Pre-Algebra Examples

$\frac{6}{z} - \frac{9}{z - 2} = \frac{1}{7}$

Step 1

Simplify $\frac{6}{z} - \frac{9}{z - 2}$ .

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Step 1.1

To write $\frac{6}{z}$ as a fraction with a common denominator, multiply by $\frac{z - 2}{z - 2}$ .

$\frac{6}{z} \cdot \frac{z - 2}{z - 2} - \frac{9}{z - 2} = \frac{1}{7}$

Step 1.2

To write $- \frac{9}{z - 2}$ as a fraction with a common denominator, multiply by $\frac{z}{z}$ .

$\frac{6}{z} \cdot \frac{z - 2}{z - 2} - \frac{9}{z - 2} \cdot \frac{z}{z} = \frac{1}{7}$

Step 1.3

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 1.3.1

Multiply $\frac{6}{z}$ by $\frac{z - 2}{z - 2}$ .

$\frac{6 (z - 2)}{z (z - 2)} - \frac{9}{z - 2} \cdot \frac{z}{z} = \frac{1}{7}$

Step 1.3.2

Multiply $\frac{9}{z - 2}$ by $\frac{z}{z}$ .

$\frac{6 (z - 2)}{z (z - 2)} - \frac{9 z}{(z - 2) z} = \frac{1}{7}$

Step 1.3.3

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

$\frac{6 (z - 2)}{z (z - 2)} - \frac{9 z}{z (z - 2)} = \frac{1}{7}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{6 (z - 2) - 9 z}{z (z - 2)} = \frac{1}{7}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Factor $3$ out of $6 (z - 2) - 9 z$ .

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Step 1.5.1.1

Factor $3$ out of $6 (z - 2)$ .

$\frac{3 (2 (z - 2)) - 9 z}{z (z - 2)} = \frac{1}{7}$

Step 1.5.1.2

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

$\frac{3 (2 (z - 2)) + 3 (- 3 z)}{z (z - 2)} = \frac{1}{7}$

Step 1.5.1.3

Factor $3$ out of $3 (2 (z - 2)) + 3 (- 3 z)$ .

$\frac{3 (2 (z - 2) - 3 z)}{z (z - 2)} = \frac{1}{7}$

Step 1.5.2

Apply the distributive property.

$\frac{3 (2 z + 2 \cdot - 2 - 3 z)}{z (z - 2)} = \frac{1}{7}$

Step 1.5.3

Multiply $2$ by $- 2$ .

$\frac{3 (2 z - 4 - 3 z)}{z (z - 2)} = \frac{1}{7}$

Step 1.5.4

Subtract $3 z$ from $2 z$ .

$\frac{3 (- z - 4)}{z (z - 2)} = \frac{1}{7}$

Step 1.6

Simplify with factoring out.

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Step 1.6.1

Factor $- 1$ out of $- z$ .

$\frac{3 (- (z) - 4)}{z (z - 2)} = \frac{1}{7}$

Step 1.6.2

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{3 (- (z) - 1 (4))}{z (z - 2)} = \frac{1}{7}$

Step 1.6.3

Factor $- 1$ out of $- (z) - 1 (4)$ .

$\frac{3 (- (z + 4))}{z (z - 2)} = \frac{1}{7}$

Step 1.6.4

Simplify the expression.

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Step 1.6.4.1

Rewrite $- (z + 4)$ as $- 1 (z + 4)$ .

$\frac{3 (- 1 (z + 4))}{z (z - 2)} = \frac{1}{7}$

Step 1.6.4.2

Move the negative in front of the fraction.

$- \frac{3 (z + 4)}{z (z - 2)} = \frac{1}{7}$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$z = - 12, - 7$

Step 3