Examples

6 x - y + 4 z = 0

$6 x - y + 4 z = 0$ ,

x - 7 y + z = 0

$x - 7 y + z = 0$

Step 1

Solve the equation for

x

$x$ .

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Step 1.1

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 1.1.1

Add

y

$y$ to both sides of the equation.

6 x + 4 z = y

$6 x + 4 z = y$

x - 7 y + z = 0

$x - 7 y + z = 0$

Step 1.1.2

Subtract

4 z

$4 z$ from both sides of the equation.

6 x = y - 4 z

$6 x = y - 4 z$

x - 7 y + z = 0

$x - 7 y + z = 0$

6 x = y - 4 z

$6 x = y - 4 z$

x - 7 y + z = 0

$x - 7 y + z = 0$

Step 1.2

Divide each term in

6 x = y - 4 z

$6 x = y - 4 z$ by

6

$6$ and simplify.

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Step 1.2.1

Divide each term in

6 x = y - 4 z

$6 x = y - 4 z$ by

6

$6$ .

\frac{6 x}{6} = \frac{y}{6} + \frac{- 4 z}{6}

$\frac{6 x}{6} = \frac{y}{6} + \frac{- 4 z}{6}$

x - 7 y + z = 0

$x - 7 y + z = 0$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of

6

$6$ .

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Step 1.2.2.1.1

Cancel the common factor.

\frac{6 x}{6} = \frac{y}{6} + \frac{- 4 z}{6}

$\frac{6 x}{6} = \frac{y}{6} + \frac{- 4 z}{6}$

x - 7 y + z = 0

$x - 7 y + z = 0$

Step 1.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{y}{6} + \frac{- 4 z}{6}

$x = \frac{y}{6} + \frac{- 4 z}{6}$

x - 7 y + z = 0

$x - 7 y + z = 0$

x = \frac{y}{6} + \frac{- 4 z}{6}

$x = \frac{y}{6} + \frac{- 4 z}{6}$

x - 7 y + z = 0

$x - 7 y + z = 0$

x = \frac{y}{6} + \frac{- 4 z}{6}

$x = \frac{y}{6} + \frac{- 4 z}{6}$

x - 7 y + z = 0

$x - 7 y + z = 0$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Cancel the common factor of

- 4

$- 4$ and

6

$6$ .

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Step 1.2.3.1.1.1

Factor

2

$2$ out of

- 4 z

$- 4 z$ .

x = \frac{y}{6} + \frac{2 (- 2 z)}{6}

$x = \frac{y}{6} + \frac{2 (- 2 z)}{6}$

x - 7 y + z = 0

$x - 7 y + z = 0$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor

2

$2$ out of

6

$6$ .

x = \frac{y}{6} + \frac{2 (- 2 z)}{2 (3)}

$x = \frac{y}{6} + \frac{2 (- 2 z)}{2 (3)}$

x - 7 y + z = 0

$x - 7 y + z = 0$

Step 1.2.3.1.1.2.2

Cancel the common factor.

x = \frac{y}{6} + \frac{2 (- 2 z)}{2 \cdot 3}

$x = \frac{y}{6} + \frac{2 (- 2 z)}{2 \cdot 3}$

x - 7 y + z = 0

$x - 7 y + z = 0$

Step 1.2.3.1.1.2.3

Rewrite the expression.

x = \frac{y}{6} + \frac{- 2 z}{3}

$x = \frac{y}{6} + \frac{- 2 z}{3}$

x - 7 y + z = 0

$x - 7 y + z = 0$

x = \frac{y}{6} + \frac{- 2 z}{3}

$x = \frac{y}{6} + \frac{- 2 z}{3}$

x - 7 y + z = 0

$x - 7 y + z = 0$

x = \frac{y}{6} + \frac{- 2 z}{3}

$x = \frac{y}{6} + \frac{- 2 z}{3}$

x - 7 y + z = 0

$x - 7 y + z = 0$

Step 1.2.3.1.2

Move the negative in front of the fraction.

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

x - 7 y + z = 0

$x - 7 y + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

x - 7 y + z = 0

$x - 7 y + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

x - 7 y + z = 0

$x - 7 y + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

x - 7 y + z = 0

$x - 7 y + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

x - 7 y + z = 0

$x - 7 y + z = 0$

Step 2

Solve the equation for

z

$z$ .

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Step 2.1

Simplify

(\frac{y}{6} - \frac{2 z}{3}) - 7 y + z

$(\frac{y}{6} - \frac{2 z}{3}) - 7 y + z$ .

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Step 2.1.1

To write

- 7 y

$- 7 y$ as a fraction with a common denominator, multiply by

\frac{6}{6}

$\frac{6}{6}$ .

