Algebra Examples

$2 x + 2 y + 3 z = 4$ $5 x + 3 y + 5 z = 5$ $3 x + 4 y + 6 z = 5$

Step 1

Solve for $x$ in $2 x + 2 y + 3 z = 4$ .

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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

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

$2 x + 3 z = 4 - 2 y$

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

Step 1.1.2

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

$2 x = 4 - 2 y - 3 z$

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

$2 x = 4 - 2 y - 3 z$

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

Step 1.2

Divide each term in $2 x = 4 - 2 y - 3 z$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x = 4 - 2 y - 3 z$ by $2$ .

$\frac{2 x}{2} = \frac{4}{2} + \frac{- 2 y}{2} + \frac{- 3 z}{2}$

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{4}{2} + \frac{- 2 y}{2} + \frac{- 3 z}{2}$

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{2} + \frac{- 2 y}{2} + \frac{- 3 z}{2}$

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

$x = \frac{4}{2} + \frac{- 2 y}{2} + \frac{- 3 z}{2}$

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

$x = \frac{4}{2} + \frac{- 2 y}{2} + \frac{- 3 z}{2}$

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

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

Divide $4$ by $2$ .

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

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

Step 1.2.3.1.2

Cancel the common factor of $- 2$ and $2$ .

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Step 1.2.3.1.2.1

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

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

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $2$ out of $2$ .

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

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

Step 1.2.3.1.2.2.2

Cancel the common factor.

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

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

Step 1.2.3.1.2.2.3

Rewrite the expression.

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

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

Step 1.2.3.1.2.2.4

Divide $- y$ by $1$ .

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

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

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

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

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

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

Step 1.2.3.1.3

Move the negative in front of the fraction.

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

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

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

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

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

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

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

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

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

$5 x + 3 y + 5 z = 5$

$3 x + 4 y + 6 z = 5$

Step 2

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

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Step 2.1

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

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

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

$3 x + 4 y + 6 z = 5$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

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

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

$3 x + 4 y + 6 z = 5$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $5$ by $2$ .

$10 + 5 (- y) + 5 (- \frac{3 z}{2}) + 3 y + 5 z = 5$

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

$3 x + 4 y + 6 z = 5$

Step 2.2.1.1.2.2

Multiply $- 1$ by $5$ .

$10 - 5 y + 5 (- \frac{3 z}{2}) + 3 y + 5 z = 5$

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

$3 x + 4 y + 6 z = 5$

Step 2.2.1.1.2.3

Multiply $5 (- \frac{3 z}{2})$ .

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Step 2.2.1.1.2.3.1

Multiply $- 1$ by $5$ .

$10 - 5 y - 5 \frac{3 z}{2} + 3 y + 5 z = 5$

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

$3 x + 4 y + 6 z = 5$

Step 2.2.1.1.2.3.2

Combine $- 5$ and $\frac{3 z}{2}$ .

$10 - 5 y + \frac{- 5 (3 z)}{2} + 3 y + 5 z = 5$

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

$3 x + 4 y + 6 z = 5$

Step 2.2.1.1.2.3.3

Multiply $3$ by $- 5$ .

$10 - 5 y + \frac{- 15 z}{2} + 3 y + 5 z = 5$

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

$3 x + 4 y + 6 z = 5$

$10 - 5 y + \frac{- 15 z}{2} + 3 y + 5 z = 5$

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

$3 x + 4 y + 6 z = 5$

$10 - 5 y + \frac{- 15 z}{2} + 3 y + 5 z = 5$

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

$3 x + 4 y + 6 z = 5$

Step 2.2.1.1.3

Move the negative in front of the fraction.

$10 - 5 y - \frac{15 z}{2} + 3 y + 5 z = 5$

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

$3 x + 4 y + 6 z = 5$

$10 - 5 y - \frac{15 z}{2} + 3 y + 5 z = 5$

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

$3 x + 4 y + 6 z = 5$

Step 2.2.1.2

Add $- 5 y$ and $3 y$ .

$10 - 2 y - \frac{15 z}{2} + 5 z = 5$

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

$3 x + 4 y + 6 z = 5$

Step 2.2.1.3

To write $5 z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$10 - 2 y - \frac{15 z}{2} + 5 z \cdot \frac{2}{2} = 5$

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

$3 x + 4 y + 6 z = 5$

Step 2.2.1.4

Simplify terms.

