Basic Math Examples

$y = \frac{(z - 3) (z - 1)}{(z - 1) (z - 5)}$

Step 1

Rewrite the equation as $\frac{(z - 3) (z - 1)}{(z - 1) (z - 5)} = y$ .

$\frac{(z - 3) (z - 1)}{(z - 1) (z - 5)} = y$

Step 2

Reduce the expression $\frac{(z - 3) (z - 1)}{(z - 1) (z - 5)}$ by cancelling the common factors.

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Step 2.1

Cancel the common factor.

$\frac{(z - 3) (z - 1)}{(z - 1) (z - 5)} = y$

Step 2.2

Rewrite the expression.

$\frac{z - 3}{z - 5} = y$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

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

$z - 5, 1$

Step 3.2

Remove parentheses.

$z - 5, 1$

Step 3.3

The LCM of one and any expression is the expression.

$z - 5$

Step 4

Multiply each term in $\frac{z - 3}{z - 5} = y$ by $z - 5$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{z - 3}{z - 5} = y$ by $z - 5$ .

$\frac{z - 3}{z - 5} (z - 5) = y (z - 5)$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $z - 5$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{z - 3}{z - 5} (z - 5) = y (z - 5)$

Step 4.2.1.2

Rewrite the expression.

$z - 3 = y (z - 5)$

Step 4.3

Simplify the right side.

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Step 4.3.1

Apply the distributive property.

$z - 3 = y z + y \cdot - 5$

Step 4.3.2

Move $- 5$ to the left of $y$ .

$z - 3 = y z - 5 y$

Step 5

Solve the equation.

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Step 5.1

Subtract $y z$ from both sides of the equation.

$z - 3 - y z = - 5 y$

Step 5.2

Add $3$ to both sides of the equation.

$z - y z = - 5 y + 3$

Step 5.3

Factor $z$ out of $z - y z$ .

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Step 5.3.1

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

$z \cdot 1 - y z = - 5 y + 3$

Step 5.3.2

Factor $z$ out of $- y z$ .

$z \cdot 1 + z (- y) = - 5 y + 3$

Step 5.3.3

Factor $z$ out of $z \cdot 1 + z (- y)$ .

$z (1 - y) = - 5 y + 3$

Step 5.4

Divide each term in $z (1 - y) = - 5 y + 3$ by $1 - y$ and simplify.

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Step 5.4.1

Divide each term in $z (1 - y) = - 5 y + 3$ by $1 - y$ .

$\frac{z (1 - y)}{1 - y} = \frac{- 5 y}{1 - y} + \frac{3}{1 - y}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $1 - y$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{z (1 - y)}{1 - y} = \frac{- 5 y}{1 - y} + \frac{3}{1 - y}$

Step 5.4.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 5 y}{1 - y} + \frac{3}{1 - y}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Combine the numerators over the common denominator.

$z = \frac{- 5 y + 3}{1 - y}$

Step 5.4.3.2

Factor $- 1$ out of $- 5 y$ .

$z = \frac{- (5 y) + 3}{1 - y}$

Step 5.4.3.3

Rewrite $3$ as $- 1 (- 3)$ .

$z = \frac{- (5 y) - 1 (- 3)}{1 - y}$

Step 5.4.3.4

Factor $- 1$ out of $- (5 y) - 1 (- 3)$ .

$z = \frac{- (5 y - 3)}{1 - y}$

Step 5.4.3.5

Simplify the expression.

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Step 5.4.3.5.1

Rewrite $- (5 y - 3)$ as $- 1 (5 y - 3)$ .

$z = \frac{- 1 (5 y - 3)}{1 - y}$

Step 5.4.3.5.2

Move the negative in front of the fraction.

$z = - \frac{5 y - 3}{1 - y}$