Algebra Examples

$x + 2 y + z = 6$ $- \frac{1}{2} x + 5 y + \frac{1}{3} z = 8$ $2 x + 10 y - 3 z = - 12$

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.

$x + z = 6 - 2 y$

$- \frac{1}{2} x + 5 y + \frac{1}{3} z = 8$

$2 x + 10 y - 3 z = - 12$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = 6 - 2 y - z$

$- \frac{1}{2} x + 5 y + \frac{1}{3} z = 8$

$2 x + 10 y - 3 z = - 12$

$x = 6 - 2 y - z$

$- \frac{1}{2} x + 5 y + \frac{1}{3} z = 8$

$2 x + 10 y - 3 z = - 12$

Step 2

Replace all occurrences of $x$ with $6 - 2 y - z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- \frac{1}{2} x + 5 y + \frac{1}{3} z = 8$ with $6 - 2 y - z$ .

$- \frac{1}{2} \cdot (6 - 2 y - z) + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- \frac{1}{2} \cdot (6 - 2 y - z) + 5 y + \frac{1}{3} z$ .

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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.

$- \frac{1}{2} \cdot 6 - \frac{1}{2} \cdot (- 2 y) - \frac{1}{2} \cdot (- z) + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1.1

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

$\frac{- 1}{2} \cdot 6 - \frac{1}{2} \cdot (- 2 y) - \frac{1}{2} \cdot (- z) + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.1.2.1.2

Factor $2$ out of $6$ .

$\frac{- 1}{2} \cdot (2 (3)) - \frac{1}{2} \cdot (- 2 y) - \frac{1}{2} \cdot (- z) + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.1.2.1.3

Cancel the common factor.

$\frac{- 1}{2} \cdot (2 \cdot 3) - \frac{1}{2} \cdot (- 2 y) - \frac{1}{2} \cdot (- z) + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.1.2.1.4

Rewrite the expression.

$- 1 \cdot 3 - \frac{1}{2} \cdot (- 2 y) - \frac{1}{2} \cdot (- z) + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

$- 1 \cdot 3 - \frac{1}{2} \cdot (- 2 y) - \frac{1}{2} \cdot (- z) + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.1.2.2

Multiply $- 1$ by $3$ .

$- 3 - \frac{1}{2} \cdot (- 2 y) - \frac{1}{2} \cdot (- z) + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.1.2.3

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.3.1

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

$- 3 + \frac{- 1}{2} \cdot (- 2 y) - \frac{1}{2} \cdot (- z) + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.1.2.3.2

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

$- 3 + \frac{- 1}{2} \cdot (2 (- y)) - \frac{1}{2} \cdot (- z) + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.1.2.3.3

Cancel the common factor.

$- 3 + \frac{- 1}{2} \cdot (2 (- y)) - \frac{1}{2} \cdot (- z) + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.1.2.3.4

Rewrite the expression.

$- 3 - 1 (- y) - \frac{1}{2} \cdot (- z) + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

$- 3 - 1 (- y) - \frac{1}{2} \cdot (- z) + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.1.2.4

Multiply $- 1$ by $- 1$ .

$- 3 + 1 y - \frac{1}{2} \cdot (- z) + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.1.2.5

Multiply $y$ by $1$ .

$- 3 + y - \frac{1}{2} \cdot (- z) + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.1.2.6

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

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Step 2.2.1.1.2.6.1

Multiply $- 1$ by $- 1$ .

$- 3 + y + 1 (\frac{1}{2}) \cdot z + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.1.2.6.2

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

$- 3 + y + \frac{1}{2} z + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.1.2.6.3

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

$- 3 + y + \frac{z}{2} + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

$- 3 + y + \frac{z}{2} + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

$- 3 + y + \frac{z}{2} + 5 y + \frac{1}{3} z = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.1.3

Combine $\frac{1}{3}$ and $z$ .

$- 3 + y + \frac{z}{2} + 5 y + \frac{z}{3} = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

$- 3 + y + \frac{z}{2} + 5 y + \frac{z}{3} = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.2

Add $y$ and $5 y$ .

