Basic Math Examples

$x + y = 2 x y$ , $x + z = 4 x z$ , $y + z = 8 y z$

Step 1

Solve for $x$ in $x + y = 2 x y$ .

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Step 1.1

Subtract $2 x y$ from both sides of the equation.

$x + y - 2 x y = 0$

$x + z = 4 x z$

$y + z = 8 y z$

Step 1.2

Subtract $y$ from both sides of the equation.

$x - 2 x y = - y$

$x + z = 4 x z$

$y + z = 8 y z$

Step 1.3

Factor $x$ out of $x - 2 x y$ .

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Step 1.3.1

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

$x \cdot 1 - 2 x y = - y$

$x + z = 4 x z$

$y + z = 8 y z$

Step 1.3.2

Factor $x$ out of $- 2 x y$ .

$x \cdot 1 + x (- 2 y) = - y$

$x + z = 4 x z$

$y + z = 8 y z$

Step 1.3.3

Factor $x$ out of $x \cdot 1 + x (- 2 y)$ .

$x \cdot (1 - 2 y) = - y$

$x + z = 4 x z$

$y + z = 8 y z$

Step 1.3.4

Multiply $x$ by $1 - 2 y$ .

$x (1 - 2 y) = - y$

$x + z = 4 x z$

$y + z = 8 y z$

$x (1 - 2 y) = - y$

$x + z = 4 x z$

$y + z = 8 y z$

Step 1.4

Divide each term in $x (1 - 2 y) = - y$ by $1 - 2 y$ and simplify.

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Step 1.4.1

Divide each term in $x (1 - 2 y) = - y$ by $1 - 2 y$ .

$\frac{x (1 - 2 y)}{1 - 2 y} = \frac{- y}{1 - 2 y}$

$x + z = 4 x z$

$y + z = 8 y z$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Cancel the common factor of $1 - 2 y$ .

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Step 1.4.2.1.1

Cancel the common factor.

$\frac{x (1 - 2 y)}{1 - 2 y} = \frac{- y}{1 - 2 y}$

$x + z = 4 x z$

$y + z = 8 y z$

Step 1.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- y}{1 - 2 y}$

$x + z = 4 x z$

$y + z = 8 y z$

$x = \frac{- y}{1 - 2 y}$

$x + z = 4 x z$

$y + z = 8 y z$

$x = \frac{- y}{1 - 2 y}$

$x + z = 4 x z$

$y + z = 8 y z$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Move the negative in front of the fraction.

$x = - \frac{y}{1 - 2 y}$

$x + z = 4 x z$

$y + z = 8 y z$

$x = - \frac{y}{1 - 2 y}$

$x + z = 4 x z$

$y + z = 8 y z$

$x = - \frac{y}{1 - 2 y}$

$x + z = 4 x z$

$y + z = 8 y z$

$x = - \frac{y}{1 - 2 y}$

$x + z = 4 x z$

$y + z = 8 y z$

Step 2

Replace all occurrences of $x$ with $- \frac{y}{1 - 2 y}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x + z = 4 x z$ with $- \frac{y}{1 - 2 y}$ .

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2

Simplify $(- \frac{y}{1 - 2 y}) + z = 4 (- \frac{y}{1 - 2 y}) \cdot z$ .

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify $(- \frac{y}{1 - 2 y}) + z$ .

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Step 2.2.1.1.1

To write $z$ as a fraction with a common denominator, multiply by $\frac{1 - 2 y}{1 - 2 y}$ .

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2.1.1.2

Simplify terms.

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Step 2.2.1.1.2.1

Combine $z$ and $\frac{1 - 2 y}{1 - 2 y}$ .

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2.1.1.2.2

Combine the numerators over the common denominator.

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2.1.1.3

Simplify the numerator.

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Step 2.2.1.1.3.1

Apply the distributive property.

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2.1.1.3.2

Multiply $z$ by $1$ .

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2.1.1.3.3

Rewrite using the commutative property of multiplication.

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2.1.1.4

Simplify with factoring out.

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Step 2.2.1.1.4.1

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

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2.1.1.4.2

Factor $- 1$ out of $z$ .

$\frac{- (y) - 1 (- z) - 2 z y}{1 - 2 y} = 4 (- \frac{y}{1 - 2 y}) \cdot z$

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2.1.1.4.3

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

$\frac{- (y - z) - 2 z y}{1 - 2 y} = 4 (- \frac{y}{1 - 2 y}) \cdot z$

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2.1.1.4.4

Factor $- 1$ out of $- 2 z y$ .

$\frac{- (y - z) - (2 z y)}{1 - 2 y} = 4 (- \frac{y}{1 - 2 y}) \cdot z$

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2.1.1.4.5

Factor $- 1$ out of $- (y - z) - (2 z y)$ .

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2.1.1.4.6

Simplify the expression.

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Step 2.2.1.1.4.6.1

Rewrite $- (y - z + 2 z y)$ as $- 1 (y - z + 2 z y)$ .

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2.1.1.4.6.2

Move the negative in front of the fraction.

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $4 (- \frac{y}{1 - 2 y}) \cdot z$ .

