Linear Algebra Examples

$y - 3 = y (x + 4)$

Step 1

Simplify $y (x + 4)$ .

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Step 1.1

Rewrite.

$y - 3 = 0 + 0 + y (x + 4)$

Step 1.2

Simplify by adding zeros.

$y - 3 = y (x + 4)$

Step 1.3

Apply the distributive property.

$y - 3 = y x + y \cdot 4$

Step 1.4

Move $4$ to the left of $y$ .

$y - 3 = y x + 4 y$

Step 2

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$y x + 4 y = y - 3$

Step 3

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

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Step 3.1

Subtract $y$ from both sides of the equation.

$y x + 4 y - y = - 3$

Step 3.2

Subtract $y$ from $4 y$ .

$y x + 3 y = - 3$

Step 4

Factor $y$ out of $y x + 3 y$ .

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Step 4.1

Factor $y$ out of $y x$ .

$y (x) + 3 y = - 3$

Step 4.2

Factor $y$ out of $3 y$ .

$y (x) + y \cdot 3 = - 3$

Step 4.3

Factor $y$ out of $y (x) + y \cdot 3$ .

$y (x + 3) = - 3$

Step 5

Divide each term in $y (x + 3) = - 3$ by $x + 3$ and simplify.

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Step 5.1

Divide each term in $y (x + 3) = - 3$ by $x + 3$ .

$\frac{y (x + 3)}{x + 3} = \frac{- 3}{x + 3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $x + 3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{y (x + 3)}{x + 3} = \frac{- 3}{x + 3}$

Step 5.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 3}{x + 3}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$y = - \frac{3}{x + 3}$

Step 6

Set the denominator in $\frac{3}{x + 3}$ equal to $0$ to find where the expression is undefined.

$x + 3 = 0$

Step 7

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 8

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

Interval Notation:

$(- \infty, - 3) \cup (- 3, \infty)$

Set-Builder Notation:

${x | x \neq - 3}$

Step 9