Linear Algebra Examples

$6 x - 4 y = 13 + x y$

Step 1

Subtract $x y$ from both sides of the equation.

$6 x - 4 y - x y = 13$

Step 2

Subtract $6 x$ from both sides of the equation.

$- 4 y - x y = 13 - 6 x$

Step 3

Factor $y$ out of $- 4 y - x y$ .

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Step 3.1

Factor $y$ out of $- 4 y$ .

$y \cdot - 4 - x y = 13 - 6 x$

Step 3.2

Factor $y$ out of $- x y$ .

$y \cdot - 4 + y (- x) = 13 - 6 x$

Step 3.3

Factor $y$ out of $y \cdot - 4 + y (- x)$ .

$y (- 4 - x) = 13 - 6 x$

Step 4

Divide each term in $y (- 4 - x) = 13 - 6 x$ by $- 4 - x$ and simplify.

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Step 4.1

Divide each term in $y (- 4 - x) = 13 - 6 x$ by $- 4 - x$ .

$\frac{y (- 4 - x)}{- 4 - x} = \frac{13}{- 4 - x} + \frac{- 6 x}{- 4 - x}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 4 - x$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{y (- 4 - x)}{- 4 - x} = \frac{13}{- 4 - x} + \frac{- 6 x}{- 4 - x}$

Step 4.2.1.2

Divide $y$ by $1$ .

$y = \frac{13}{- 4 - x} + \frac{- 6 x}{- 4 - x}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Combine the numerators over the common denominator.

$y = \frac{13 - 6 x}{- 4 - x}$

Step 4.3.2

Rewrite $- 4$ as $- 1 (4)$ .

$y = \frac{13 - 6 x}{- 1 (4) - x}$

Step 4.3.3

Factor $- 1$ out of $- x$ .

$y = \frac{13 - 6 x}{- 1 (4) - (x)}$

Step 4.3.4

Factor $- 1$ out of $- 1 (4) - (x)$ .

$y = \frac{13 - 6 x}{- 1 (4 + x)}$

Step 4.3.5

Move the negative in front of the fraction.

$y = - \frac{13 - 6 x}{4 + x}$

Step 5

Set the denominator in $\frac{13 - 6 x}{4 + x}$ equal to $0$ to find where the expression is undefined.

$4 + x = 0$

Step 6

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 7

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

Interval Notation:

$(- \infty, - 4) \cup (- 4, \infty)$

Set-Builder Notation:

${x | x \neq - 4}$

Step 8