Linear Algebra Examples

$x + y = - 10$ , ${(x + 3)}^{2} + {(y + 9)}^{2} = 10$

Step 1

Subtract $y$ from both sides of the equation.

$x = - 10 - y$

${(x + 3)}^{2} + {(y + 9)}^{2} = 10$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in ${(x + 3)}^{2} + {(y + 9)}^{2} = 10$ with $- 10 - y$ .

${((- 10 - y) + 3)}^{2} + {(y + 9)}^{2} = 10$

$x = - 10 - y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${((- 10 - y) + 3)}^{2} + {(y + 9)}^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Add $- 10$ and $3$ .

${(- y - 7)}^{2} + {(y + 9)}^{2} = 10$

$x = - 10 - y$

Step 2.2.1.1.2

Rewrite ${(- y - 7)}^{2}$ as $(- y - 7) (- y - 7)$ .

$(- y - 7) (- y - 7) + {(y + 9)}^{2} = 10$

$x = - 10 - y$

Step 2.2.1.1.3

Expand $(- y - 7) (- y - 7)$ using the FOIL Method.

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Step 2.2.1.1.3.1

Apply the distributive property.

$- y (- y - 7) - 7 (- y - 7) + {(y + 9)}^{2} = 10$

$x = - 10 - y$

Step 2.2.1.1.3.2

Apply the distributive property.

$- y (- y) - y \cdot - 7 - 7 (- y - 7) + {(y + 9)}^{2} = 10$

$x = - 10 - y$

Step 2.2.1.1.3.3

Apply the distributive property.

$- y (- y) - y \cdot - 7 - 7 (- y) - 7 \cdot - 7 + {(y + 9)}^{2} = 10$

$x = - 10 - y$

$- y (- y) - y \cdot - 7 - 7 (- y) - 7 \cdot - 7 + {(y + 9)}^{2} = 10$

$x = - 10 - y$

Step 2.2.1.1.4

Simplify and combine like terms.

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Step 2.2.1.1.4.1

Simplify each term.

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Step 2.2.1.1.4.1.1

Rewrite using the commutative property of multiplication.

$- 1 \cdot (- 1 y \cdot y) - y \cdot - 7 - 7 (- y) - 7 \cdot - 7 + {(y + 9)}^{2} = 10$

$x = - 10 - y$

Step 2.2.1.1.4.1.2

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

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Step 2.2.1.1.4.1.2.1

Move $y$ .

$- 1 \cdot (- 1 (y \cdot y)) - y \cdot - 7 - 7 (- y) - 7 \cdot - 7 + {(y + 9)}^{2} = 10$

$x = - 10 - y$

Step 2.2.1.1.4.1.2.2

Multiply $y$ by $y$ .

$- 1 \cdot (- 1 y^{2}) - y \cdot - 7 - 7 (- y) - 7 \cdot - 7 + {(y + 9)}^{2} = 10$

$x = - 10 - y$

$- 1 \cdot (- 1 y^{2}) - y \cdot - 7 - 7 (- y) - 7 \cdot - 7 + {(y + 9)}^{2} = 10$

$x = - 10 - y$

Step 2.2.1.1.4.1.3

Multiply $- 1$ by $- 1$ .

$1 y^{2} - y \cdot - 7 - 7 (- y) - 7 \cdot - 7 + {(y + 9)}^{2} = 10$

$x = - 10 - y$

Step 2.2.1.1.4.1.4

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

$y^{2} - y \cdot - 7 - 7 (- y) - 7 \cdot - 7 + {(y + 9)}^{2} = 10$

$x = - 10 - y$

Step 2.2.1.1.4.1.5

Multiply $- 7$ by $- 1$ .

$y^{2} + 7 y - 7 (- y) - 7 \cdot - 7 + {(y + 9)}^{2} = 10$

$x = - 10 - y$

Step 2.2.1.1.4.1.6

Multiply $- 1$ by $- 7$ .

$y^{2} + 7 y + 7 y - 7 \cdot - 7 + {(y + 9)}^{2} = 10$

$x = - 10 - y$

Step 2.2.1.1.4.1.7

Multiply $- 7$ by $- 7$ .