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Step 2.2.2.1.1

Multiply $4 (- \frac{y}{1 - 2 y})$ .

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Step 2.2.2.1.1.1

Multiply $- 1$ by $4$ .

$- \frac{y - z + 2 z y}{1 - 2 y} = - 4 \frac{y}{1 - 2 y} \cdot z$

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2.2.1.1.2

Combine $- 4$ and $\frac{y}{1 - 2 y}$ .

$- \frac{y - z + 2 z y}{1 - 2 y} = \frac{- 4 y}{1 - 2 y} \cdot z$

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

$- \frac{y - z + 2 z y}{1 - 2 y} = \frac{- 4 y}{1 - 2 y} \cdot z$

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2.2.1.2

Move the negative in front of the fraction.

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 y}{1 - 2 y} \cdot z$

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2.2.1.3

Combine $z$ and $\frac{4 y}{1 - 2 y}$ .

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

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 2.2.2.1.4

Move $4$ to the left of $z$ .

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

$y + z = 8 y z$

Step 3

Solve for $y$ in $y + z = 8 y z$ .

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Step 3.1

Subtract $8 y z$ from both sides of the equation.

$y + z - 8 y z = 0$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

Step 3.2

Subtract $z$ from both sides of the equation.

$y - 8 y z = - z$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

Step 3.3

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

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Step 3.3.1

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

$y \cdot 1 - 8 y z = - z$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

Step 3.3.2

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

$y \cdot 1 + y (- 8 z) = - z$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

Step 3.3.3

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

$y \cdot (1 - 8 z) = - z$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

Step 3.3.4

Multiply $y$ by $1 - 8 z$ .

$y (1 - 8 z) = - z$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

$y (1 - 8 z) = - z$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

Step 3.4

Divide each term in $y (1 - 8 z) = - z$ by $1 - 8 z$ and simplify.

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Step 3.4.1

Divide each term in $y (1 - 8 z) = - z$ by $1 - 8 z$ .

$\frac{y (1 - 8 z)}{1 - 8 z} = \frac{- z}{1 - 8 z}$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $1 - 8 z$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{y (1 - 8 z)}{1 - 8 z} = \frac{- z}{1 - 8 z}$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- z}{1 - 8 z}$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

$y = \frac{- z}{1 - 8 z}$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

$y = \frac{- z}{1 - 8 z}$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Move the negative in front of the fraction.

$y = - \frac{z}{1 - 8 z}$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

$y = - \frac{z}{1 - 8 z}$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

$y = - \frac{z}{1 - 8 z}$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

$y = - \frac{z}{1 - 8 z}$

$- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$

$x = - \frac{y}{1 - 2 y}$

Step 4

Replace all occurrences of $y$ with $- \frac{z}{1 - 8 z}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $- \frac{y - z + 2 z y}{1 - 2 y} = - \frac{4 z y}{1 - 2 y}$ with $- \frac{z}{1 - 8 z}$ .

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2

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

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify $- \frac{(- \frac{z}{1 - 8 z}) - z + 2 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$ .

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Step 4.2.1.1.1

Multiply the numerator and denominator of the fraction by $1 - 8 z$ .

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Step 4.2.1.1.1.1

Multiply $\frac{- \frac{z}{1 - 8 z} - z + 2 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$ by $\frac{1 - 8 z}{1 - 8 z}$ .

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.1.2

Combine.

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.2

Apply the distributive property.

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.3

Cancel the common factor of $1 - 8 z$ .

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Step 4.2.1.1.3.1

Move the leading negative in $- \frac{z}{1 - 8 z}$ into the numerator.

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.3.2

Cancel the common factor.

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.3.3

Rewrite the expression.

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4

Simplify the numerator.

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Step 4.2.1.1.4.1

Factor $z$ out of $- z + (1 - 8 z) (- z) + (1 - 8 z) \cdot 2 z (- \frac{z}{1 - 8 z})$ .

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Step 4.2.1.1.4.1.1

Factor $z$ out of $- z$ .

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4.1.2

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

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4.1.3

Factor $z$ out of $(1 - 8 z) \cdot 2 z (- \frac{z}{1 - 8 z})$ .

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4.1.4

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

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4.1.5

Factor $z$ out of $z \cdot (- 1 + (1 - 8 z) \cdot (- 1)) + z ((1 - 8 z) \cdot 2 (- \frac{z}{1 - 8 z}))$ .

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4.2

Apply the distributive property.

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4.3

Multiply $- 1$ by $1$ .

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4.4

Multiply $- 1$ by $- 8$ .

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4.5

Cancel the common factor of $1 - 8 z$ .

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Step 4.2.1.1.4.5.1

Move the leading negative in $- \frac{z}{1 - 8 z}$ into the numerator.

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4.5.2

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

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4.5.3

Cancel the common factor.

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4.5.4

Rewrite the expression.

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4.6

Multiply $- 1$ by $2$ .

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4.7

Subtract $1$ from $- 1$ .

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

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4.8

Subtract $2 z$ from $8 z$ .

