Algebra Examples

$2 x - y + z = - 3$ $x + 3 y - z = - 2$ $- 4 x + 2 y - 3 z = 1$

Step 1

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

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Step 1.1

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

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

$x + 3 y - z = - 2$

$- 4 x + 2 y - 3 z = 1$

Step 1.2

Add $y$ to both sides of the equation.

$z = - 3 - 2 x + y$

$x + 3 y - z = - 2$

$- 4 x + 2 y - 3 z = 1$

$z = - 3 - 2 x + y$

$x + 3 y - z = - 2$

$- 4 x + 2 y - 3 z = 1$

Step 2

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

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Step 2.1

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

$x + 3 y - (- 3 - 2 x + y) = - 2$

$z = - 3 - 2 x + y$

$- 4 x + 2 y - 3 z = 1$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$x + 3 y + 3 - (- 2 x) - y = - 2$

$z = - 3 - 2 x + y$

$- 4 x + 2 y - 3 z = 1$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 1$ by $- 3$ .

$x + 3 y + 3 - (- 2 x) - y = - 2$

$z = - 3 - 2 x + y$

$- 4 x + 2 y - 3 z = 1$

Step 2.2.1.1.2.2

Multiply $- 2$ by $- 1$ .

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

$z = - 3 - 2 x + y$

$- 4 x + 2 y - 3 z = 1$

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

$z = - 3 - 2 x + y$

$- 4 x + 2 y - 3 z = 1$

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

$z = - 3 - 2 x + y$

$- 4 x + 2 y - 3 z = 1$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $x$ and $2 x$ .

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

$z = - 3 - 2 x + y$

$- 4 x + 2 y - 3 z = 1$

Step 2.2.1.2.2

Subtract $y$ from $3 y$ .

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

$z = - 3 - 2 x + y$

$- 4 x + 2 y - 3 z = 1$

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

$z = - 3 - 2 x + y$

$- 4 x + 2 y - 3 z = 1$

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

$z = - 3 - 2 x + y$

$- 4 x + 2 y - 3 z = 1$

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

$z = - 3 - 2 x + y$

$- 4 x + 2 y - 3 z = 1$

Step 2.3

Replace all occurrences of $z$ in $- 4 x + 2 y - 3 z = 1$ with $- 3 - 2 x + y$ .

$- 4 x + 2 y - 3 (- 3 - 2 x + y) = 1$

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

$z = - 3 - 2 x + y$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- 4 x + 2 y - 3 (- 3 - 2 x + y)$ .

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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.

$- 4 x + 2 y - 3 \cdot - 3 - 3 (- 2 x) - 3 y = 1$

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

$z = - 3 - 2 x + y$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 3$ by $- 3$ .

$- 4 x + 2 y + 9 - 3 (- 2 x) - 3 y = 1$

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

$z = - 3 - 2 x + y$

Step 2.4.1.1.2.2

Multiply $- 2$ by $- 3$ .

$- 4 x + 2 y + 9 + 6 x - 3 y = 1$

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

$z = - 3 - 2 x + y$

$- 4 x + 2 y + 9 + 6 x - 3 y = 1$

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

$z = - 3 - 2 x + y$

$- 4 x + 2 y + 9 + 6 x - 3 y = 1$

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

$z = - 3 - 2 x + y$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $- 4 x$ and $6 x$ .

$2 x + 2 y + 9 - 3 y = 1$

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

$z = - 3 - 2 x + y$

Step 2.4.1.2.2

Subtract $3 y$ from $2 y$ .

$2 x - y + 9 = 1$

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

$z = - 3 - 2 x + y$

$2 x - y + 9 = 1$

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

$z = - 3 - 2 x + y$

$2 x - y + 9 = 1$

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

$z = - 3 - 2 x + y$

$2 x - y + 9 = 1$

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

$z = - 3 - 2 x + y$

$2 x - y + 9 = 1$

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

$z = - 3 - 2 x + y$

Step 3

Solve for $x$ in $2 x - y + 9 = 1$ .

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Step 3.1

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

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Step 3.1.1

Add $y$ to both sides of the equation.

$2 x + 9 = 1 + y$

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

$z = - 3 - 2 x + y$

Step 3.1.2

Subtract $9$ from both sides of the equation.

$2 x = 1 + y - 9$

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

$z = - 3 - 2 x + y$

Step 3.1.3

Subtract $9$ from $1$ .