$y^{2} + 7 y + 7 y + 49 + {(y + 9)}^{2} = 10$

$x = - 10 - y$

$y^{2} + 7 y + 7 y + 49 + {(y + 9)}^{2} = 10$

$x = - 10 - y$

Step 2.2.1.1.4.2

Add $7 y$ and $7 y$ .

$y^{2} + 14 y + 49 + {(y + 9)}^{2} = 10$

$x = - 10 - y$

$y^{2} + 14 y + 49 + {(y + 9)}^{2} = 10$

$x = - 10 - y$

Step 2.2.1.1.5

Rewrite ${(y + 9)}^{2}$ as $(y + 9) (y + 9)$ .

$y^{2} + 14 y + 49 + (y + 9) (y + 9) = 10$

$x = - 10 - y$

Step 2.2.1.1.6

Expand $(y + 9) (y + 9)$ using the FOIL Method.

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Step 2.2.1.1.6.1

Apply the distributive property.

$y^{2} + 14 y + 49 + y (y + 9) + 9 (y + 9) = 10$

$x = - 10 - y$

Step 2.2.1.1.6.2

Apply the distributive property.

$y^{2} + 14 y + 49 + y \cdot y + y \cdot 9 + 9 (y + 9) = 10$

$x = - 10 - y$

Step 2.2.1.1.6.3

Apply the distributive property.

$y^{2} + 14 y + 49 + y \cdot y + y \cdot 9 + 9 y + 9 \cdot 9 = 10$

$x = - 10 - y$

$y^{2} + 14 y + 49 + y \cdot y + y \cdot 9 + 9 y + 9 \cdot 9 = 10$

$x = - 10 - y$

Step 2.2.1.1.7

Simplify and combine like terms.

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Step 2.2.1.1.7.1

Simplify each term.

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Step 2.2.1.1.7.1.1

Multiply $y$ by $y$ .

$y^{2} + 14 y + 49 + y^{2} + y \cdot 9 + 9 y + 9 \cdot 9 = 10$

$x = - 10 - y$

Step 2.2.1.1.7.1.2

Move $9$ to the left of $y$ .

$y^{2} + 14 y + 49 + y^{2} + 9 \cdot y + 9 y + 9 \cdot 9 = 10$

$x = - 10 - y$

Step 2.2.1.1.7.1.3

Multiply $9$ by $9$ .

$y^{2} + 14 y + 49 + y^{2} + 9 y + 9 y + 81 = 10$

$x = - 10 - y$

$y^{2} + 14 y + 49 + y^{2} + 9 y + 9 y + 81 = 10$

$x = - 10 - y$

Step 2.2.1.1.7.2

Add $9 y$ and $9 y$ .

$y^{2} + 14 y + 49 + y^{2} + 18 y + 81 = 10$

$x = - 10 - y$

$y^{2} + 14 y + 49 + y^{2} + 18 y + 81 = 10$

$x = - 10 - y$

$y^{2} + 14 y + 49 + y^{2} + 18 y + 81 = 10$

$x = - 10 - y$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

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

$2 y^{2} + 14 y + 49 + 18 y + 81 = 10$

$x = - 10 - y$

Step 2.2.1.2.2

Add $14 y$ and $18 y$ .

$2 y^{2} + 32 y + 49 + 81 = 10$

$x = - 10 - y$

Step 2.2.1.2.3

Add $49$ and $81$ .

$2 y^{2} + 32 y + 130 = 10$

$x = - 10 - y$

$2 y^{2} + 32 y + 130 = 10$

$x = - 10 - y$

$2 y^{2} + 32 y + 130 = 10$

$x = - 10 - y$

$2 y^{2} + 32 y + 130 = 10$

$x = - 10 - y$

$2 y^{2} + 32 y + 130 = 10$

$x = - 10 - y$

Step 3

Solve for $y$ in $2 y^{2} + 32 y + 130 = 10$ .

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Step 3.1

Subtract $10$ from both sides of the equation.

$2 y^{2} + 32 y + 130 - 10 = 0$

$x = - 10 - y$

Step 3.2

Subtract $10$ from $130$ .

$2 y^{2} + 32 y + 120 = 0$

$x = - 10 - y$

Step 3.3

Factor the left side of the equation.