- \frac{2 z}{3} + \frac{y}{6} - 7 y \cdot \frac{6}{6} + z = 0

$- \frac{2 z}{3} + \frac{y}{6} - 7 y \cdot \frac{6}{6} + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.1.2

Simplify terms.

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Step 2.1.2.1

Combine

- 7 y

$- 7 y$ and

\frac{6}{6}

$\frac{6}{6}$ .

- \frac{2 z}{3} + \frac{y}{6} + \frac{- 7 y \cdot 6}{6} + z = 0

$- \frac{2 z}{3} + \frac{y}{6} + \frac{- 7 y \cdot 6}{6} + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.1.2.2

Combine the numerators over the common denominator.

- \frac{2 z}{3} + \frac{y - 7 y \cdot 6}{6} + z = 0

$- \frac{2 z}{3} + \frac{y - 7 y \cdot 6}{6} + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

- \frac{2 z}{3} + \frac{y - 7 y \cdot 6}{6} + z = 0

$- \frac{2 z}{3} + \frac{y - 7 y \cdot 6}{6} + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.1.3

Simplify each term.

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Step 2.1.3.1

Simplify the numerator.

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Step 2.1.3.1.1

Factor

y

$y$ out of

y - 7 y \cdot 6

$y - 7 y \cdot 6$ .

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Step 2.1.3.1.1.1

Raise

y

$y$ to the power of

1

$1$ .

- \frac{2 z}{3} + \frac{y - 7 y \cdot 6}{6} + z = 0

$- \frac{2 z}{3} + \frac{y - 7 y \cdot 6}{6} + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.1.3.1.1.2

Factor

y

$y$ out of

y^{1}

$y^{1}$ .

- \frac{2 z}{3} + \frac{y \cdot 1 - 7 y \cdot 6}{6} + z = 0

$- \frac{2 z}{3} + \frac{y \cdot 1 - 7 y \cdot 6}{6} + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.1.3.1.1.3

Factor

y

$y$ out of

- 7 y \cdot 6

$- 7 y \cdot 6$ .

- \frac{2 z}{3} + \frac{y \cdot 1 + y (- 7 \cdot 6)}{6} + z = 0

$- \frac{2 z}{3} + \frac{y \cdot 1 + y (- 7 \cdot 6)}{6} + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.1.3.1.1.4

Factor

y

$y$ out of

y \cdot 1 + y (- 7 \cdot 6)

$y \cdot 1 + y (- 7 \cdot 6)$ .

- \frac{2 z}{3} + \frac{y (1 - 7 \cdot 6)}{6} + z = 0

$- \frac{2 z}{3} + \frac{y (1 - 7 \cdot 6)}{6} + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

- \frac{2 z}{3} + \frac{y (1 - 7 \cdot 6)}{6} + z = 0

$- \frac{2 z}{3} + \frac{y (1 - 7 \cdot 6)}{6} + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.1.3.1.2

Multiply

- 7

$- 7$ by

6

$6$ .

- \frac{2 z}{3} + \frac{y (1 - 42)}{6} + z = 0

$- \frac{2 z}{3} + \frac{y (1 - 42)}{6} + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.1.3.1.3

Subtract

42

$42$ from

1

$1$ .

- \frac{2 z}{3} + \frac{y \cdot - 41}{6} + z = 0

$- \frac{2 z}{3} + \frac{y \cdot - 41}{6} + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

- \frac{2 z}{3} + \frac{y \cdot - 41}{6} + z = 0

$- \frac{2 z}{3} + \frac{y \cdot - 41}{6} + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.1.3.2

Move

- 41

$- 41$ to the left of

y

$y$ .

- \frac{2 z}{3} + \frac{- 41 \cdot y}{6} + z = 0

$- \frac{2 z}{3} + \frac{- 41 \cdot y}{6} + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.1.3.3

Move the negative in front of the fraction.

- \frac{2 z}{3} - \frac{41 y}{6} + z = 0

$- \frac{2 z}{3} - \frac{41 y}{6} + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

- \frac{2 z}{3} - \frac{41 y}{6} + z = 0

$- \frac{2 z}{3} - \frac{41 y}{6} + z = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.1.4

To write

z

$z$ as a fraction with a common denominator, multiply by

\frac{3}{3}

$\frac{3}{3}$ .

- \frac{41 y}{6} - \frac{2 z}{3} + z \cdot \frac{3}{3} = 0

$- \frac{41 y}{6} - \frac{2 z}{3} + z \cdot \frac{3}{3} = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.1.5

Combine

z

$z$ and

\frac{3}{3}

$\frac{3}{3}$ .

- \frac{41 y}{6} - \frac{2 z}{3} + \frac{z \cdot 3}{3} = 0

$- \frac{41 y}{6} - \frac{2 z}{3} + \frac{z \cdot 3}{3} = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.1.6

Combine the numerators over the common denominator.