$- \frac{z (- 2 + 6 z)}{(1 - 8 z) \cdot 1 + (1 - 8 z) (- 2 (- \frac{z}{1 - 8 z}))} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4.9

Factor $2$ out of $- 2 + 6 z$ .

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Step 4.2.1.1.4.9.1

Factor $2$ out of $- 2$ .

$- \frac{z (2 (- 1) + 6 z)}{(1 - 8 z) \cdot 1 + (1 - 8 z) (- 2 (- \frac{z}{1 - 8 z}))} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4.9.2

Factor $2$ out of $6 z$ .

$- \frac{z (2 (- 1) + 2 (3 z))}{(1 - 8 z) \cdot 1 + (1 - 8 z) (- 2 (- \frac{z}{1 - 8 z}))} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.4.9.3

Factor $2$ out of $2 (- 1) + 2 (3 z)$ .

$- \frac{z (2 (- 1 + 3 z))}{(1 - 8 z) \cdot 1 + (1 - 8 z) (- 2 (- \frac{z}{1 - 8 z}))} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$- \frac{z \cdot (2 (- 1 + 3 z))}{(1 - 8 z) \cdot 1 + (1 - 8 z) (- 2 (- \frac{z}{1 - 8 z}))} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$- \frac{z \cdot (2 (- 1 + 3 z))}{(1 - 8 z) \cdot 1 + (1 - 8 z) (- 2 (- \frac{z}{1 - 8 z}))} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.5

Simplify the denominator.

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Step 4.2.1.1.5.1

Factor $1 - 8 z$ out of $(1 - 8 z) \cdot 1 + (1 - 8 z) (- 2 (- \frac{z}{1 - 8 z}))$ .

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Step 4.2.1.1.5.1.1

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

$- \frac{z \cdot (2 (- 1 + 3 z))}{(1 - 8 z) (1) + (1 - 8 z) (- 2 (- \frac{z}{1 - 8 z}))} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.5.1.2

Factor $1 - 8 z$ out of $(1 - 8 z) (1) + (1 - 8 z) (- 2 (- \frac{z}{1 - 8 z}))$ .

$- \frac{z \cdot (2 (- 1 + 3 z))}{(1 - 8 z) (1 - 2 (- \frac{z}{1 - 8 z}))} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$- \frac{z \cdot (2 (- 1 + 3 z))}{(1 - 8 z) (1 - 2 (- \frac{z}{1 - 8 z}))} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.5.2

Multiply $- 2 (- \frac{z}{1 - 8 z})$ .

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Step 4.2.1.1.5.2.1

Multiply $- 1$ by $- 2$ .

$- \frac{z \cdot (2 (- 1 + 3 z))}{(1 - 8 z) (1 + 2 (\frac{z}{1 - 8 z}))} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.5.2.2

Combine $2$ and $\frac{z}{1 - 8 z}$ .

$- \frac{z \cdot (2 (- 1 + 3 z))}{(1 - 8 z) (1 + \frac{2 z}{1 - 8 z})} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$- \frac{z \cdot (2 (- 1 + 3 z))}{(1 - 8 z) (1 + \frac{2 z}{1 - 8 z})} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.5.3

Write $1$ as a fraction with a common denominator.

$- \frac{z \cdot (2 (- 1 + 3 z))}{(1 - 8 z) (\frac{1 - 8 z}{1 - 8 z} + \frac{2 z}{1 - 8 z})} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.5.4

Combine the numerators over the common denominator.

$- \frac{z \cdot (2 (- 1 + 3 z))}{(1 - 8 z) (\frac{1 - 8 z + 2 z}{1 - 8 z})} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.5.5

Add $- 8 z$ and $2 z$ .

$- \frac{z \cdot (2 (- 1 + 3 z))}{(1 - 8 z) (\frac{1 - 6 z}{1 - 8 z})} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$- \frac{z \cdot (2 (- 1 + 3 z))}{(1 - 8 z) (\frac{1 - 6 z}{1 - 8 z})} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.6

Move $2$ to the left of $z$ .

$- \frac{2 \cdot (z (- 1 + 3 z))}{(1 - 8 z) (\frac{1 - 6 z}{1 - 8 z})} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.7

Factor $\frac{1 - 6 z}{1 - 8 z}$ out of $\frac{2 \cdot z (- 1 + 3 z)}{(1 - 8 z) \frac{1 - 6 z}{1 - 8 z}}$ .

$- (\frac{1 - 8 z}{1 - 6 z} \cdot \frac{2 \cdot (z (- 1 + 3 z))}{1 - 8 z}) = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.8

Cancel the common factor of $1 - 8 z$ .

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Step 4.2.1.1.8.1

Cancel the common factor.

$- (\frac{1 - 8 z}{1 - 6 z} \cdot \frac{2 \cdot (z (- 1 + 3 z))}{1 - 8 z}) = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.8.2

Rewrite the expression.

$- (\frac{1}{1 - 6 z} \cdot (2 \cdot (z (- 1 + 3 z)))) = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$- (\frac{1}{1 - 6 z} \cdot (2 \cdot (z (- 1 + 3 z)))) = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.9

Combine $2$ and $\frac{1}{1 - 6 z}$ .

