Algebra Examples

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

Step 1

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

$3 - z - \frac{2}{z} = 0$

Step 2

Reorder terms.

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

Step 3

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

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

Step 4

Combine $- z$ and $\frac{z}{z}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Multiply $z$ by $z$ by adding the exponents.

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Step 6.1

Move $z$ .

$\frac{- (z \cdot z) - 2}{z} + 3 = 0$

Step 6.2

Multiply $z$ by $z$ .

$\frac{- z^{2} - 2}{z} + 3 = 0$

Step 7

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

$\frac{- z^{2} - 2}{z} + \frac{3 z}{z} = 0$

Step 8

Combine the numerators over the common denominator.

$\frac{- z^{2} - 2 + 3 z}{z} = 0$

Step 9

Factor by grouping.

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Step 9.1

Reorder terms.

$\frac{- z^{2} + 3 z - 2}{z} = 0$

Step 9.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 2 = 2$ and whose sum is $b = 3$ .

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Step 9.2.1

Factor $3$ out of $3 z$ .

$\frac{- z^{2} + 3 (z) - 2}{z} = 0$

Step 9.2.2

Rewrite $3$ as $1$ plus $2$

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

Step 9.2.3

Apply the distributive property.

$\frac{- z^{2} + 1 z + 2 z - 2}{z} = 0$

Step 9.2.4

Multiply $z$ by $1$ .

$\frac{- z^{2} + z + 2 z - 2}{z} = 0$

Step 9.3

Factor out the greatest common factor from each group.

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Step 9.3.1

Group the first two terms and the last two terms.

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

Step 9.3.2

Factor out the greatest common factor (GCF) from each group.

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

Step 9.4

Factor the polynomial by factoring out the greatest common factor, $- z + 1$ .

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

Step 10

Set the numerator equal to zero.

$(- z + 1) (z - 2) = 0$

Step 11

Solve the equation for $z$ .

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Step 11.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$- z + 1 = 0$

$z - 2 = 0$

Step 11.2

Set $- z + 1$ equal to $0$ and solve for $z$ .

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Step 11.2.1

Set $- z + 1$ equal to $0$ .

$- z + 1 = 0$

Step 11.2.2

Solve $- z + 1 = 0$ for $z$ .

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Step 11.2.2.1

Subtract $1$ from both sides of the equation.

$- z = - 1$

Step 11.2.2.2

Divide each term in $- z = - 1$ by $- 1$ and simplify.

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Step 11.2.2.2.1

Divide each term in $- z = - 1$ by $- 1$ .

$\frac{- z}{- 1} = \frac{- 1}{- 1}$

Step 11.2.2.2.2

Simplify the left side.

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Step 11.2.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{z}{1} = \frac{- 1}{- 1}$

Step 11.2.2.2.2.2

Divide $z$ by $1$ .

$z = \frac{- 1}{- 1}$

Step 11.2.2.2.3

Simplify the right side.

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Step 11.2.2.2.3.1

Divide $- 1$ by $- 1$ .

$z = 1$

Step 11.3

Set $z - 2$ equal to $0$ and solve for $z$ .

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Step 11.3.1

Set $z - 2$ equal to $0$ .

$z - 2 = 0$

Step 11.3.2

Add $2$ to both sides of the equation.

$z = 2$

Step 11.4

The final solution is all the values that make $(- z + 1) (z - 2) = 0$ true.

$z = 1, 2$