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Step 2.2.1.4.1

Combine $5 z$ and $\frac{2}{2}$ .

$10 - 2 y - \frac{15 z}{2} + \frac{5 z \cdot 2}{2} = 5$

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

$3 x + 4 y + 6 z = 5$

Step 2.2.1.4.2

Combine the numerators over the common denominator.

$10 - 2 y + \frac{- 15 z + 5 z \cdot 2}{2} = 5$

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

$3 x + 4 y + 6 z = 5$

$10 - 2 y + \frac{- 15 z + 5 z \cdot 2}{2} = 5$

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

$3 x + 4 y + 6 z = 5$

Step 2.2.1.5

Simplify each term.

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Step 2.2.1.5.1

Simplify the numerator.

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Step 2.2.1.5.1.1

Factor $5 z$ out of $- 15 z + 5 z \cdot 2$ .

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Step 2.2.1.5.1.1.1

Factor $5 z$ out of $- 15 z$ .

$10 - 2 y + \frac{5 z (- 3) + 5 z \cdot 2}{2} = 5$

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

$3 x + 4 y + 6 z = 5$

Step 2.2.1.5.1.1.2

Factor $5 z$ out of $5 z \cdot 2$ .

$10 - 2 y + \frac{5 z (- 3) + 5 z (2)}{2} = 5$

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

$3 x + 4 y + 6 z = 5$

Step 2.2.1.5.1.1.3

Factor $5 z$ out of $5 z (- 3) + 5 z (2)$ .

$10 - 2 y + \frac{5 z (- 3 + 2)}{2} = 5$

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

$3 x + 4 y + 6 z = 5$

$10 - 2 y + \frac{5 z (- 3 + 2)}{2} = 5$

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

$3 x + 4 y + 6 z = 5$

Step 2.2.1.5.1.2

Add $- 3$ and $2$ .

$10 - 2 y + \frac{5 z \cdot - 1}{2} = 5$

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

$3 x + 4 y + 6 z = 5$

Step 2.2.1.5.1.3

Multiply $- 1$ by $5$ .

$10 - 2 y + \frac{- 5 z}{2} = 5$

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

$3 x + 4 y + 6 z = 5$

$10 - 2 y + \frac{- 5 z}{2} = 5$

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

$3 x + 4 y + 6 z = 5$

Step 2.2.1.5.2

Move the negative in front of the fraction.

$10 - 2 y - \frac{5 z}{2} = 5$

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

$3 x + 4 y + 6 z = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

$3 x + 4 y + 6 z = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

$3 x + 4 y + 6 z = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

$3 x + 4 y + 6 z = 5$

Step 2.3

Replace all occurrences of $x$ in $3 x + 4 y + 6 z = 5$ with $2 - y - \frac{3 z}{2}$ .

$3 (2 - y - \frac{3 z}{2}) + 4 y + 6 z = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $3 (2 - y - \frac{3 z}{2}) + 4 y + 6 z$ .

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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.

$3 \cdot 2 + 3 (- y) + 3 (- \frac{3 z}{2}) + 4 y + 6 z = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $3$ by $2$ .

$6 + 3 (- y) + 3 (- \frac{3 z}{2}) + 4 y + 6 z = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 2.4.1.1.2.2

Multiply $- 1$ by $3$ .

$6 - 3 y + 3 (- \frac{3 z}{2}) + 4 y + 6 z = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 2.4.1.1.2.3

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

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Step 2.4.1.1.2.3.1

Multiply $- 1$ by $3$ .

$6 - 3 y - 3 \frac{3 z}{2} + 4 y + 6 z = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 2.4.1.1.2.3.2

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

$6 - 3 y + \frac{- 3 (3 z)}{2} + 4 y + 6 z = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 2.4.1.1.2.3.3

Multiply $3$ by $- 3$ .

$6 - 3 y + \frac{- 9 z}{2} + 4 y + 6 z = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

$6 - 3 y + \frac{- 9 z}{2} + 4 y + 6 z = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

$6 - 3 y + \frac{- 9 z}{2} + 4 y + 6 z = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 2.4.1.1.3

Move the negative in front of the fraction.

$6 - 3 y - \frac{9 z}{2} + 4 y + 6 z = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

$6 - 3 y - \frac{9 z}{2} + 4 y + 6 z = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 2.4.1.2

Add $- 3 y$ and $4 y$ .