$- (\frac{2}{1 - 6 z} \cdot (z (- 1 + 3 z))) = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.10

Combine $z$ and $\frac{2}{1 - 6 z}$ .

$- (\frac{z \cdot 2}{1 - 6 z} \cdot (- 1 + 3 z)) = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.11

Move $2$ to the left of $z$ .

$- (\frac{2 \cdot z}{1 - 6 z} \cdot (- 1 + 3 z)) = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.12

Multiply $\frac{2 z}{1 - 6 z}$ by $- 1 + 3 z$ .

$- \frac{2 z (- 1 + 3 z)}{1 - 6 z} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.13

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

$- \frac{2 z (- 1 \cdot 1 + 3 z)}{1 - 6 z} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.14

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

$- \frac{2 z (- 1 \cdot 1 - (- 3 z))}{1 - 6 z} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.15

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

$- \frac{2 z (- 1 (1 - 3 z))}{1 - 6 z} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.16

Simplify the expression.

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Step 4.2.1.1.16.1

Move the negative in front of the fraction.

$\frac{(2 z) (1 - 3 z)}{1 - 6 z} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.16.2

Multiply $- 1$ by $- 1$ .

$1 (\frac{(2 z) (1 - 3 z)}{1 - 6 z}) = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.16.3

Multiply $\frac{(2 z) (1 - 3 z)}{1 - 6 z}$ by $1$ .

$\frac{(2 z) (1 - 3 z)}{1 - 6 z} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.1.1.16.4

Reorder factors in $\frac{(2 z) (1 - 3 z)}{1 - 6 z}$ .

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $- \frac{4 z (- \frac{z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$ .

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Step 4.2.2.1.1

Simplify the numerator.

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Step 4.2.2.1.1.1

Multiply $- 1$ by $4$ .

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{- 4 z \frac{z}{1 - 8 z}}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.1.2

Combine $- 4$ and $\frac{z}{1 - 8 z}$ .

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{z (\frac{- 4 z}{1 - 8 z})}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.1.3

Combine $z$ and $\frac{- 4 z}{1 - 8 z}$ .

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{\frac{z (- 4 z)}{1 - 8 z}}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{\frac{z (- 4 z)}{1 - 8 z}}{1 - 2 (- \frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.2

Simplify the denominator.

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Step 4.2.2.1.2.1

Multiply $- 2 (- \frac{z}{1 - 8 z})$ .

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Step 4.2.2.1.2.1.1

Multiply $- 1$ by $- 2$ .

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{\frac{z (- 4 z)}{1 - 8 z}}{1 + 2 (\frac{z}{1 - 8 z})}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.2.1.2

Combine $2$ and $\frac{z}{1 - 8 z}$ .

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{\frac{z (- 4 z)}{1 - 8 z}}{1 + \frac{2 z}{1 - 8 z}}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{\frac{z (- 4 z)}{1 - 8 z}}{1 + \frac{2 z}{1 - 8 z}}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.2.2

Write $1$ as a fraction with a common denominator.

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{\frac{z (- 4 z)}{1 - 8 z}}{\frac{1 - 8 z}{1 - 8 z} + \frac{2 z}{1 - 8 z}}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.2.3

Combine the numerators over the common denominator.

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{\frac{z (- 4 z)}{1 - 8 z}}{\frac{1 - 8 z + 2 z}{1 - 8 z}}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.2.4

Add $- 8 z$ and $2 z$ .

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{\frac{z (- 4 z)}{1 - 8 z}}{\frac{1 - 6 z}{1 - 8 z}}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{\frac{z (- 4 z)}{1 - 8 z}}{\frac{1 - 6 z}{1 - 8 z}}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.3

Simplify the numerator.

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Step 4.2.2.1.3.1

Raise $z$ to the power of $1$ .

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{\frac{- 4 (z \cdot z)}{1 - 8 z}}{\frac{1 - 6 z}{1 - 8 z}}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.3.2

Raise $z$ to the power of $1$ .

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{\frac{- 4 (z \cdot z)}{1 - 8 z}}{\frac{1 - 6 z}{1 - 8 z}}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{\frac{- 4 z^{1 + 1}}{1 - 8 z}}{\frac{1 - 6 z}{1 - 8 z}}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.3.4

Add $1$ and $1$ .

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{\frac{- 4 z^{2}}{1 - 8 z}}{\frac{1 - 6 z}{1 - 8 z}}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{\frac{- 4 z^{2}}{1 - 8 z}}{\frac{1 - 6 z}{1 - 8 z}}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.4

Move the negative in front of the fraction.

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{- \frac{4 z^{2}}{1 - 8 z}}{\frac{1 - 6 z}{1 - 8 z}}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - (- \frac{4 z^{2}}{1 - 8 z} \cdot \frac{1 - 8 z}{1 - 6 z})$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.6

Cancel the common factor of $1 - 8 z$ .

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Step 4.2.2.1.6.1

Move the leading negative in $- \frac{4 z^{2}}{1 - 8 z}$ into the numerator.