$2 x = y - 8$

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

$z = - 3 - 2 x + y$

$2 x = y - 8$

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

$z = - 3 - 2 x + y$

Step 3.2

Divide each term in $2 x = y - 8$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 x = y - 8$ by $2$ .

$\frac{2 x}{2} = \frac{y}{2} + \frac{- 8}{2}$

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

$z = - 3 - 2 x + y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{y}{2} + \frac{- 8}{2}$

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

$z = - 3 - 2 x + y$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{y}{2} + \frac{- 8}{2}$

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

$z = - 3 - 2 x + y$

$x = \frac{y}{2} + \frac{- 8}{2}$

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

$z = - 3 - 2 x + y$

$x = \frac{y}{2} + \frac{- 8}{2}$

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

$z = - 3 - 2 x + y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 8$ by $2$ .

$x = \frac{y}{2} - 4$

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

$z = - 3 - 2 x + y$

$x = \frac{y}{2} - 4$

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

$z = - 3 - 2 x + y$

$x = \frac{y}{2} - 4$

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

$z = - 3 - 2 x + y$

$x = \frac{y}{2} - 4$

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

$z = - 3 - 2 x + y$

Step 4

Replace all occurrences of $x$ with $\frac{y}{2} - 4$ in each equation.

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Step 4.1

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

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

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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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.

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

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2.1.1.2

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

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

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2.1.1.3

Multiply $3$ by $- 4$ .

$\frac{3 y}{2} - 12 + 2 y + 3 = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

$\frac{3 y}{2} - 12 + 2 y + 3 = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2.1.2

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

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

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2.1.3

Combine $2 y$ and $\frac{2}{2}$ .

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

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2.1.4

Combine the numerators over the common denominator.

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

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2.1.5

Find the common denominator.

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Step 4.2.1.5.1

Write $- 12$ as a fraction with denominator $1$ .

$\frac{3 y + 2 y \cdot 2}{2} + \frac{- 12}{1} + 3 = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2.1.5.2

Multiply $\frac{- 12}{1}$ by $\frac{2}{2}$ .

$\frac{3 y + 2 y \cdot 2}{2} + \frac{- 12}{1} \cdot \frac{2}{2} + 3 = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2.1.5.3

Multiply $\frac{- 12}{1}$ by $\frac{2}{2}$ .

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

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2.1.5.4

Write $3$ as a fraction with denominator $1$ .

$\frac{3 y + 2 y \cdot 2}{2} + \frac{- 12 \cdot 2}{2} + \frac{3}{1} = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2.1.5.5

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

$\frac{3 y + 2 y \cdot 2}{2} + \frac{- 12 \cdot 2}{2} + \frac{3}{1} \cdot \frac{2}{2} = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2.1.5.6

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

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

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

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

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2.1.6

Combine the numerators over the common denominator.

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

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2.1.7

Simplify each term.

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Step 4.2.1.7.1

Multiply $2$ by $2$ .

$\frac{3 y + 4 y - 12 \cdot 2 + 3 \cdot 2}{2} = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2.1.7.2

Multiply $- 12$ by $2$ .

$\frac{3 y + 4 y - 24 + 3 \cdot 2}{2} = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2.1.7.3

Multiply $3$ by $2$ .

$\frac{3 y + 4 y - 24 + 6}{2} = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

$\frac{3 y + 4 y - 24 + 6}{2} = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2.1.8

Simplify by adding terms.

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Step 4.2.1.8.1

Add $3 y$ and $4 y$ .

$\frac{7 y - 24 + 6}{2} = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.2.1.8.2

Add $- 24$ and $6$ .

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 - 2 x + y$

Step 4.3

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

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

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

Step 4.4

Simplify the right side.

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Step 4.4.1

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

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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.

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

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

Step 4.4.1.1.2

Cancel the common factor of $2$ .

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Step 4.4.1.1.2.1

Factor $2$ out of $- 2$ .

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

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

Step 4.4.1.1.2.2

Cancel the common factor.

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

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

Step 4.4.1.1.2.3

Rewrite the expression.

$z = - 3 - 1 y - 2 \cdot - 4 + y$

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 - 1 y - 2 \cdot - 4 + y$

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

Step 4.4.1.1.3

Multiply $- 2$ by $- 4$ .