$6 + y - \frac{9 z}{2} + 6 z = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 2.4.1.3

To write $6 z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$6 + y - \frac{9 z}{2} + 6 z \cdot \frac{2}{2} = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 2.4.1.4

Simplify terms.

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Step 2.4.1.4.1

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

$6 + y - \frac{9 z}{2} + \frac{6 z \cdot 2}{2} = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 2.4.1.4.2

Combine the numerators over the common denominator.

$6 + y + \frac{- 9 z + 6 z \cdot 2}{2} = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

$6 + y + \frac{- 9 z + 6 z \cdot 2}{2} = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 2.4.1.5

Simplify the numerator.

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Step 2.4.1.5.1

Factor $3 z$ out of $- 9 z + 6 z \cdot 2$ .

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Step 2.4.1.5.1.1

Factor $3 z$ out of $- 9 z$ .

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

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 2.4.1.5.1.2

Factor $3 z$ out of $6 z \cdot 2$ .

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

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 2.4.1.5.1.3

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

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

$10 - 2 y - \frac{5 z}{2} = 5$

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

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

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 2.4.1.5.2

Multiply $2$ by $2$ .

$6 + y + \frac{3 z (- 3 + 4)}{2} = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 2.4.1.5.3

Add $- 3$ and $4$ .

$6 + y + \frac{3 z \cdot 1}{2} = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 2.4.1.5.4

Multiply $3$ by $1$ .

$6 + y + \frac{3 z}{2} = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

$6 + y + \frac{3 z}{2} = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

$6 + y + \frac{3 z}{2} = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

$6 + y + \frac{3 z}{2} = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

$6 + y + \frac{3 z}{2} = 5$

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 3

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

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Step 3.1

Subtract $6$ from both sides of the equation.

$y + \frac{3 z}{2} = 5 - 6$

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 3.2

Subtract $\frac{3 z}{2}$ from both sides of the equation.

$y = 5 - 6 - \frac{3 z}{2}$

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 3.3

Subtract $6$ from $5$ .

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

$10 - 2 y - \frac{5 z}{2} = 5$

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

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

$10 - 2 y - \frac{5 z}{2} = 5$

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

Step 4

Replace all occurrences of $y$ with $- 1 - \frac{3 z}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $10 - 2 y - \frac{5 z}{2} = 5$ with $- 1 - \frac{3 z}{2}$ .

$10 - 2 (- 1 - \frac{3 z}{2}) - \frac{5 z}{2} = 5$

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $10 - 2 (- 1 - \frac{3 z}{2}) - \frac{5 z}{2}$ .

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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.

$10 - 2 \cdot - 1 - 2 (- \frac{3 z}{2}) - \frac{5 z}{2} = 5$

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

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

Step 4.2.1.1.2

Multiply $- 2$ by $- 1$ .

$10 + 2 - 2 (- \frac{3 z}{2}) - \frac{5 z}{2} = 5$

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

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

Step 4.2.1.1.3

Cancel the common factor of $2$ .

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Step 4.2.1.1.3.1

Move the leading negative in $- \frac{3 z}{2}$ into the numerator.

$10 + 2 - 2 \frac{- 3 z}{2} - \frac{5 z}{2} = 5$

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

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

Step 4.2.1.1.3.2

Factor $2$ out of $- 2$ .

$10 + 2 + 2 (- 1) (\frac{- 3 z}{2}) - \frac{5 z}{2} = 5$

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

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

Step 4.2.1.1.3.3

Cancel the common factor.

$10 + 2 + 2 \cdot (- 1 \frac{- 3 z}{2}) - \frac{5 z}{2} = 5$

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

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

Step 4.2.1.1.3.4

Rewrite the expression.

$10 + 2 - 1 (- 3 z) - \frac{5 z}{2} = 5$

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

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

$10 + 2 - 1 (- 3 z) - \frac{5 z}{2} = 5$

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

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

Step 4.2.1.1.4

Multiply $- 3$ by $- 1$ .

$10 + 2 + 3 z - \frac{5 z}{2} = 5$

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

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

$10 + 2 + 3 z - \frac{5 z}{2} = 5$

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

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

Step 4.2.1.2

Add $10$ and $2$ .

$12 + 3 z - \frac{5 z}{2} = 5$

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

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

Step 4.2.1.3

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

$12 + 3 z \cdot \frac{2}{2} - \frac{5 z}{2} = 5$

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

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

Step 4.2.1.4

Simplify terms.