$- 3 + 6 y + \frac{z}{2} + \frac{z}{3} = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.3

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

$- 3 + 6 y + \frac{z}{2} \cdot \frac{3}{3} + \frac{z}{3} = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.4

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

$- 3 + 6 y + \frac{z}{2} \cdot \frac{3}{3} + \frac{z}{3} \cdot \frac{2}{2} = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.5.1

Multiply $\frac{z}{2}$ by $\frac{3}{3}$ .

$- 3 + 6 y + \frac{z \cdot 3}{2 \cdot 3} + \frac{z}{3} \cdot \frac{2}{2} = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.5.2

Multiply $2$ by $3$ .

$- 3 + 6 y + \frac{z \cdot 3}{6} + \frac{z}{3} \cdot \frac{2}{2} = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.5.3

Multiply $\frac{z}{3}$ by $\frac{2}{2}$ .

$- 3 + 6 y + \frac{z \cdot 3}{6} + \frac{z \cdot 2}{3 \cdot 2} = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.5.4

Multiply $3$ by $2$ .

$- 3 + 6 y + \frac{z \cdot 3}{6} + \frac{z \cdot 2}{6} = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

$- 3 + 6 y + \frac{z \cdot 3}{6} + \frac{z \cdot 2}{6} = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.6

Combine the numerators over the common denominator.

$- 3 + 6 y + \frac{z \cdot 3 + z \cdot 2}{6} = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.7

Reorder $z$ and $2$ .

$- 3 + 6 y + \frac{3 \cdot z + 2 \cdot z}{6} = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.2.1.8

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

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

$2 x + 10 y - 3 z = - 12$

Step 2.3

Replace all occurrences of $x$ in $2 x + 10 y - 3 z = - 12$ with $6 - 2 y - z$ .

$2 (6 - 2 y - z) + 10 y - 3 z = - 12$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $2 (6 - 2 y - z) + 10 y - 3 z$ .

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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.

$2 \cdot 6 + 2 (- 2 y) + 2 (- z) + 10 y - 3 z = - 12$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $2$ by $6$ .

$12 + 2 (- 2 y) + 2 (- z) + 10 y - 3 z = - 12$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

Step 2.4.1.1.2.2

Multiply $- 2$ by $2$ .

$12 - 4 y + 2 (- z) + 10 y - 3 z = - 12$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

Step 2.4.1.1.2.3

Multiply $- 1$ by $2$ .

$12 - 4 y - 2 z + 10 y - 3 z = - 12$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

$12 - 4 y - 2 z + 10 y - 3 z = - 12$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

$12 - 4 y - 2 z + 10 y - 3 z = - 12$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $- 4 y$ and $10 y$ .

$12 + 6 y - 2 z - 3 z = - 12$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

Step 2.4.1.2.2

Subtract $3 z$ from $- 2 z$ .

$12 + 6 y - 5 z = - 12$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

$12 + 6 y - 5 z = - 12$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

$12 + 6 y - 5 z = - 12$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

$12 + 6 y - 5 z = - 12$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

$12 + 6 y - 5 z = - 12$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

Step 3

Solve for $y$ in $12 + 6 y - 5 z = - 12$ .

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Step 3.1

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

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Step 3.1.1

Subtract $12$ from both sides of the equation.

$6 y - 5 z = - 12 - 12$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

Step 3.1.2

Add $5 z$ to both sides of the equation.

$6 y = - 12 - 12 + 5 z$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

Step 3.1.3

Subtract $12$ from $- 12$ .

$6 y = - 24 + 5 z$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

$6 y = - 24 + 5 z$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

Step 3.2

Divide each term in $6 y = - 24 + 5 z$ by $6$ and simplify.

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Step 3.2.1

Divide each term in $6 y = - 24 + 5 z$ by $6$ .

$\frac{6 y}{6} = \frac{- 24}{6} + \frac{5 z}{6}$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

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.

$\frac{6 y}{6} = \frac{- 24}{6} + \frac{5 z}{6}$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 24}{6} + \frac{5 z}{6}$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

$y = \frac{- 24}{6} + \frac{5 z}{6}$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

$y = \frac{- 24}{6} + \frac{5 z}{6}$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 24$ by $6$ .

