Algebra Examples

$x + 2 y + 3 z = 0$ $2 x + y + 3 z = 0$ $3 x + 2 z = 1$

Step 1

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

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Step 1.1

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

$x + 3 z = - 2 y$

$2 x + y + 3 z = 0$

$3 x + 2 z = 1$

Step 1.2

Subtract $3 z$ from both sides of the equation.

$x = - 2 y - 3 z$

$2 x + y + 3 z = 0$

$3 x + 2 z = 1$

$x = - 2 y - 3 z$

$2 x + y + 3 z = 0$

$3 x + 2 z = 1$

Step 2

Replace all occurrences of $x$ with $- 2 y - 3 z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 x + y + 3 z = 0$ with $- 2 y - 3 z$ .

$2 (- 2 y - 3 z) + y + 3 z = 0$

$x = - 2 y - 3 z$

$3 x + 2 z = 1$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (- 2 y - 3 z) + y + 3 z$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$2 (- 2 y) + 2 (- 3 z) + y + 3 z = 0$

$x = - 2 y - 3 z$

$3 x + 2 z = 1$

Step 2.2.1.1.2

Multiply $- 2$ by $2$ .

$- 4 y + 2 (- 3 z) + y + 3 z = 0$

$x = - 2 y - 3 z$

$3 x + 2 z = 1$

Step 2.2.1.1.3

Multiply $- 3$ by $2$ .

$- 4 y - 6 z + y + 3 z = 0$

$x = - 2 y - 3 z$

$3 x + 2 z = 1$

$- 4 y - 6 z + y + 3 z = 0$

$x = - 2 y - 3 z$

$3 x + 2 z = 1$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $- 4 y$ and $y$ .

$- 3 y - 6 z + 3 z = 0$

$x = - 2 y - 3 z$

$3 x + 2 z = 1$

Step 2.2.1.2.2

Add $- 6 z$ and $3 z$ .

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

$3 x + 2 z = 1$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

$3 x + 2 z = 1$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

$3 x + 2 z = 1$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

$3 x + 2 z = 1$

Step 2.3

Replace all occurrences of $x$ in $3 x + 2 z = 1$ with $- 2 y - 3 z$ .

$3 (- 2 y - 3 z) + 2 z = 1$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $3 (- 2 y - 3 z) + 2 z$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Apply the distributive property.

$3 (- 2 y) + 3 (- 3 z) + 2 z = 1$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

Step 2.4.1.1.2

Multiply $- 2$ by $3$ .

$- 6 y + 3 (- 3 z) + 2 z = 1$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

Step 2.4.1.1.3

Multiply $- 3$ by $3$ .

$- 6 y - 9 z + 2 z = 1$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

$- 6 y - 9 z + 2 z = 1$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

Step 2.4.1.2

Add $- 9 z$ and $2 z$ .

$- 6 y - 7 z = 1$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

$- 6 y - 7 z = 1$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

$- 6 y - 7 z = 1$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

$- 6 y - 7 z = 1$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

Step 3

Solve for $y$ in $- 6 y - 7 z = 1$ .

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Step 3.1

Add $7 z$ to both sides of the equation.

$- 6 y = 1 + 7 z$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

Step 3.2

Divide each term in $- 6 y = 1 + 7 z$ by $- 6$ and simplify.

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Step 3.2.1

Divide each term in $- 6 y = 1 + 7 z$ by $- 6$ .

$\frac{- 6 y}{- 6} = \frac{1}{- 6} + \frac{7 z}{- 6}$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 6$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 6 y}{- 6} = \frac{1}{- 6} + \frac{7 z}{- 6}$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{1}{- 6} + \frac{7 z}{- 6}$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

$y = \frac{1}{- 6} + \frac{7 z}{- 6}$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

$y = \frac{1}{- 6} + \frac{7 z}{- 6}$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{1}{6} + \frac{7 z}{- 6}$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

Step 3.2.3.1.2

Move the negative in front of the fraction.

$y = - \frac{1}{6} - \frac{7 z}{6}$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$- 3 y - 3 z = 0$

$x = - 2 y - 3 z$

Step 4

Replace all occurrences of $y$ with $- \frac{1}{6} - \frac{7 z}{6}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $- 3 y - 3 z = 0$ with $- \frac{1}{6} - \frac{7 z}{6}$ .

$- 3 (- \frac{1}{6} - \frac{7 z}{6}) - 3 z = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 3 (- \frac{1}{6} - \frac{7 z}{6}) - 3 z$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$- 3 (- \frac{1}{6}) - 3 (- \frac{7 z}{6}) - 3 z = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.1.2

Cancel the common factor of $3$ .

