Linear Algebra Examples

$\frac{z}{6} + 6 = 8 - \frac{z}{6}$

Step 1

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

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Step 1.1

Add $\frac{z}{6}$ to both sides of the equation.

$\frac{z}{6} + 6 + \frac{z}{6} = 8$

Step 1.2

Combine the numerators over the common denominator.

$6 + \frac{z + z}{6} = 8$

Step 1.3

Add $z$ and $z$ .

$6 + \frac{2 z}{6} = 8$

Step 1.4

Cancel the common factor of $2$ and $6$ .

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Step 1.4.1

Factor $2$ out of $2 z$ .

$6 + \frac{2 (z)}{6} = 8$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $2$ out of $6$ .

$6 + \frac{2 z}{2 \cdot 3} = 8$

Step 1.4.2.2

Cancel the common factor.

$6 + \frac{2 z}{2 \cdot 3} = 8$

Step 1.4.2.3

Rewrite the expression.

$6 + \frac{z}{3} = 8$

Step 2

Move all terms not containing $z$ to the right side of the equation.

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Step 2.1

Subtract $6$ from both sides of the equation.

$\frac{z}{3} = 8 - 6$

Step 2.2

Subtract $6$ from $8$ .

$\frac{z}{3} = 2$

Step 3

Multiply both sides of the equation by $3$ .

$3 \frac{z}{3} = 3 \cdot 2$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $3$ .

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Step 4.1.1.1

Cancel the common factor.

$3 \frac{z}{3} = 3 \cdot 2$

Step 4.1.1.2

Rewrite the expression.

$z = 3 \cdot 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $3$ by $2$ .

$z = 6$

Step 5

The domain is the set of all valid $z$ values.

${z | z = 6}$

Step 6