Linear Algebra Examples

$v = \frac{x - y}{z}$

Step 1

Rewrite the equation as $\frac{x - y}{z} = v$ .

$\frac{x - y}{z} = v$

Step 2

Multiply both sides by $z$ .

$\frac{x - y}{z} z = v z$

Step 3

Simplify the left side.

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Step 3.1

Cancel the common factor of $z$ .

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Step 3.1.1

Cancel the common factor.

$\frac{x - y}{z} z = v z$

Step 3.1.2

Rewrite the expression.

$x - y = v z$

Step 4

Solve for $y$ .

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Step 4.1

Subtract $x$ from both sides of the equation.

$- y = v z - x$

Step 4.2

Divide each term in $- y = v z - x$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- y = v z - x$ by $- 1$ .

$\frac{- y}{- 1} = \frac{v z}{- 1} + \frac{- x}{- 1}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{v z}{- 1} + \frac{- x}{- 1}$

Step 4.2.2.2

Divide $y$ by $1$ .

$y = \frac{v z}{- 1} + \frac{- x}{- 1}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Move the negative one from the denominator of $\frac{v z}{- 1}$ .

$y = - 1 \cdot (v z) + \frac{- x}{- 1}$

Step 4.2.3.1.2

Rewrite $- 1 \cdot (v z)$ as $- (v z)$ .

$y = - (v z) + \frac{- x}{- 1}$

Step 4.2.3.1.3

Dividing two negative values results in a positive value.

$y = - v z + \frac{x}{1}$

Step 4.2.3.1.4

Divide $x$ by $1$ .

$y = - v z + x$

Step 5

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:

${x | x \in ℝ}$