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - (\frac{- 4 z^{2}}{1 - 8 z} \cdot \frac{1 - 8 z}{1 - 6 z})$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.6.2

Cancel the common factor.

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - (\frac{- 4 z^{2}}{1 - 8 z} \cdot \frac{1 - 8 z}{1 - 6 z})$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.6.3

Rewrite the expression.

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - (- 4 z^{2} \frac{1}{1 - 6 z})$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - (- 4 z^{2} \frac{1}{1 - 6 z})$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.7

Combine $- 4$ and $\frac{1}{1 - 6 z}$ .

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - (z^{2} (\frac{- 4}{1 - 6 z}))$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.8

Combine $z^{2}$ and $\frac{- 4}{1 - 6 z}$ .

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{z^{2} \cdot - 4}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.9

Simplify the expression.

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Step 4.2.2.1.9.1

Move $- 4$ to the left of $z^{2}$ .

$\frac{2 z (1 - 3 z)}{1 - 6 z} = - \frac{- 4 \cdot z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.9.2

Move the negative in front of the fraction.

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 \cdot z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 \cdot z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.10

Multiply $- - \frac{4 z^{2}}{1 - 6 z}$ .

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Step 4.2.2.1.10.1

Multiply $- 1$ by $- 1$ .

$\frac{2 z (1 - 3 z)}{1 - 6 z} = 1 (\frac{4 z^{2}}{1 - 6 z})$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.2.2.1.10.2

Multiply $\frac{4 z^{2}}{1 - 6 z}$ by $1$ .

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$x = - \frac{y}{1 - 2 y}$

Step 4.3

Replace all occurrences of $y$ in $x = - \frac{y}{1 - 2 y}$ with $- \frac{z}{1 - 8 z}$ .

$x = - \frac{- \frac{z}{1 - 8 z}}{1 - 2 (- \frac{z}{1 - 8 z})}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- \frac{- \frac{z}{1 - 8 z}}{1 - 2 (- \frac{z}{1 - 8 z})}$ .

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Step 4.4.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = - (- \frac{z}{1 - 8 z} \cdot \frac{1}{1 - 2 (- \frac{z}{1 - 8 z})})$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 4.4.1.2

Simplify the denominator.

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Step 4.4.1.2.1

Multiply $- 2 (- \frac{z}{1 - 8 z})$ .

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Step 4.4.1.2.1.1

Multiply $- 1$ by $- 2$ .

$x = - (- \frac{z}{1 - 8 z} \cdot \frac{1}{1 + 2 (\frac{z}{1 - 8 z})})$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 4.4.1.2.1.2

Combine $2$ and $\frac{z}{1 - 8 z}$ .

$x = - (- \frac{z}{1 - 8 z} \cdot \frac{1}{1 + \frac{2 z}{1 - 8 z}})$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$x = - (- \frac{z}{1 - 8 z} \cdot \frac{1}{1 + \frac{2 z}{1 - 8 z}})$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 4.4.1.2.2

Write $1$ as a fraction with a common denominator.

$x = - (- \frac{z}{1 - 8 z} \cdot \frac{1}{\frac{1 - 8 z}{1 - 8 z} + \frac{2 z}{1 - 8 z}})$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 4.4.1.2.3

Combine the numerators over the common denominator.

$x = - (- \frac{z}{1 - 8 z} \cdot \frac{1}{\frac{1 - 8 z + 2 z}{1 - 8 z}})$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 4.4.1.2.4

Add $- 8 z$ and $2 z$ .

$x = - (- \frac{z}{1 - 8 z} \cdot \frac{1}{\frac{1 - 6 z}{1 - 8 z}})$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$x = - (- \frac{z}{1 - 8 z} \cdot \frac{1}{\frac{1 - 6 z}{1 - 8 z}})$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 4.4.1.3

Multiply the numerator by the reciprocal of the denominator.

$x = - (- \frac{z}{1 - 8 z} \cdot (1 (\frac{1 - 8 z}{1 - 6 z})))$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 4.4.1.4

Multiply $\frac{1 - 8 z}{1 - 6 z}$ by $1$ .

$x = - (- \frac{z}{1 - 8 z} \cdot \frac{1 - 8 z}{1 - 6 z})$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 4.4.1.5

Cancel the common factor of $1 - 8 z$ .

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Step 4.4.1.5.1

Move the leading negative in $- \frac{z}{1 - 8 z}$ into the numerator.

$x = - (\frac{- z}{1 - 8 z} \cdot \frac{1 - 8 z}{1 - 6 z})$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 4.4.1.5.2

Cancel the common factor.

$x = - (\frac{- z}{1 - 8 z} \cdot \frac{1 - 8 z}{1 - 6 z})$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 4.4.1.5.3

Rewrite the expression.

$x = - (- z \frac{1}{1 - 6 z})$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$x = - (- z \frac{1}{1 - 6 z})$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 4.4.1.6

Combine $\frac{1}{1 - 6 z}$ and $z$ .

$x = \frac{z}{1 - 6 z}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 4.4.1.7

Multiply $- - \frac{z}{1 - 6 z}$ .