$y = - 4 + \frac{5 z}{6}$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

$y = - 4 + \frac{5 z}{6}$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

$y = - 4 + \frac{5 z}{6}$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

$y = - 4 + \frac{5 z}{6}$

$- 3 + 6 y + \frac{5 z}{6} = 8$

$x = 6 - 2 y - z$

Step 4

Replace all occurrences of $y$ with $- 4 + \frac{5 z}{6}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $- 3 + 6 y + \frac{5 z}{6} = 8$ with $- 4 + \frac{5 z}{6}$ .

$- 3 + 6 (- 4 + \frac{5 z}{6}) + \frac{5 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 3 + 6 (- 4 + \frac{5 z}{6}) + \frac{5 z}{6}$ .

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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.

$- 3 + 6 \cdot - 4 + 6 (\frac{5 z}{6}) + \frac{5 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

Step 4.2.1.1.2

Multiply $6$ by $- 4$ .

$- 3 - 24 + 6 (\frac{5 z}{6}) + \frac{5 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

Step 4.2.1.1.3

Cancel the common factor of $6$ .

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Step 4.2.1.1.3.1

Cancel the common factor.

$- 3 - 24 + 6 (\frac{5 z}{6}) + \frac{5 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

Step 4.2.1.1.3.2

Rewrite the expression.

$- 3 - 24 + 5 z + \frac{5 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

$- 3 - 24 + 5 z + \frac{5 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

$- 3 - 24 + 5 z + \frac{5 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

Step 4.2.1.2

Subtract $24$ from $- 3$ .

$- 27 + 5 z + \frac{5 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

Step 4.2.1.3

To write $5 z$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- 27 + 5 z \cdot \frac{6}{6} + \frac{5 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

Step 4.2.1.4

Simplify terms.

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Step 4.2.1.4.1

Combine $5 z$ and $\frac{6}{6}$ .

$- 27 + \frac{5 z \cdot 6}{6} + \frac{5 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

Step 4.2.1.4.2

Combine the numerators over the common denominator.

$- 27 + \frac{5 z \cdot 6 + 5 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

$- 27 + \frac{5 z \cdot 6 + 5 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Factor $5 z$ out of $5 z \cdot 6 + 5 z$ .

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Step 4.2.1.5.1.1

Factor $5 z$ out of $5 z \cdot 6$ .

$- 27 + \frac{5 z (6) + 5 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

Step 4.2.1.5.1.2

Factor $5 z$ out of $5 z$ .

$- 27 + \frac{5 z (6) + 5 z (1)}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

Step 4.2.1.5.1.3

Factor $5 z$ out of $5 z (6) + 5 z (1)$ .

$- 27 + \frac{5 z (6 + 1)}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

$- 27 + \frac{5 z (6 + 1)}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

Step 4.2.1.5.2

Add $6$ and $1$ .

$- 27 + \frac{5 z \cdot 7}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

Step 4.2.1.5.3

Multiply $7$ by $5$ .

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 - 2 y - z$

Step 4.3

Replace all occurrences of $y$ in $x = 6 - 2 y - z$ with $- 4 + \frac{5 z}{6}$ .

$x = 6 - 2 (- 4 + \frac{5 z}{6}) - z$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $6 - 2 (- 4 + \frac{5 z}{6}) - z$ .

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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.

$x = 6 - 2 \cdot - 4 - 2 \frac{5 z}{6} - z$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4.1.1.2

Multiply $- 2$ by $- 4$ .

$x = 6 + 8 - 2 \frac{5 z}{6} - z$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4.1.1.3

Cancel the common factor of $2$ .

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Step 4.4.1.1.3.1

Factor $2$ out of $- 2$ .

$x = 6 + 8 + 2 (- 1) (\frac{5 z}{6}) - z$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4.1.1.3.2

Factor $2$ out of $6$ .

$x = 6 + 8 + 2 \cdot (- 1 \frac{5 z}{2 \cdot 3}) - z$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4.1.1.3.3

Cancel the common factor.

$x = 6 + 8 + 2 \cdot (- 1 \frac{5 z}{2 \cdot 3}) - z$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4.1.1.3.4

Rewrite the expression.

$x = 6 + 8 - 1 \frac{5 z}{3} - z$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 + 8 - 1 \frac{5 z}{3} - z$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4.1.1.4

Rewrite $- 1 \frac{5 z}{3}$ as $- \frac{5 z}{3}$ .

