Linear Algebra Examples

$x^{2} + y^{2} - 20 y + 100 = y^{2} - 12 y + 36$

Step 1

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

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Step 1.1

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

$x^{2} + y^{2} - 20 y + 100 - y^{2} = - 12 y + 36$

Step 1.2

Add $12 y$ to both sides of the equation.

$x^{2} + y^{2} - 20 y + 100 - y^{2} + 12 y = 36$

Step 1.3

Combine the opposite terms in $x^{2} + y^{2} - 20 y + 100 - y^{2} + 12 y$ .

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Step 1.3.1

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

$x^{2} - 20 y + 100 + 0 + 12 y = 36$

Step 1.3.2

Add $x^{2} - 20 y + 100$ and $0$ .

$x^{2} - 20 y + 100 + 12 y = 36$

Step 1.4

Add $- 20 y$ and $12 y$ .

$x^{2} - 8 y + 100 = 36$

Step 2

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

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Step 2.1

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

$- 8 y + 100 = 36 - x^{2}$

Step 2.2

Subtract $100$ from both sides of the equation.

$- 8 y = 36 - x^{2} - 100$

Step 2.3

Subtract $100$ from $36$ .

$- 8 y = - x^{2} - 64$

Step 3

Divide each term in $- 8 y = - x^{2} - 64$ by $- 8$ and simplify.

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Step 3.1

Divide each term in $- 8 y = - x^{2} - 64$ by $- 8$ .

$\frac{- 8 y}{- 8} = \frac{- x^{2}}{- 8} + \frac{- 64}{- 8}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 8$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 8 y}{- 8} = \frac{- x^{2}}{- 8} + \frac{- 64}{- 8}$

Step 3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{- 8} + \frac{- 64}{- 8}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Dividing two negative values results in a positive value.

$y = \frac{x^{2}}{8} + \frac{- 64}{- 8}$

Step 3.3.1.2

Divide $- 64$ by $- 8$ .

$y = \frac{x^{2}}{8} + 8$

Step 4

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 5