Algebra Examples

$8 - \frac{4}{z} = \frac{8 z - 8}{z + 2}$

Step 1

Subtract $\frac{8 z - 8}{z + 2}$ from both sides of the equation.

$8 - \frac{4}{z} - \frac{8 z - 8}{z + 2} = 0$

Step 2

Simplify $8 - \frac{4}{z} - \frac{8 z - 8}{z + 2}$ .

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Step 2.1

Factor $8$ out of $8 z - 8$ .

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Step 2.1.1

Factor $8$ out of $8 z$ .

$8 - \frac{4}{z} - \frac{8 (z) - 8}{z + 2} = 0$

Step 2.1.2

Factor $8$ out of $- 8$ .

$8 - \frac{4}{z} - \frac{8 (z) + 8 (- 1)}{z + 2} = 0$

Step 2.1.3

Factor $8$ out of $8 (z) + 8 (- 1)$ .

$8 - \frac{4}{z} - \frac{8 (z - 1)}{z + 2} = 0$

Step 2.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{z + 2}{z + 2}$ .

$- \frac{4}{z} + \frac{8 (z + 2)}{z + 2} - \frac{8 (z - 1)}{z + 2} = 0$

Step 2.3

Combine the numerators over the common denominator.

$- \frac{4}{z} + \frac{8 (z + 2) - 8 (z - 1)}{z + 2} = 0$

Step 2.4

Simplify each term.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Factor $8$ out of $8 (z + 2) - 8 (z - 1)$ .

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Step 2.4.1.1.1

Factor $8$ out of $- 8 (z - 1)$ .

$- \frac{4}{z} + \frac{8 (z + 2) + 8 (- (z - 1))}{z + 2} = 0$

Step 2.4.1.1.2

Factor $8$ out of $8 (z + 2) + 8 (- (z - 1))$ .

$- \frac{4}{z} + \frac{8 (z + 2 - (z - 1))}{z + 2} = 0$

Step 2.4.1.2

Apply the distributive property.

$- \frac{4}{z} + \frac{8 (z + 2 - z - - 1)}{z + 2} = 0$

Step 2.4.1.3

Multiply $- 1$ by $- 1$ .

$- \frac{4}{z} + \frac{8 (z + 2 - z + 1)}{z + 2} = 0$

Step 2.4.1.4

Subtract $z$ from $z$ .

$- \frac{4}{z} + \frac{8 (0 + 2 + 1)}{z + 2} = 0$

Step 2.4.1.5

Add $0$ and $2$ .

$- \frac{4}{z} + \frac{8 (2 + 1)}{z + 2} = 0$

Step 2.4.1.6

Add $2$ and $1$ .

$- \frac{4}{z} + \frac{8 \cdot 3}{z + 2} = 0$

Step 2.4.2

Multiply $8$ by $3$ .

$- \frac{4}{z} + \frac{24}{z + 2} = 0$

Step 2.5

To write $- \frac{4}{z}$ as a fraction with a common denominator, multiply by $\frac{z + 2}{z + 2}$ .

$- \frac{4}{z} \cdot \frac{z + 2}{z + 2} + \frac{24}{z + 2} = 0$

Step 2.6

To write $\frac{24}{z + 2}$ as a fraction with a common denominator, multiply by $\frac{z}{z}$ .

$- \frac{4}{z} \cdot \frac{z + 2}{z + 2} + \frac{24}{z + 2} \cdot \frac{z}{z} = 0$

Step 2.7

Write each expression with a common denominator of $z (z + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.7.1

Multiply $\frac{4}{z}$ by $\frac{z + 2}{z + 2}$ .

$- \frac{4 (z + 2)}{z (z + 2)} + \frac{24}{z + 2} \cdot \frac{z}{z} = 0$

Step 2.7.2

Multiply $\frac{24}{z + 2}$ by $\frac{z}{z}$ .

$- \frac{4 (z + 2)}{z (z + 2)} + \frac{24 z}{(z + 2) z} = 0$

Step 2.7.3

Reorder the factors of $(z + 2) z$ .

$- \frac{4 (z + 2)}{z (z + 2)} + \frac{24 z}{z (z + 2)} = 0$

Step 2.8

Combine the numerators over the common denominator.

$\frac{- 4 (z + 2) + 24 z}{z (z + 2)} = 0$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Factor $4$ out of $- 4 (z + 2) + 24 z$ .

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Step 2.9.1.1

Factor $4$ out of $- 4 (z + 2)$ .

$\frac{4 (- (z + 2)) + 24 z}{z (z + 2)} = 0$

Step 2.9.1.2

Factor $4$ out of $24 z$ .

$\frac{4 (- (z + 2)) + 4 (6 z)}{z (z + 2)} = 0$

Step 2.9.1.3

Factor $4$ out of $4 (- (z + 2)) + 4 (6 z)$ .

$\frac{4 (- (z + 2) + 6 z)}{z (z + 2)} = 0$

Step 2.9.2

Apply the distributive property.

$\frac{4 (- z - 1 \cdot 2 + 6 z)}{z (z + 2)} = 0$

Step 2.9.3

Multiply $- 1$ by $2$ .

$\frac{4 (- z - 2 + 6 z)}{z (z + 2)} = 0$

Step 2.9.4

Add $- z$ and $6 z$ .

$\frac{4 (5 z - 2)}{z (z + 2)} = 0$

Step 3

Set the numerator equal to zero.

$4 (5 z - 2) = 0$

Step 4

Solve the equation for $z$ .

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Step 4.1

Divide each term in $4 (5 z - 2) = 0$ by $4$ and simplify.

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Step 4.1.1

Divide each term in $4 (5 z - 2) = 0$ by $4$ .

$\frac{4 (5 z - 2)}{4} = \frac{0}{4}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $4$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{4 (5 z - 2)}{4} = \frac{0}{4}$

Step 4.1.2.1.2

Divide $5 z - 2$ by $1$ .

$5 z - 2 = \frac{0}{4}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $4$ .

$5 z - 2 = 0$

Step 4.2

Add $2$ to both sides of the equation.

$5 z = 2$

Step 4.3

Divide each term in $5 z = 2$ by $5$ and simplify.

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Step 4.3.1

Divide each term in $5 z = 2$ by $5$ .

$\frac{5 z}{5} = \frac{2}{5}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $5$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{5 z}{5} = \frac{2}{5}$

Step 4.3.2.1.2

Divide $z$ by $1$ .

$z = \frac{2}{5}$