Algebra Examples

$\frac{2 z}{3 z - 9} - \frac{z + 1}{2 z - 6}$

Step 1

Simplify each term.

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Step 1.1

Factor $3$ out of $3 z - 9$ .

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Step 1.1.1

Factor $3$ out of $3 z$ .

$\frac{2 z}{3 (z) - 9} - \frac{z + 1}{2 z - 6}$

Step 1.1.2

Factor $3$ out of $- 9$ .

$\frac{2 z}{3 z + 3 \cdot - 3} - \frac{z + 1}{2 z - 6}$

Step 1.1.3

Factor $3$ out of $3 z + 3 \cdot - 3$ .

$\frac{2 z}{3 (z - 3)} - \frac{z + 1}{2 z - 6}$

Step 1.2

Factor $2$ out of $2 z - 6$ .

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Step 1.2.1

Factor $2$ out of $2 z$ .

$\frac{2 z}{3 (z - 3)} - \frac{z + 1}{2 (z) - 6}$

Step 1.2.2

Factor $2$ out of $- 6$ .

$\frac{2 z}{3 (z - 3)} - \frac{z + 1}{2 z + 2 \cdot - 3}$

Step 1.2.3

Factor $2$ out of $2 z + 2 \cdot - 3$ .

$\frac{2 z}{3 (z - 3)} - \frac{z + 1}{2 (z - 3)}$

Step 2

To write $\frac{2 z}{3 (z - 3)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 z}{3 (z - 3)} \cdot \frac{2}{2} - \frac{z + 1}{2 (z - 3)}$

Step 3

To write $- \frac{z + 1}{2 (z - 3)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 z}{3 (z - 3)} \cdot \frac{2}{2} - \frac{z + 1}{2 (z - 3)} \cdot \frac{3}{3}$

Step 4

Write each expression with a common denominator of $6 (z - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2 z}{3 (z - 3)}$ by $\frac{2}{2}$ .

$\frac{2 z \cdot 2}{3 (z - 3) \cdot 2} - \frac{z + 1}{2 (z - 3)} \cdot \frac{3}{3}$

Step 4.2

Multiply $2$ by $3$ .

$\frac{2 z \cdot 2}{6 (z - 3)} - \frac{z + 1}{2 (z - 3)} \cdot \frac{3}{3}$

Step 4.3

Multiply $\frac{z + 1}{2 (z - 3)}$ by $\frac{3}{3}$ .

$\frac{2 z \cdot 2}{6 (z - 3)} - \frac{(z + 1) \cdot 3}{2 (z - 3) \cdot 3}$

Step 4.4

Multiply $3$ by $2$ .

$\frac{2 z \cdot 2}{6 (z - 3)} - \frac{(z + 1) \cdot 3}{6 (z - 3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 z \cdot 2 - (z + 1) \cdot 3}{6 (z - 3)}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $2$ by $2$ .

$\frac{4 z - (z + 1) \cdot 3}{6 (z - 3)}$

Step 6.2

Apply the distributive property.

$\frac{4 z + (- z - 1 \cdot 1) \cdot 3}{6 (z - 3)}$

Step 6.3

Multiply $- 1$ by $1$ .

$\frac{4 z + (- z - 1) \cdot 3}{6 (z - 3)}$

Step 6.4

Apply the distributive property.

$\frac{4 z - z \cdot 3 - 1 \cdot 3}{6 (z - 3)}$

Step 6.5

Multiply $3$ by $- 1$ .

$\frac{4 z - 3 z - 1 \cdot 3}{6 (z - 3)}$

Step 6.6

Multiply $- 1$ by $3$ .

$\frac{4 z - 3 z - 3}{6 (z - 3)}$

Step 6.7

Subtract $3 z$ from $4 z$ .

$\frac{z - 3}{6 (z - 3)}$

Step 7

Cancel the common factor of $z - 3$ .

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Step 7.1

Cancel the common factor.

$\frac{z - 3}{6 (z - 3)}$

Step 7.2

Rewrite the expression.

$\frac{1}{6}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{6}$

Decimal Form:

$0.1\overline{6}$