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Step 4.4.1.7.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{z}{1 - 6 z})$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 4.4.1.7.2

Multiply $\frac{z}{1 - 6 z}$ by $1$ .

$x = \frac{z}{1 - 6 z}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$x = \frac{z}{1 - 6 z}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$x = \frac{z}{1 - 6 z}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$x = \frac{z}{1 - 6 z}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$x = \frac{z}{1 - 6 z}$

$\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5

Solve for $z$ in $\frac{2 z (1 - 3 z)}{1 - 6 z} = \frac{4 z^{2}}{1 - 6 z}$ .

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Step 5.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$2 z (1 - 3 z) = 4 z^{2}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.2

Simplify $2 z (1 - 3 z)$ .

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Step 5.2.1

Rewrite.

$0 + 0 + 2 z (1 - 3 z) = 4 z^{2}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.2.2

Simplify by multiplying through.

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Step 5.2.2.1

Apply the distributive property.

$2 z \cdot 1 + 2 z (- 3 z) = 4 z^{2}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.2.2.2

Simplify the expression.

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Step 5.2.2.2.1

Multiply $2$ by $1$ .

$2 z + 2 z (- 3 z) = 4 z^{2}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.2.2.2.2

Rewrite using the commutative property of multiplication.

$2 z + 2 \cdot (- 3 z \cdot z) = 4 z^{2}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$2 z + 2 \cdot (- 3 z \cdot z) = 4 z^{2}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$2 z + 2 \cdot (- 3 z \cdot z) = 4 z^{2}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

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

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Step 5.2.3.1.1

Move $z$ .

$2 z + 2 \cdot (- 3 (z \cdot z)) = 4 z^{2}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.2.3.1.2

Multiply $z$ by $z$ .

$2 z + 2 \cdot (- 3 z^{2}) = 4 z^{2}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$2 z + 2 \cdot (- 3 z^{2}) = 4 z^{2}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.2.3.2

Multiply $2$ by $- 3$ .

$2 z - 6 z^{2} = 4 z^{2}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$2 z - 6 z^{2} = 4 z^{2}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$2 z - 6 z^{2} = 4 z^{2}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.3

Move all terms containing $z$ to the left side of the equation.

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Step 5.3.1

Subtract $4 z^{2}$ from both sides of the equation.

$2 z - 6 z^{2} - 4 z^{2} = 0$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.3.2

Subtract $4 z^{2}$ from $- 6 z^{2}$ .

$2 z - 10 z^{2} = 0$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$2 z - 10 z^{2} = 0$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.4

Factor the left side of the equation.

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Step 5.4.1

Let $u = z$ . Substitute $u$ for all occurrences of $z$ .

$2 u - 10 u^{2} = 0$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.4.2

Factor $2 u$ out of $2 u - 10 u^{2}$ .

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Step 5.4.2.1

Factor $2 u$ out of $2 u$ .

$2 u (1) - 10 u^{2} = 0$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.4.2.2

Factor $2 u$ out of $- 10 u^{2}$ .

$2 u (1) + 2 u (- 5 u) = 0$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.4.2.3

Factor $2 u$ out of $2 u (1) + 2 u (- 5 u)$ .

$2 u (1 - 5 u) = 0$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$2 u (1 - 5 u) = 0$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.4.3

Replace all occurrences of $u$ with $z$ .

$2 z (1 - 5 z) = 0$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$2 z (1 - 5 z) = 0$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.5

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

$z = 0$

$1 - 5 z = 0$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.6

Set $z$ equal to $0$ .

$z = 0$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.7

Set $1 - 5 z$ equal to $0$ and solve for $z$ .

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Step 5.7.1

Set $1 - 5 z$ equal to $0$ .

$1 - 5 z = 0$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.7.2

Solve $1 - 5 z = 0$ for $z$ .

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Step 5.7.2.1

Subtract $1$ from both sides of the equation.

$- 5 z = - 1$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.7.2.2

Divide each term in $- 5 z = - 1$ by $- 5$ and simplify.

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Step 5.7.2.2.1

Divide each term in $- 5 z = - 1$ by $- 5$ .

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

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.7.2.2.2

Simplify the left side.

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Step 5.7.2.2.2.1

Cancel the common factor of $- 5$ .

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Step 5.7.2.2.2.1.1

Cancel the common factor.

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

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.7.2.2.2.1.2

Divide $z$ by $1$ .

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

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

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

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

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

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.7.2.2.3

Simplify the right side.

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Step 5.7.2.2.3.1

Dividing two negative values results in a positive value.

$z = \frac{1}{5}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$z = \frac{1}{5}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$z = \frac{1}{5}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$z = \frac{1}{5}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$z = \frac{1}{5}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 5.8

The final solution is all the values that make $2 z (1 - 5 z) = 0$ true.

$z = 0, \frac{1}{5}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

$z = 0, \frac{1}{5}$

$x = \frac{z}{1 - 6 z}$

$y = - \frac{z}{1 - 8 z}$

Step 6

Replace all occurrences of $z$ with $0$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = \frac{z}{1 - 6 z}$ with $0$ .