$z = - 3 - 1 y + 8 + y$

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

Step 4.4.1.1.4

Rewrite $- 1 y$ as $- y$ .

$z = - 3 - y + 8 + y$

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 - y + 8 + y$

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Combine the opposite terms in $- 3 - y + 8 + y$ .

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Step 4.4.1.2.1.1

Add $- y$ and $y$ .

$z = - 3 + 0 + 8$

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

Step 4.4.1.2.1.2

Add $- 3$ and $0$ .

$z = - 3 + 8$

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

$z = - 3 + 8$

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

Step 4.4.1.2.2

Add $- 3$ and $8$ .

$z = 5$

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

$z = 5$

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

$z = 5$

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

$z = 5$

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

$z = 5$

$\frac{7 y - 18}{2} = - 2$

$x = \frac{y}{2} - 4$

Step 5

Solve for $y$ in $\frac{7 y - 18}{2} = - 2$ .

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Step 5.1

Multiply both sides by $2$ .

$\frac{7 y - 18}{2} \cdot 2 = - 2 \cdot 2$

$z = 5$

$x = \frac{y}{2} - 4$

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1

Cancel the common factor.

$\frac{7 y - 18}{2} \cdot 2 = - 2 \cdot 2$

$z = 5$

$x = \frac{y}{2} - 4$

Step 5.2.1.1.2

Rewrite the expression.

$7 y - 18 = - 2 \cdot 2$

$z = 5$

$x = \frac{y}{2} - 4$

$7 y - 18 = - 2 \cdot 2$

$z = 5$

$x = \frac{y}{2} - 4$

$7 y - 18 = - 2 \cdot 2$

$z = 5$

$x = \frac{y}{2} - 4$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $- 2$ by $2$ .

$7 y - 18 = - 4$

$z = 5$

$x = \frac{y}{2} - 4$

$7 y - 18 = - 4$

$z = 5$

$x = \frac{y}{2} - 4$

$7 y - 18 = - 4$

$z = 5$

$x = \frac{y}{2} - 4$

Step 5.3

Solve for $y$ .

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Step 5.3.1

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

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Step 5.3.1.1

Add $18$ to both sides of the equation.

$7 y = - 4 + 18$

$z = 5$

$x = \frac{y}{2} - 4$

Step 5.3.1.2

Add $- 4$ and $18$ .

$7 y = 14$

$z = 5$

$x = \frac{y}{2} - 4$

$7 y = 14$

$z = 5$

$x = \frac{y}{2} - 4$

Step 5.3.2

Divide each term in $7 y = 14$ by $7$ and simplify.

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Step 5.3.2.1

Divide each term in $7 y = 14$ by $7$ .

$\frac{7 y}{7} = \frac{14}{7}$

$z = 5$

$x = \frac{y}{2} - 4$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Cancel the common factor of $7$ .

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Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{7 y}{7} = \frac{14}{7}$

$z = 5$

$x = \frac{y}{2} - 4$

Step 5.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{14}{7}$

$z = 5$

$x = \frac{y}{2} - 4$

$y = \frac{14}{7}$

$z = 5$

$x = \frac{y}{2} - 4$

$y = \frac{14}{7}$

$z = 5$

$x = \frac{y}{2} - 4$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Divide $14$ by $7$ .

$y = 2$

$z = 5$

$x = \frac{y}{2} - 4$

$y = 2$

$z = 5$

$x = \frac{y}{2} - 4$

$y = 2$

$z = 5$

$x = \frac{y}{2} - 4$

$y = 2$

$z = 5$

$x = \frac{y}{2} - 4$

$y = 2$

$z = 5$

$x = \frac{y}{2} - 4$

Step 6

Replace all occurrences of $y$ with $2$ in each equation.

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Step 6.1

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

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

$y = 2$

$z = 5$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{2}{2} - 4$ .

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Step 6.2.1.1

Divide $2$ by $2$ .

$x = 1 - 4$

$y = 2$

$z = 5$

Step 6.2.1.2

Subtract $4$ from $1$ .

$x = - 3$

$y = 2$

$z = 5$

$x = - 3$

$y = 2$

$z = 5$

$x = - 3$

$y = 2$

$z = 5$

$x = - 3$

$y = 2$

$z = 5$

Step 7

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

$(- 3, 2, 5)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 3, 2, 5)$

Equation Form:

$x = - 3, y = 2, z = 5$