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Step 4.2.1.1.2.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$- 3 (\frac{- 1}{6}) - 3 (- \frac{7 z}{6}) - 3 z = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.1.2.2

Factor $3$ out of $- 3$ .

$3 (- 1) (\frac{- 1}{6}) - 3 (- \frac{7 z}{6}) - 3 z = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.1.2.3

Factor $3$ out of $6$ .

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.1.2.4

Cancel the common factor.

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.1.2.5

Rewrite the expression.

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.1.3

Cancel the common factor of $3$ .

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Step 4.2.1.1.3.1

Move the leading negative in $- \frac{7 z}{6}$ into the numerator.

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.1.3.2

Factor $3$ out of $- 3$ .

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.1.3.3

Factor $3$ out of $6$ .

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.1.3.4

Cancel the common factor.

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.1.3.5

Rewrite the expression.

$- 1 (\frac{- 1}{2}) - 1 \frac{- 7 z}{2} - 3 z = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

$- 1 (\frac{- 1}{2}) - 1 \frac{- 7 z}{2} - 3 z = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.1.4

Simplify each term.

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Step 4.2.1.1.4.1

Move the negative in front of the fraction.

$- 1 (- \frac{1}{2}) - 1 \frac{- 7 z}{2} - 3 z = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.1.4.2

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

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Step 4.2.1.1.4.2.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{2}) - 1 \frac{- 7 z}{2} - 3 z = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.1.4.2.2

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2} - 1 \frac{- 7 z}{2} - 3 z = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

$\frac{1}{2} - 1 \frac{- 7 z}{2} - 3 z = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.1.4.3

Move the negative in front of the fraction.

$\frac{1}{2} - 1 (- \frac{7 z}{2}) - 3 z = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.1.4.4

Multiply $- 1 (- \frac{7 z}{2})$ .

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Step 4.2.1.1.4.4.1

Multiply $- 1$ by $- 1$ .

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.1.4.4.2

Multiply $\frac{7 z}{2}$ by $1$ .

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.2

To write $- 3 z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.3

Combine $- 3 z$ and $\frac{2}{2}$ .

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.4

Combine the numerators over the common denominator.

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.5

Combine the numerators over the common denominator.

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.6

Multiply $2$ by $- 3$ .

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

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.2.1.7

Subtract $6 z$ from $7 z$ .

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 2 y - 3 z$

Step 4.3

Replace all occurrences of $y$ in $x = - 2 y - 3 z$ with $- \frac{1}{6} - \frac{7 z}{6}$ .

$x = - 2 (- \frac{1}{6} - \frac{7 z}{6}) - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- 2 (- \frac{1}{6} - \frac{7 z}{6}) - 3 z$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$x = - 2 (- \frac{1}{6}) - 2 (- \frac{7 z}{6}) - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.1.2

Cancel the common factor of $2$ .

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Step 4.4.1.1.2.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$x = - 2 (\frac{- 1}{6}) - 2 (- \frac{7 z}{6}) - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.1.2.2

Factor $2$ out of $- 2$ .

$x = 2 (- 1) (\frac{- 1}{6}) - 2 (- \frac{7 z}{6}) - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.1.2.3

Factor $2$ out of $6$ .

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

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.1.2.4

Cancel the common factor.

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

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.1.2.5

Rewrite the expression.

$x = - 1 (\frac{- 1}{3}) - 2 (- \frac{7 z}{6}) - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 1 (\frac{- 1}{3}) - 2 (- \frac{7 z}{6}) - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.1.3

Cancel the common factor of $2$ .

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Step 4.4.1.1.3.1

Move the leading negative in $- \frac{7 z}{6}$ into the numerator.

$x = - 1 (\frac{- 1}{3}) - 2 \frac{- 7 z}{6} - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.1.3.2

Factor $2$ out of $- 2$ .

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

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.1.3.3

Factor $2$ out of $6$ .

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

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.1.3.4

Cancel the common factor.

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

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.1.3.5

Rewrite the expression.

$x = - 1 (\frac{- 1}{3}) - 1 \frac{- 7 z}{3} - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = - 1 (\frac{- 1}{3}) - 1 \frac{- 7 z}{3} - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.1.4

Simplify each term.

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Step 4.4.1.1.4.1

Move the negative in front of the fraction.

$x = - 1 (- \frac{1}{3}) - 1 \frac{- 7 z}{3} - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.1.4.2

Multiply $- 1 (- \frac{1}{3})$ .

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Step 4.4.1.1.4.2.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{1}{3}) - 1 \frac{- 7 z}{3} - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.1.4.2.2

Multiply $\frac{1}{3}$ by $1$ .