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Step 4.2.1.4.1

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

$12 + \frac{3 z \cdot 2}{2} - \frac{5 z}{2} = 5$

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

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

Step 4.2.1.4.2

Combine the numerators over the common denominator.

$12 + \frac{3 z \cdot 2 - 5 z}{2} = 5$

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

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

$12 + \frac{3 z \cdot 2 - 5 z}{2} = 5$

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

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

Step 4.2.1.5

Simplify each term.

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Step 4.2.1.5.1

Simplify the numerator.

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Step 4.2.1.5.1.1

Factor $z$ out of $3 z \cdot 2 - 5 z$ .

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Step 4.2.1.5.1.1.1

Factor $z$ out of $3 z \cdot 2$ .

$12 + \frac{z (3 \cdot 2) - 5 z}{2} = 5$

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

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

Step 4.2.1.5.1.1.2

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

$12 + \frac{z (3 \cdot 2) + z \cdot - 5}{2} = 5$

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

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

Step 4.2.1.5.1.1.3

Factor $z$ out of $z (3 \cdot 2) + z \cdot - 5$ .

$12 + \frac{z (3 \cdot 2 - 5)}{2} = 5$

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

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

$12 + \frac{z (3 \cdot 2 - 5)}{2} = 5$

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

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

Step 4.2.1.5.1.2

Multiply $3$ by $2$ .

$12 + \frac{z (6 - 5)}{2} = 5$

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

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

Step 4.2.1.5.1.3

Subtract $5$ from $6$ .

$12 + \frac{z \cdot 1}{2} = 5$

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

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

$12 + \frac{z \cdot 1}{2} = 5$

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

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

Step 4.2.1.5.2

Multiply $z$ by $1$ .

$12 + \frac{z}{2} = 5$

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

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

$12 + \frac{z}{2} = 5$

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

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

$12 + \frac{z}{2} = 5$

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

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

$12 + \frac{z}{2} = 5$

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

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

Step 4.3

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

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

$12 + \frac{z}{2} = 5$

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

Step 4.4

Simplify the right side.

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Step 4.4.1

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

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Step 4.4.1.1

Find the common denominator.

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Step 4.4.1.1.1

Write $2$ as a fraction with denominator $1$ .

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

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.1.2

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

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

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.1.3

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

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

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.1.4

Write $- (- 1 - \frac{3 z}{2})$ as a fraction with denominator $1$ .

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

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.1.5

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

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

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.1.6

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

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

$12 + \frac{z}{2} = 5$

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

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

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.2

Combine the numerators over the common denominator.

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

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.3

Simplify each term.

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Step 4.4.1.3.1

Multiply $2$ by $2$ .

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

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.3.2

Apply the distributive property.

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

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.3.3

Multiply $- 1$ by $- 1$ .

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

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.3.4

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

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Step 4.4.1.3.4.1

Multiply $- 1$ by $- 1$ .

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

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.3.4.2

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

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

$12 + \frac{z}{2} = 5$

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

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

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.3.5

Apply the distributive property.

$x = \frac{4 + 1 \cdot 2 + \frac{3 z}{2} \cdot 2 - 3 z}{2}$

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.3.6

Multiply $2$ by $1$ .

$x = \frac{4 + 2 + \frac{3 z}{2} \cdot 2 - 3 z}{2}$

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.3.7

Cancel the common factor of $2$ .

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Step 4.4.1.3.7.1

Cancel the common factor.

$x = \frac{4 + 2 + \frac{3 z}{2} \cdot 2 - 3 z}{2}$

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.3.7.2

Rewrite the expression.

$x = \frac{4 + 2 + 3 z - 3 z}{2}$

$12 + \frac{z}{2} = 5$

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

$x = \frac{4 + 2 + 3 z - 3 z}{2}$

$12 + \frac{z}{2} = 5$

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

$x = \frac{4 + 2 + 3 z - 3 z}{2}$

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.4

Simplify by adding terms.

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Step 4.4.1.4.1

Combine the opposite terms in $4 + 2 + 3 z - 3 z$ .

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Step 4.4.1.4.1.1

Subtract $3 z$ from $3 z$ .

$x = \frac{4 + 2 + 0}{2}$

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.4.1.2

Add $4 + 2$ and $0$ .

$x = \frac{4 + 2}{2}$

$12 + \frac{z}{2} = 5$

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

$x = \frac{4 + 2}{2}$

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.4.2

Simplify the expression.