- \frac{41 y}{6} + \frac{- 2 z + z \cdot 3}{3} = 0

$- \frac{41 y}{6} + \frac{- 2 z + z \cdot 3}{3} = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.1.7

Add

- 2 z

$- 2 z$ and

z \cdot 3

$z \cdot 3$ .

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Step 2.1.7.1

Reorder

z

$z$ and

3

$3$ .

- \frac{41 y}{6} + \frac{- 2 z + 3 \cdot z}{3} = 0

$- \frac{41 y}{6} + \frac{- 2 z + 3 \cdot z}{3} = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.1.7.2

Add

- 2 z

$- 2 z$ and

3 \cdot z

$3 \cdot z$ .

- \frac{41 y}{6} + \frac{z}{3} = 0

$- \frac{41 y}{6} + \frac{z}{3} = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

- \frac{41 y}{6} + \frac{z}{3} = 0

$- \frac{41 y}{6} + \frac{z}{3} = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

- \frac{41 y}{6} + \frac{z}{3} = 0

$- \frac{41 y}{6} + \frac{z}{3} = 0$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.2

Add

\frac{41 y}{6}

$\frac{41 y}{6}$ to both sides of the equation.

\frac{z}{3} = \frac{41 y}{6}

$\frac{z}{3} = \frac{41 y}{6}$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.3

Multiply both sides of the equation by

3

$3$ .

3 (\frac{z}{3}) = 3 (\frac{41 y}{6})

$3 (\frac{z}{3}) = 3 (\frac{41 y}{6})$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.4

Simplify both sides of the equation.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Cancel the common factor of

3

$3$ .

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Step 2.4.1.1.1

Cancel the common factor.

3 (\frac{z}{3}) = 3 (\frac{41 y}{6})

$3 (\frac{z}{3}) = 3 (\frac{41 y}{6})$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.4.1.1.2

Rewrite the expression.

z = 3 (\frac{41 y}{6})

$z = 3 (\frac{41 y}{6})$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

z = 3 (\frac{41 y}{6})

$z = 3 (\frac{41 y}{6})$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

z = 3 (\frac{41 y}{6})

$z = 3 (\frac{41 y}{6})$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Cancel the common factor of

3

$3$ .

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Step 2.4.2.1.1

Factor

3

$3$ out of

6

$6$ .

z = 3 (\frac{41 y}{3 (2)})

$z = 3 (\frac{41 y}{3 (2)})$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.4.2.1.2

Cancel the common factor.

z = 3 (\frac{41 y}{3 \cdot 2})

$z = 3 (\frac{41 y}{3 \cdot 2})$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 2.4.2.1.3

Rewrite the expression.

z = \frac{41 y}{2}

$z = \frac{41 y}{2}$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

z = \frac{41 y}{2}

$z = \frac{41 y}{2}$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

z = \frac{41 y}{2}

$z = \frac{41 y}{2}$

x = \frac{y}{6} - \frac{2 z}{3}

$x = \frac{y}{6} - \frac{2 z}{3}$

$z = \frac{41 y}{2}$

$x = \frac{y}{6} - \frac{2 z}{3}$

$z = \frac{41 y}{2}$

$x = \frac{y}{6} - \frac{2 z}{3}$

Step 3

Simplify the right side.

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Step 3.1

Simplify

$\frac{y}{6} - \frac{2 (\frac{41 y}{2})}{3}$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Combine

$2$ and

$\frac{41 y}{2}$ .

$x = \frac{y}{6} - \frac{\frac{2 (41 y)}{2}}{3}$

$z = \frac{41 y}{2}$

Step 3.1.1.2

Multiply

$2$ by

$41$ .

$x = \frac{y}{6} - \frac{\frac{82 y}{2}}{3}$

$z = \frac{41 y}{2}$

Step 3.1.1.3

Reduce the expression by cancelling the common factors.

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Step 3.1.1.3.1

Reduce the expression

$\frac{82 y}{2}$ by cancelling the common factors.

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Step 3.1.1.3.1.1

Factor

$2$ out of

$82 y$ .

$x = \frac{y}{6} - \frac{\frac{2 (41 y)}{2}}{3}$

$z = \frac{41 y}{2}$

Step 3.1.1.3.1.2

Factor

$2$ out of

$2$ .

$x = \frac{y}{6} - \frac{\frac{2 (41 y)}{2 (1)}}{3}$

$z = \frac{41 y}{2}$

Step 3.1.1.3.1.3

Cancel the common factor.

$x = \frac{y}{6} - \frac{\frac{2 (41 y)}{2 \cdot 1}}{3}$

$z = \frac{41 y}{2}$

Step 3.1.1.3.1.4

Rewrite the expression.

$x = \frac{y}{6} - \frac{\frac{41 y}{1}}{3}$

$z = \frac{41 y}{2}$

$x = \frac{y}{6} - \frac{\frac{41 y}{1}}{3}$

$z = \frac{41 y}{2}$

Step 3.1.1.3.2

Divide

$41 y$ by

$1$ .