$x = \frac{1}{3} - 1 \frac{- 7 z}{3} - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = \frac{1}{3} - 1 \frac{- 7 z}{3} - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.1.4.3

Move the negative in front of the fraction.

$x = \frac{1}{3} - 1 (- \frac{7 z}{3}) - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.1.4.4

Multiply $- 1 (- \frac{7 z}{3})$ .

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Step 4.4.1.1.4.4.1

Multiply $- 1$ by $- 1$ .

$x = \frac{1}{3} + 1 (\frac{7 z}{3}) - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.1.4.4.2

Multiply $\frac{7 z}{3}$ by $1$ .

$x = \frac{1}{3} + \frac{7 z}{3} - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = \frac{1}{3} + \frac{7 z}{3} - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = \frac{1}{3} + \frac{7 z}{3} - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = \frac{1}{3} + \frac{7 z}{3} - 3 z$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.2

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

$x = \frac{1}{3} + \frac{7 z}{3} - 3 z \cdot \frac{3}{3}$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.3

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

$x = \frac{1}{3} + \frac{7 z}{3} + \frac{- 3 z \cdot 3}{3}$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.4

Combine the numerators over the common denominator.

$x = \frac{1}{3} + \frac{7 z - 3 z \cdot 3}{3}$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.5

Combine the numerators over the common denominator.

$x = \frac{1 + 7 z - 3 z \cdot 3}{3}$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.6

Multiply $3$ by $- 3$ .

$x = \frac{1 + 7 z - 9 z}{3}$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 4.4.1.7

Subtract $9 z$ from $7 z$ .

$x = \frac{1 - 2 z}{3}$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = \frac{1 - 2 z}{3}$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = \frac{1 - 2 z}{3}$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = \frac{1 - 2 z}{3}$

$\frac{1 + z}{2} = 0$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 5

Solve for $z$ in $\frac{1 + z}{2} = 0$ .

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Step 5.1

Set the numerator equal to zero.

$1 + z = 0$

$x = \frac{1 - 2 z}{3}$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 5.2

Subtract $1$ from both sides of the equation.

$z = - 1$

$x = \frac{1 - 2 z}{3}$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$z = - 1$

$x = \frac{1 - 2 z}{3}$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 6

Replace all occurrences of $z$ with $- 1$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = \frac{1 - 2 z}{3}$ with $- 1$ .

$x = \frac{1 - 2 \cdot - 1}{3}$

$z = - 1$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{1 - 2 \cdot - 1}{3}$ .

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Step 6.2.1.1

Simplify the numerator.

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Step 6.2.1.1.1

Multiply $- 2$ by $- 1$ .

$x = \frac{1 + 2}{3}$

$z = - 1$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 6.2.1.1.2

Add $1$ and $2$ .

$x = \frac{3}{3}$

$z = - 1$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = \frac{3}{3}$

$z = - 1$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 6.2.1.2

Divide $3$ by $3$ .

$x = 1$

$z = - 1$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = 1$

$z = - 1$

$y = - \frac{1}{6} - \frac{7 z}{6}$

$x = 1$

$z = - 1$

$y = - \frac{1}{6} - \frac{7 z}{6}$

Step 6.3

Replace all occurrences of $z$ in $y = - \frac{1}{6} - \frac{7 z}{6}$ with $- 1$ .

$y = - \frac{1}{6} - \frac{7 (- 1)}{6}$

$x = 1$

$z = - 1$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- \frac{1}{6} - \frac{7 (- 1)}{6}$ .

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Step 6.4.1.1

Combine the numerators over the common denominator.

$y = \frac{- 1 - 7 \cdot - 1}{6}$

$x = 1$

$z = - 1$

Step 6.4.1.2

Simplify the expression.

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Step 6.4.1.2.1

Multiply $- 7$ by $- 1$ .

$y = \frac{- 1 + 7}{6}$

$x = 1$

$z = - 1$

Step 6.4.1.2.2

Add $- 1$ and $7$ .

$y = \frac{6}{6}$

$x = 1$

$z = - 1$

Step 6.4.1.2.3

Divide $6$ by $6$ .

$y = 1$

$x = 1$

$z = - 1$

$y = 1$

$x = 1$

$z = - 1$

$y = 1$

$x = 1$

$z = - 1$

$y = 1$

$x = 1$

$z = - 1$

$y = 1$

$x = 1$

$z = - 1$

Step 7

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(1, 1, - 1)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(1, 1, - 1)$

Equation Form:

$x = 1, y = 1, z = - 1$