Linear Algebra Examples

$x y + y = x^{2}$

Step 1

Factor $y$ out of $x y + y$ .

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Step 1.1

Factor $y$ out of $x y$ .

$y x + y = x^{2}$

Step 1.2

Raise $y$ to the power of $1$ .

$y x + y = x^{2}$

Step 1.3

Factor $y$ out of $y^{1}$ .

$y x + y \cdot 1 = x^{2}$

Step 1.4

Factor $y$ out of $y x + y \cdot 1$ .

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

Step 2

Divide each term in $y (x + 1) = x^{2}$ by $x + 1$ and simplify.

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Step 2.1

Divide each term in $y (x + 1) = x^{2}$ by $x + 1$ .

$\frac{y (x + 1)}{x + 1} = \frac{x^{2}}{x + 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $x + 1$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{y (x + 1)}{x + 1} = \frac{x^{2}}{x + 1}$

Step 2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{x + 1}$

Step 3

Set the denominator in $\frac{x^{2}}{x + 1}$ equal to $0$ to find where the expression is undefined.

$x + 1 = 0$

Step 4

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 1) \cup (- 1, \infty)$

Set-Builder Notation:

${x | x \neq - 1}$

Step 6