Linear Algebra Examples

$\frac{3 a x - n}{5} = - 4$

Step 1

Multiply both sides by $5$ .

$\frac{3 a x - n}{5} \cdot 5 = - 4 \cdot 5$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $5$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{3 a x - n}{5} \cdot 5 = - 4 \cdot 5$

Step 2.1.1.2

Rewrite the expression.

$3 a x - n = - 4 \cdot 5$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $- 4$ by $5$ .

$3 a x - n = - 20$

Step 3

Solve for $a$ .

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Step 3.1

Add $n$ to both sides of the equation.

$3 a x = - 20 + n$

Step 3.2

Divide each term in $3 a x = - 20 + n$ by $3 x$ and simplify.

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Step 3.2.1

Divide each term in $3 a x = - 20 + n$ by $3 x$ .

$\frac{3 a x}{3 x} = \frac{- 20}{3 x} + \frac{n}{3 x}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{3 a x}{3 x} = \frac{- 20}{3 x} + \frac{n}{3 x}$

Step 3.2.2.1.2

Rewrite the expression.

$\frac{a x}{x} = \frac{- 20}{3 x} + \frac{n}{3 x}$

Step 3.2.2.2

Cancel the common factor of $x$ .

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Step 3.2.2.2.1

Cancel the common factor.

$\frac{a x}{x} = \frac{- 20}{3 x} + \frac{n}{3 x}$

Step 3.2.2.2.2

Divide $a$ by $1$ .

$a = \frac{- 20}{3 x} + \frac{n}{3 x}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

$a = - \frac{20}{3 x} + \frac{n}{3 x}$

Step 4

Set the denominator in $\frac{20}{3 x}$ equal to $0$ to find where the expression is undefined.

$3 x = 0$

Step 5

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $3$ .

$x = 0$

Step 6

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$