$x = 6 + 8 - \frac{5 z}{3} - z$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 6 + 8 - \frac{5 z}{3} - z$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4.1.2

Add $6$ and $8$ .

$x = 14 - \frac{5 z}{3} - z$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4.1.3

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

$x = 14 - \frac{5 z}{3} - z \cdot \frac{3}{3}$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4.1.4

Simplify terms.

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Step 4.4.1.4.1

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

$x = 14 - \frac{5 z}{3} + \frac{- z \cdot 3}{3}$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4.1.4.2

Combine the numerators over the common denominator.

$x = 14 + \frac{- 5 z - z \cdot 3}{3}$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 14 + \frac{- 5 z - z \cdot 3}{3}$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4.1.5

Simplify each term.

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Step 4.4.1.5.1

Simplify the numerator.

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Step 4.4.1.5.1.1

Factor $z$ out of $- 5 z - z \cdot 3$ .

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Step 4.4.1.5.1.1.1

Factor $z$ out of $- 5 z$ .

$x = 14 + \frac{z \cdot - 5 - z \cdot 3}{3}$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4.1.5.1.1.2

Factor $z$ out of $- z \cdot 3$ .

$x = 14 + \frac{z \cdot - 5 + z (- 1 \cdot 3)}{3}$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4.1.5.1.1.3

Factor $z$ out of $z \cdot - 5 + z (- 1 \cdot 3)$ .

$x = 14 + \frac{z (- 5 - 1 \cdot 3)}{3}$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 14 + \frac{z (- 5 - 1 \cdot 3)}{3}$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4.1.5.1.2

Multiply $- 1$ by $3$ .

$x = 14 + \frac{z (- 5 - 3)}{3}$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4.1.5.1.3

Subtract $3$ from $- 5$ .

$x = 14 + \frac{z \cdot - 8}{3}$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 14 + \frac{z \cdot - 8}{3}$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4.1.5.2

Move $- 8$ to the left of $z$ .

$x = 14 + \frac{- 8 \cdot z}{3}$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 4.4.1.5.3

Move the negative in front of the fraction.

$x = 14 - \frac{8 z}{3}$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 14 - \frac{8 z}{3}$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 14 - \frac{8 z}{3}$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 14 - \frac{8 z}{3}$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

$x = 14 - \frac{8 z}{3}$

$- 27 + \frac{35 z}{6} = 8$

$y = - 4 + \frac{5 z}{6}$

Step 5

Solve for $z$ in $- 27 + \frac{35 z}{6} = 8$ .

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Step 5.1

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

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Step 5.1.1

Add $27$ to both sides of the equation.

$\frac{35 z}{6} = 8 + 27$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

Step 5.1.2

Add $8$ and $27$ .

$\frac{35 z}{6} = 35$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

$\frac{35 z}{6} = 35$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

Step 5.2

Multiply both sides of the equation by $\frac{6}{35}$ .

$\frac{6}{35} \cdot \frac{35 z}{6} = \frac{6}{35} \cdot 35$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

Step 5.3

Simplify both sides of the equation.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Simplify $\frac{6}{35} \cdot \frac{35 z}{6}$ .

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Step 5.3.1.1.1

Cancel the common factor of $6$ .

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Step 5.3.1.1.1.1

Cancel the common factor.

$\frac{6}{35} \cdot \frac{35 z}{6} = \frac{6}{35} \cdot 35$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

Step 5.3.1.1.1.2

Rewrite the expression.

$\frac{1}{35} \cdot (35 z) = \frac{6}{35} \cdot 35$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

$\frac{1}{35} \cdot (35 z) = \frac{6}{35} \cdot 35$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

Step 5.3.1.1.2

Cancel the common factor of $35$ .

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Step 5.3.1.1.2.1

Factor $35$ out of $35 z$ .

$\frac{1}{35} \cdot (35 (z)) = \frac{6}{35} \cdot 35$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

Step 5.3.1.1.2.2

Cancel the common factor.

$\frac{1}{35} \cdot (35 z) = \frac{6}{35} \cdot 35$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

Step 5.3.1.1.2.3

Rewrite the expression.

$z = \frac{6}{35} \cdot 35$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

$z = \frac{6}{35} \cdot 35$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

$z = \frac{6}{35} \cdot 35$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

$z = \frac{6}{35} \cdot 35$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Cancel the common factor of $35$ .