$x = \frac{0}{1 - 6 \cdot 0}$

$z = 0$

$y = - \frac{z}{1 - 8 z}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{0}{1 - 6 \cdot 0}$ .

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Step 6.2.1.1

Simplify the denominator.

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Step 6.2.1.1.1

Multiply $- 6$ by $0$ .

$x = \frac{0}{1 + 0}$

$z = 0$

$y = - \frac{z}{1 - 8 z}$

Step 6.2.1.1.2

Add $1$ and $0$ .

$x = \frac{0}{1}$

$z = 0$

$y = - \frac{z}{1 - 8 z}$

$x = \frac{0}{1}$

$z = 0$

$y = - \frac{z}{1 - 8 z}$

Step 6.2.1.2

Divide $0$ by $1$ .

$x = 0$

$z = 0$

$y = - \frac{z}{1 - 8 z}$

$x = 0$

$z = 0$

$y = - \frac{z}{1 - 8 z}$

$x = 0$

$z = 0$

$y = - \frac{z}{1 - 8 z}$

Step 6.3

Replace all occurrences of $z$ in $y = - \frac{z}{1 - 8 z}$ with $0$ .

$y = - \frac{0}{1 - 8 \cdot 0}$

$x = 0$

$z = 0$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- \frac{0}{1 - 8 \cdot 0}$ .

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Step 6.4.1.1

Simplify the denominator.

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Step 6.4.1.1.1

Multiply $- 8$ by $0$ .

$y = - \frac{0}{1 + 0}$

$x = 0$

$z = 0$

Step 6.4.1.1.2

Add $1$ and $0$ .

$y = - \frac{0}{1}$

$x = 0$

$z = 0$

$y = - \frac{0}{1}$

$x = 0$

$z = 0$

Step 6.4.1.2

Simplify the expression.

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Step 6.4.1.2.1

Divide $0$ by $1$ .

$y = - 0$

$x = 0$

$z = 0$

Step 6.4.1.2.2

Multiply $- 1$ by $0$ .

$y = 0$

$x = 0$

$z = 0$

$y = 0$

$x = 0$

$z = 0$

$y = 0$

$x = 0$

$z = 0$

$y = 0$

$x = 0$

$z = 0$

$y = 0$

$x = 0$

$z = 0$

Step 7

Replace all occurrences of $z$ with $\frac{1}{5}$ in each equation.

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Step 7.1

Replace all occurrences of $z$ in $x = \frac{z}{1 - 6 z}$ with $\frac{1}{5}$ .

$x = \frac{\frac{1}{5}}{1 - 6 (\frac{1}{5})}$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{\frac{1}{5}}{1 - 6 (\frac{1}{5})}$ .

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Step 7.2.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{1}{5} \cdot \frac{1}{1 - 6 (\frac{1}{5})}$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

Step 7.2.1.2

Simplify the denominator.

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Step 7.2.1.2.1

Combine $- 6$ and $\frac{1}{5}$ .

$x = \frac{1}{5} \cdot \frac{1}{1 + \frac{- 6}{5}}$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

Step 7.2.1.2.2

Move the negative in front of the fraction.

$x = \frac{1}{5} \cdot \frac{1}{1 - \frac{6}{5}}$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

Step 7.2.1.2.3

Write $1$ as a fraction with a common denominator.

$x = \frac{1}{5} \cdot \frac{1}{\frac{5}{5} - \frac{6}{5}}$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

Step 7.2.1.2.4

Combine the numerators over the common denominator.

$x = \frac{1}{5} \cdot \frac{1}{\frac{5 - 6}{5}}$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

Step 7.2.1.2.5

Subtract $6$ from $5$ .

$x = \frac{1}{5} \cdot \frac{1}{\frac{- 1}{5}}$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

Step 7.2.1.2.6

Move the negative in front of the fraction.

$x = \frac{1}{5} \cdot \frac{1}{- \frac{1}{5}}$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

$x = \frac{1}{5} \cdot \frac{1}{- \frac{1}{5}}$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

Step 7.2.1.3

Reduce the expression by cancelling the common factors.

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Step 7.2.1.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 7.2.1.3.1.1

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

$x = \frac{1}{5} \cdot \frac{- 1 \cdot - 1}{- \frac{1}{5}}$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

Step 7.2.1.3.1.2

Move the negative in front of the fraction.

$x = \frac{1}{5} \cdot (- \frac{1}{\frac{1}{5}})$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

$x = \frac{1}{5} \cdot (- \frac{1}{\frac{1}{5}})$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

Step 7.2.1.3.2

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{1}{5} \cdot (- (1 \cdot 5))$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

Step 7.2.1.3.3

Cancel the common factor of $5$ .

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Step 7.2.1.3.3.1

Factor $5$ out of $- (1 \cdot 5)$ .

$x = \frac{1}{5} \cdot (5 (- 1 \cdot 1))$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

Step 7.2.1.3.3.2

Cancel the common factor.