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Step 3.3.1

Factor $2$ out of $2 y^{2} + 32 y + 120$ .

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Step 3.3.1.1

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

$2 (y^{2}) + 32 y + 120 = 0$

$x = - 10 - y$

Step 3.3.1.2

Factor $2$ out of $32 y$ .

$2 (y^{2}) + 2 (16 y) + 120 = 0$

$x = - 10 - y$

Step 3.3.1.3

Factor $2$ out of $120$ .

$2 y^{2} + 2 (16 y) + 2 \cdot 60 = 0$

$x = - 10 - y$

Step 3.3.1.4

Factor $2$ out of $2 y^{2} + 2 (16 y)$ .

$2 (y^{2} + 16 y) + 2 \cdot 60 = 0$

$x = - 10 - y$

Step 3.3.1.5

Factor $2$ out of $2 (y^{2} + 16 y) + 2 \cdot 60$ .

$2 (y^{2} + 16 y + 60) = 0$

$x = - 10 - y$

$2 (y^{2} + 16 y + 60) = 0$

$x = - 10 - y$

Step 3.3.2

Factor.

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Step 3.3.2.1

Factor $y^{2} + 16 y + 60$ using the AC method.

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Step 3.3.2.1.1

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 $60$ and whose sum is $16$ .

$6, 10$

$x = - 10 - y$

Step 3.3.2.1.2

Write the factored form using these integers.

$2 ((y + 6) (y + 10)) = 0$

$x = - 10 - y$

$2 ((y + 6) (y + 10)) = 0$

$x = - 10 - y$

Step 3.3.2.2

Remove unnecessary parentheses.

$2 (y + 6) (y + 10) = 0$

$x = - 10 - y$

$2 (y + 6) (y + 10) = 0$

$x = - 10 - y$

$2 (y + 6) (y + 10) = 0$

$x = - 10 - y$

Step 3.4

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

$y + 6 = 0$

$y + 10 = 0$

$x = - 10 - y$

Step 3.5

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

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Step 3.5.1

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

$y + 6 = 0$

$x = - 10 - y$

Step 3.5.2

Subtract $6$ from both sides of the equation.

$y = - 6$

$x = - 10 - y$

$y = - 6$

$x = - 10 - y$

Step 3.6

Set $y + 10$ equal to $0$ and solve for $y$ .

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Step 3.6.1

Set $y + 10$ equal to $0$ .

$y + 10 = 0$

$x = - 10 - y$

Step 3.6.2

Subtract $10$ from both sides of the equation.

$y = - 10$

$x = - 10 - y$

$y = - 10$

$x = - 10 - y$

Step 3.7

The final solution is all the values that make $2 (y + 6) (y + 10) = 0$ true.

$y = - 6, - 10$

$x = - 10 - y$

$y = - 6, - 10$

$x = - 10 - y$

Step 4

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

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Step 4.1

Replace all occurrences of $y$ in $x = - 10 - y$ with $- 6$ .

$x = - 10 - (- 6)$

$y = - 6$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 10 - (- 6)$ .

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Step 4.2.1.1

Multiply $- 1$ by $- 6$ .

$x = - 10 + 6$

$y = - 6$

Step 4.2.1.2

Add $- 10$ and $6$ .

$x = - 4$

$y = - 6$

$x = - 4$

$y = - 6$

$x = - 4$

$y = - 6$

$x = - 4$

$y = - 6$

Step 5

Replace all occurrences of $y$ with $- 10$ in each equation.

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Step 5.1

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

$x = - 10 - (- 10)$

$y = - 10$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 10 - (- 10)$ .

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Step 5.2.1.1

Multiply $- 1$ by $- 10$ .

$x = - 10 + 10$

$y = - 10$

Step 5.2.1.2

Add $- 10$ and $10$ .

$x = 0$

$y = - 10$

$x = 0$

$y = - 10$

$x = 0$

$y = - 10$

$x = 0$

$y = - 10$

Step 6

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

$(- 4, - 6)$

$(0, - 10)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(- 4, - 6), (0, - 10)$

Equation Form:

$x = - 4, y = - 6$

$x = 0, y = - 10$

Step 8