Solve in Terms of the Arbitrary Variable x

$6 x - y + 4 z = 0$ , $x - 7 y + z = 0$

Step 1

Solve the equation for $x$ .

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Step 1.1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1.1

Add $y$ to both sides of the equation.

$6 x + 4 z = y$

$x - 7 y + z = 0$

Step 1.1.2

Subtract $4 z$ from both sides of the equation.

$6 x = y - 4 z$

$x - 7 y + z = 0$

$6 x = y - 4 z$

$x - 7 y + z = 0$

Step 1.2

Divide each term in $6 x = y - 4 z$ by $6$ and simplify.

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Step 1.2.1

Divide each term in $6 x = y - 4 z$ by $6$ .

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

$x - 7 y + z = 0$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $6$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

$x - 7 y + z = 0$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$x - 7 y + z = 0$

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

$x - 7 y + z = 0$

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

$x - 7 y + z = 0$

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 $- 4$ and $6$ .

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Step 1.2.3.1.1.1

Factor $2$ out of $- 4 z$ .

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

$x - 7 y + z = 0$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $2$ out of $6$ .

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

$x - 7 y + z = 0$

Step 1.2.3.1.1.2.2

Cancel the common factor.

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

$x - 7 y + z = 0$

Step 1.2.3.1.1.2.3

Rewrite the expression.

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

$x - 7 y + z = 0$

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

$x - 7 y + z = 0$

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

$x - 7 y + z = 0$

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

$x - 7 y + z = 0$

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

$x - 7 y + z = 0$

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

$x - 7 y + z = 0$

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

$x - 7 y + z = 0$

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

$x - 7 y + z = 0$

Step 2

Solve the equation for $z$ .

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Step 2.1

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

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Step 2.1.1

To write $- 7 y$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

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

Step 2.1.2

Simplify terms.

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Step 2.1.2.1

Combine $- 7 y$ and $\frac{6}{6}$ .

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

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

Step 2.1.2.2

Combine the numerators over the common denominator.

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

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

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

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

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 $y$ out of $y - 7 y \cdot 6$ .

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Step 2.1.3.1.1.1

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

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

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

Step 2.1.3.1.1.2

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

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

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

Step 2.1.3.1.1.3

Factor $y$ out of $- 7 y \cdot 6$ .

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

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

Step 2.1.3.1.1.4

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

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

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

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

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

Step 2.1.3.1.2

Multiply $- 7$ by $6$ .

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

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

Step 2.1.3.1.3

Subtract $42$ from $1$ .

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

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

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

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

Step 2.1.3.2

Move $- 41$ to the left of $y$ .

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

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

Step 2.1.3.3

Move the negative in front of the fraction.

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

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

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

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

Step 2.1.4

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

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

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

Step 2.1.5

Combine $z$ and $\frac{3}{3}$ .

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

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

Step 2.1.6

Combine the numerators over the common denominator.

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

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

Step 2.1.7

Add $- 2 z$ and $z \cdot 3$ .

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Step 2.1.7.1

Reorder $z$ and $3$ .

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

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

Step 2.1.7.2

Add $- 2 z$ and $3 \cdot z$ .

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

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

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

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

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

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

Step 2.2

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

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

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

Step 2.3

Multiply both sides of the equation by $3$ .

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

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

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 $3$ .

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Step 2.4.1.1.1

Cancel the common factor.

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

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

Step 2.4.1.1.2

Rewrite the expression.

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

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

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

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

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

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

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Cancel the common factor of $3$ .

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Step 2.4.2.1.1

Factor $3$ out of $6$ .

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

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

Step 2.4.2.1.2

Cancel the common factor.

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

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

Step 2.4.2.1.3

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

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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