$x = \frac{1}{5} \cdot (5 (- 1 \cdot 1))$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

Step 7.2.1.3.3.3

Rewrite the expression.

$x = - 1 \cdot 1$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

$x = - 1 \cdot 1$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

Step 7.2.1.3.4

Multiply $- 1$ by $1$ .

$x = - 1$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

$x = - 1$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

$x = - 1$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

$x = - 1$

$z = \frac{1}{5}$

$y = - \frac{z}{1 - 8 z}$

Step 7.3

Replace all occurrences of $z$ in $y = - \frac{z}{1 - 8 z}$ with $\frac{1}{5}$ .

$y = - \frac{\frac{1}{5}}{1 - 8 (\frac{1}{5})}$

$x = - 1$

$z = \frac{1}{5}$

Step 7.4

Simplify the right side.

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Step 7.4.1

Simplify $- \frac{\frac{1}{5}}{1 - 8 (\frac{1}{5})}$ .

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Step 7.4.1.1

Multiply the numerator by the reciprocal of the denominator.

$y = - (\frac{1}{5} \cdot \frac{1}{1 - 8 (\frac{1}{5})})$

$x = - 1$

$z = \frac{1}{5}$

Step 7.4.1.2

Simplify the denominator.

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Step 7.4.1.2.1

Combine $- 8$ and $\frac{1}{5}$ .

$y = - (\frac{1}{5} \cdot \frac{1}{1 + \frac{- 8}{5}})$

$x = - 1$

$z = \frac{1}{5}$

Step 7.4.1.2.2

Move the negative in front of the fraction.

$y = - (\frac{1}{5} \cdot \frac{1}{1 - \frac{8}{5}})$

$x = - 1$

$z = \frac{1}{5}$

Step 7.4.1.2.3

Write $1$ as a fraction with a common denominator.

$y = - (\frac{1}{5} \cdot \frac{1}{\frac{5}{5} - \frac{8}{5}})$

$x = - 1$

$z = \frac{1}{5}$

Step 7.4.1.2.4

Combine the numerators over the common denominator.

$y = - (\frac{1}{5} \cdot \frac{1}{\frac{5 - 8}{5}})$

$x = - 1$

$z = \frac{1}{5}$

Step 7.4.1.2.5

Subtract $8$ from $5$ .

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

$x = - 1$

$z = \frac{1}{5}$

Step 7.4.1.2.6

Move the negative in front of the fraction.

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

$x = - 1$

$z = \frac{1}{5}$

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

$x = - 1$

$z = \frac{1}{5}$

Step 7.4.1.3

Cancel the common factor of $1$ and $- 1$ .

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Step 7.4.1.3.1

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

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

$x = - 1$

$z = \frac{1}{5}$

Step 7.4.1.3.2

Move the negative in front of the fraction.

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

$x = - 1$

$z = \frac{1}{5}$

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

$x = - 1$

$z = \frac{1}{5}$

Step 7.4.1.4

Multiply the numerator by the reciprocal of the denominator.

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

$x = - 1$

$z = \frac{1}{5}$

Step 7.4.1.5

Multiply $\frac{5}{3}$ by $1$ .

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

$x = - 1$

$z = \frac{1}{5}$

Step 7.4.1.6

Cancel the common factor of $5$ .

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Step 7.4.1.6.1

Move the leading negative in $- \frac{5}{3}$ into the numerator.

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

$x = - 1$

$z = \frac{1}{5}$

Step 7.4.1.6.2

Factor $5$ out of $- 5$ .

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

$x = - 1$

$z = \frac{1}{5}$

Step 7.4.1.6.3

Cancel the common factor.

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

$x = - 1$

$z = \frac{1}{5}$

Step 7.4.1.6.4

Rewrite the expression.

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

$x = - 1$

$z = \frac{1}{5}$

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

$x = - 1$

$z = \frac{1}{5}$

Step 7.4.1.7

Move the negative in front of the fraction.

$y = \frac{1}{3}$

$x = - 1$

$z = \frac{1}{5}$

Step 7.4.1.8

Multiply $- - \frac{1}{3}$ .

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Step 7.4.1.8.1

Multiply $- 1$ by $- 1$ .

$y = 1 (\frac{1}{3})$

$x = - 1$

$z = \frac{1}{5}$

Step 7.4.1.8.2

Multiply $\frac{1}{3}$ by $1$ .

$y = \frac{1}{3}$

$x = - 1$

$z = \frac{1}{5}$

$y = \frac{1}{3}$

$x = - 1$

$z = \frac{1}{5}$

$y = \frac{1}{3}$

$x = - 1$

$z = \frac{1}{5}$

$y = \frac{1}{3}$

$x = - 1$

$z = \frac{1}{5}$

$y = \frac{1}{3}$

$x = - 1$

$z = \frac{1}{5}$

Step 8

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(0, 0, 0)$

$(- 1, \frac{1}{3}, \frac{1}{5})$

Step 9

The result can be shown in multiple forms.

Point Form:

$(0, 0, 0), (- 1, \frac{1}{3}, \frac{1}{5})$

Equation Form:

$x = 0, y = 0, z = 0$

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