Finite Math Examples

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

Step 1

Subtract $3$ from both sides of the equation.

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

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, z, 1$

Step 2.2

The LCM of one and any expression is the expression.

$z$

Step 3

Multiply each term in $- z = \frac{2}{z} - 3$ by $z$ to eliminate the fractions.

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Step 3.1

Multiply each term in $- z = \frac{2}{z} - 3$ by $z$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Move $z$ .

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

Step 3.2.1.2

Multiply $z$ by $z$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $z$ .

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

$- z^{2} = 2 - 3 z$

Step 4

Solve the equation.

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Step 4.1

Add $3 z$ to both sides of the equation.

$- z^{2} + 3 z = 2$

Step 4.2

Subtract $2$ from both sides of the equation.

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

Step 4.3

Factor the left side of the equation.

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Step 4.3.1

Factor $- 1$ out of $- z^{2} + 3 z - 2$ .

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Step 4.3.1.1

Factor $- 1$ out of $- z^{2}$ .

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

Step 4.3.1.2

Factor $- 1$ out of $3 z$ .

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

Step 4.3.1.3

Rewrite $- 2$ as $- 1 (2)$ .

$- (z^{2}) - (- 3 z) - 1 \cdot 2 = 0$

Step 4.3.1.4

Factor $- 1$ out of $- (z^{2}) - (- 3 z)$ .

$- (z^{2} - 3 z) - 1 \cdot 2 = 0$

Step 4.3.1.5

Factor $- 1$ out of $- (z^{2} - 3 z) - 1 (2)$ .

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

Step 4.3.2

Factor.

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Step 4.3.2.1

Factor $z^{2} - 3 z + 2$ using the AC method.

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Step 4.3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 4.3.2.1.2

Write the factored form using these integers.

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

Step 4.3.2.2

Remove unnecessary parentheses.

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

Step 4.4

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

$z - 2 = 0$

$z - 1 = 0$

Step 4.5

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

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Step 4.5.1

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

$z - 2 = 0$

Step 4.5.2

Add $2$ to both sides of the equation.

$z = 2$

Step 4.6

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

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Step 4.6.1

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

$z - 1 = 0$

Step 4.6.2

Add $1$ to both sides of the equation.

$z = 1$

Step 4.7

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

$z = 2, 1$