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Step 5.3.2.1.1

Cancel the common factor.

$z = \frac{6}{35} \cdot 35$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

Step 5.3.2.1.2

Rewrite the expression.

$z = 6$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

$z = 6$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

$z = 6$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

$z = 6$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

$z = 6$

$x = 14 - \frac{8 z}{3}$

$y = - 4 + \frac{5 z}{6}$

Step 6

Replace all occurrences of $z$ with $6$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = 14 - \frac{8 z}{3}$ with $6$ .

$x = 14 - \frac{8 (6)}{3}$

$z = 6$

$y = - 4 + \frac{5 z}{6}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $14 - \frac{8 (6)}{3}$ .

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Cancel the common factor of $6$ and $3$ .

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Step 6.2.1.1.1.1

Factor $3$ out of $8 (6)$ .

$x = 14 - \frac{3 (8 \cdot (2))}{3}$

$z = 6$

$y = - 4 + \frac{5 z}{6}$

Step 6.2.1.1.1.2

Cancel the common factors.

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Step 6.2.1.1.1.2.1

Factor $3$ out of $3$ .

$x = 14 - \frac{3 (8 \cdot (2))}{3 (1)}$

$z = 6$

$y = - 4 + \frac{5 z}{6}$

Step 6.2.1.1.1.2.2

Cancel the common factor.

$x = 14 - \frac{3 (8 \cdot (2))}{3 \cdot 1}$

$z = 6$

$y = - 4 + \frac{5 z}{6}$

Step 6.2.1.1.1.2.3

Rewrite the expression.

$x = 14 - \frac{8 \cdot (2)}{1}$

$z = 6$

$y = - 4 + \frac{5 z}{6}$

Step 6.2.1.1.1.2.4

Divide $8 \cdot (2)$ by $1$ .

$x = 14 - (8 \cdot (2))$

$z = 6$

$y = - 4 + \frac{5 z}{6}$

$x = 14 - (8 \cdot (2))$

$z = 6$

$y = - 4 + \frac{5 z}{6}$

$x = 14 - (8 \cdot (2))$

$z = 6$

$y = - 4 + \frac{5 z}{6}$

Step 6.2.1.1.2

Multiply $- (8 \cdot (2))$ .

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Step 6.2.1.1.2.1

Multiply $8$ by $2$ .

$x = 14 - 1 \cdot 16$

$z = 6$

$y = - 4 + \frac{5 z}{6}$

Step 6.2.1.1.2.2

Multiply $- 1$ by $16$ .

$x = 14 - 16$

$z = 6$

$y = - 4 + \frac{5 z}{6}$

$x = 14 - 16$

$z = 6$

$y = - 4 + \frac{5 z}{6}$

$x = 14 - 16$

$z = 6$

$y = - 4 + \frac{5 z}{6}$

Step 6.2.1.2

Subtract $16$ from $14$ .

$x = - 2$

$z = 6$

$y = - 4 + \frac{5 z}{6}$

$x = - 2$

$z = 6$

$y = - 4 + \frac{5 z}{6}$

$x = - 2$

$z = 6$

$y = - 4 + \frac{5 z}{6}$

Step 6.3

Replace all occurrences of $z$ in $y = - 4 + \frac{5 z}{6}$ with $6$ .

$y = - 4 + \frac{5 (6)}{6}$

$x = - 2$

$z = 6$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- 4 + \frac{5 (6)}{6}$ .

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Step 6.4.1.1

Multiply $5$ by $6$ .

$y = - 4 + \frac{30}{6}$

$x = - 2$

$z = 6$

Step 6.4.1.2

Divide $30$ by $6$ .

$y = - 4 + 5$

$x = - 2$

$z = 6$

Step 6.4.1.3

Add $- 4$ and $5$ .

$y = 1$

$x = - 2$

$z = 6$

$y = 1$

$x = - 2$

$z = 6$

$y = 1$

$x = - 2$

$z = 6$

$y = 1$

$x = - 2$

$z = 6$

Step 7

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

$(- 2, 1, 6)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 2, 1, 6)$

Equation Form:

$x = - 2, y = 1, z = 6$