$x = \frac{y}{6} - \frac{41 y}{3}$

$z = \frac{41 y}{2}$

$x = \frac{y}{6} - \frac{41 y}{3}$

$z = \frac{41 y}{2}$

$x = \frac{y}{6} - \frac{41 y}{3}$

$z = \frac{41 y}{2}$

Step 3.1.2

To write

$- \frac{41 y}{3}$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$x = \frac{y}{6} - \frac{41 y}{3} \cdot \frac{2}{2}$

$z = \frac{41 y}{2}$

Step 3.1.3

Write each expression with a common denominator of

$6$ , by multiplying each by an appropriate factor of

$1$ .

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Step 3.1.3.1

Multiply

$\frac{41 y}{3}$ by

$\frac{2}{2}$ .

$x = \frac{y}{6} - \frac{41 y \cdot 2}{3 \cdot 2}$

$z = \frac{41 y}{2}$

Step 3.1.3.2

Multiply

$3$ by

$2$ .

$x = \frac{y}{6} - \frac{41 y \cdot 2}{6}$

$z = \frac{41 y}{2}$

$x = \frac{y}{6} - \frac{41 y \cdot 2}{6}$

$z = \frac{41 y}{2}$

Step 3.1.4

Combine the numerators over the common denominator.

$x = \frac{y - 41 y \cdot 2}{6}$

$z = \frac{41 y}{2}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Factor

$y$ out of

$y - 41 y \cdot 2$ .

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Step 3.1.5.1.1

Raise

$y$ to the power of

$1$ .

$x = \frac{y - 41 y \cdot 2}{6}$

$z = \frac{41 y}{2}$

Step 3.1.5.1.2

Factor

$y$ out of

$y^{1}$ .

$x = \frac{y \cdot 1 - 41 y \cdot 2}{6}$

$z = \frac{41 y}{2}$

Step 3.1.5.1.3

Factor

$y$ out of

$- 41 y \cdot 2$ .

$x = \frac{y \cdot 1 + y (- 41 \cdot 2)}{6}$

$z = \frac{41 y}{2}$

Step 3.1.5.1.4

Factor

$y$ out of

$y \cdot 1 + y (- 41 \cdot 2)$ .

$x = \frac{y (1 - 41 \cdot 2)}{6}$

$z = \frac{41 y}{2}$

$x = \frac{y (1 - 41 \cdot 2)}{6}$

$z = \frac{41 y}{2}$

Step 3.1.5.2

Multiply

$- 41$ by

$2$ .

$x = \frac{y (1 - 82)}{6}$

$z = \frac{41 y}{2}$

Step 3.1.5.3

Subtract

$82$ from

$1$ .

$x = \frac{y \cdot - 81}{6}$

$z = \frac{41 y}{2}$

$x = \frac{y \cdot - 81}{6}$

$z = \frac{41 y}{2}$

Step 3.1.6

Reduce the expression by cancelling the common factors.

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Step 3.1.6.1

Cancel the common factor of

$- 81$ and

$6$ .

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Step 3.1.6.1.1

Factor

$3$ out of

$y \cdot - 81$ .

$x = \frac{3 (y \cdot - 27)}{6}$

$z = \frac{41 y}{2}$

Step 3.1.6.1.2

Cancel the common factors.

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Step 3.1.6.1.2.1

Factor

$3$ out of

$6$ .

$x = \frac{3 (y \cdot - 27)}{3 (2)}$

$z = \frac{41 y}{2}$

Step 3.1.6.1.2.2

Cancel the common factor.

$x = \frac{3 (y \cdot - 27)}{3 \cdot 2}$

$z = \frac{41 y}{2}$

Step 3.1.6.1.2.3

Rewrite the expression.

$x = \frac{y \cdot - 27}{2}$

$z = \frac{41 y}{2}$

$x = \frac{y \cdot - 27}{2}$

$z = \frac{41 y}{2}$

$x = \frac{y \cdot - 27}{2}$

$z = \frac{41 y}{2}$

Step 3.1.6.2

Simplify the expression.

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Step 3.1.6.2.1

Move

$- 27$ to the left of

$y$ .

$x = \frac{- 27 \cdot y}{2}$

$z = \frac{41 y}{2}$

Step 3.1.6.2.2

Move the negative in front of the fraction.

$x = - \frac{27 y}{2}$

$z = \frac{41 y}{2}$

$x = - \frac{27 y}{2}$

$z = \frac{41 y}{2}$

$x = - \frac{27 y}{2}$

$z = \frac{41 y}{2}$

$x = - \frac{27 y}{2}$

$z = \frac{41 y}{2}$

$x = - \frac{27 y}{2}$

$z = \frac{41 y}{2}$

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