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Step 4.4.1.4.2.1

Add $4$ and $2$ .

$x = \frac{6}{2}$

$12 + \frac{z}{2} = 5$

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

Step 4.4.1.4.2.2

Divide $6$ by $2$ .

$x = 3$

$12 + \frac{z}{2} = 5$

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

$x = 3$

$12 + \frac{z}{2} = 5$

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

$x = 3$

$12 + \frac{z}{2} = 5$

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

$x = 3$

$12 + \frac{z}{2} = 5$

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

$x = 3$

$12 + \frac{z}{2} = 5$

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

$x = 3$

$12 + \frac{z}{2} = 5$

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

Step 5

Solve for $z$ in $12 + \frac{z}{2} = 5$ .

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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

Subtract $12$ from both sides of the equation.

$\frac{z}{2} = 5 - 12$

$x = 3$

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

Step 5.1.2

Subtract $12$ from $5$ .

$\frac{z}{2} = - 7$

$x = 3$

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

$\frac{z}{2} = - 7$

$x = 3$

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

Step 5.2

Multiply both sides of the equation by $2$ .

$2 (\frac{z}{2}) = 2 \cdot - 7$

$x = 3$

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

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

Cancel the common factor of $2$ .

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Step 5.3.1.1.1

Cancel the common factor.

$2 (\frac{z}{2}) = 2 \cdot - 7$

$x = 3$

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

Step 5.3.1.1.2

Rewrite the expression.

$z = 2 \cdot - 7$

$x = 3$

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

$z = 2 \cdot - 7$

$x = 3$

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

$z = 2 \cdot - 7$

$x = 3$

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

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $2$ by $- 7$ .

$z = - 14$

$x = 3$

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

$z = - 14$

$x = 3$

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

$z = - 14$

$x = 3$

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

$z = - 14$

$x = 3$

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

Step 6

Replace all occurrences of $z$ with $- 14$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $y = - 1 - \frac{3 z}{2}$ with $- 14$ .

$y = - 1 - \frac{3 (- 14)}{2}$

$z = - 14$

$x = 3$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- 1 - \frac{3 (- 14)}{2}$ .

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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 $- 14$ and $2$ .

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Step 6.2.1.1.1.1

Factor $2$ out of $3 (- 14)$ .

$y = - 1 - \frac{2 (3 \cdot (- 7))}{2}$

$z = - 14$

$x = 3$

Step 6.2.1.1.1.2

Cancel the common factors.

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Step 6.2.1.1.1.2.1

Factor $2$ out of $2$ .

$y = - 1 - \frac{2 (3 \cdot (- 7))}{2 (1)}$

$z = - 14$

$x = 3$

Step 6.2.1.1.1.2.2

Cancel the common factor.

$y = - 1 - \frac{2 (3 \cdot (- 7))}{2 \cdot 1}$

$z = - 14$

$x = 3$

Step 6.2.1.1.1.2.3

Rewrite the expression.

$y = - 1 - \frac{3 \cdot (- 7)}{1}$

$z = - 14$

$x = 3$

Step 6.2.1.1.1.2.4

Divide $3 \cdot (- 7)$ by $1$ .

$y = - 1 - (3 \cdot (- 7))$

$z = - 14$

$x = 3$

$y = - 1 - (3 \cdot (- 7))$

$z = - 14$

$x = 3$

$y = - 1 - (3 \cdot (- 7))$

$z = - 14$

$x = 3$

Step 6.2.1.1.2

Multiply $- (3 \cdot (- 7))$ .

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Step 6.2.1.1.2.1

Multiply $3$ by $- 7$ .

$y = - 1 + 21$

$z = - 14$

$x = 3$

Step 6.2.1.1.2.2

Multiply $- 1$ by $- 21$ .

$y = - 1 + 21$

$z = - 14$

$x = 3$

$y = - 1 + 21$

$z = - 14$

$x = 3$

$y = - 1 + 21$

$z = - 14$

$x = 3$

Step 6.2.1.2

Add $- 1$ and $21$ .

$y = 20$

$z = - 14$

$x = 3$

$y = 20$

$z = - 14$

$x = 3$

$y = 20$

$z = - 14$

$x = 3$

$y = 20$

$z = - 14$

$x = 3$

Step 7

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

$(3, 20, - 14)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(3, 20, - 14)$

Equation Form:

$x = 3, y = 20, z = - 14$