2)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(10, 1), (7, - 2)$

Equation Form:

$x = 10, y = 1$

$x = 7, y = - 2$

Step 8

Solve the System of Equations (x-5)^2+(y-3)^2=29 -x+y=-9

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