Algebra Examples

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

Step 1

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

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Step 1.1

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

$y - z = - 3 - 2 x$

$x - y + 3 z = - 2$

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

Step 1.2

Add $z$ to both sides of the equation.

$y = - 3 - 2 x + z$

$x - y + 3 z = - 2$

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

$y = - 3 - 2 x + z$

$x - y + 3 z = - 2$

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

Step 2

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

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Step 2.1

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

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

$y = - 3 - 2 x + z$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

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

$y = - 3 - 2 x + z$

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

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 1$ by $- 3$ .

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

$y = - 3 - 2 x + z$

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

Step 2.2.1.1.2.2

Multiply $- 2$ by $- 1$ .

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

$y = - 3 - 2 x + z$

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

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

$y = - 3 - 2 x + z$

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

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

$y = - 3 - 2 x + z$

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

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $x$ and $2 x$ .

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

$y = - 3 - 2 x + z$

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

Step 2.2.1.2.2

Add $- z$ and $3 z$ .

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

$y = - 3 - 2 x + z$

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

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

$y = - 3 - 2 x + z$

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

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

$y = - 3 - 2 x + z$

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

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

$y = - 3 - 2 x + z$

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

Step 2.3

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

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

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

$y = - 3 - 2 x + z$

Step 2.4

Simplify the left side.

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Step 2.4.1

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

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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.

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

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

$y = - 3 - 2 x + z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 3$ by $- 3$ .

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

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

$y = - 3 - 2 x + z$

Step 2.4.1.1.2.2

Multiply $- 2$ by $- 3$ .

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

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

$y = - 3 - 2 x + z$

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

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

$y = - 3 - 2 x + z$

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

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

$y = - 3 - 2 x + z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

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

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

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

$y = - 3 - 2 x + z$

Step 2.4.1.2.2

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

$2 x + 9 - z = 1$

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

$y = - 3 - 2 x + z$

$2 x + 9 - z = 1$

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

$y = - 3 - 2 x + z$

$2 x + 9 - z = 1$

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

$y = - 3 - 2 x + z$

$2 x + 9 - z = 1$

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

$y = - 3 - 2 x + z$

$2 x + 9 - z = 1$

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

$y = - 3 - 2 x + z$

Step 3

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

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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

Subtract $9$ from both sides of the equation.

$2 x - z = 1 - 9$

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

$y = - 3 - 2 x + z$

Step 3.1.2

Add $z$ to both sides of the equation.

$2 x = 1 - 9 + z$

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

$y = - 3 - 2 x + z$

Step 3.1.3

Subtract $9$ from $1$ .

$2 x = - 8 + z$

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

$y = - 3 - 2 x + z$

$2 x = - 8 + z$

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

$y = - 3 - 2 x + z$

Step 3.2

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

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Step 3.2.1

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

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

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

$y = - 3 - 2 x + z$

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.

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

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

$y = - 3 - 2 x + z$

Step 3.2.2.1.2

Divide $x$ by $1$ .

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

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

$y = - 3 - 2 x + z$

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

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

$y = - 3 - 2 x + z$

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

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

$y = - 3 - 2 x + z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 8$ by $2$ .

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

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

$y = - 3 - 2 x + z$

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

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

$y = - 3 - 2 x + z$

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

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

$y = - 3 - 2 x + z$

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

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

$y = - 3 - 2 x + z$

Step 4

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

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Step 4.1

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

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

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

$y = - 3 - 2 x + z$

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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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.

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

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

$y = - 3 - 2 x + z$

Step 4.2.1.1.2

Multiply $3$ by $- 4$ .

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

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

$y = - 3 - 2 x + z$

Step 4.2.1.1.3

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

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

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

$y = - 3 - 2 x + z$

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

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

$y = - 3 - 2 x + z$

Step 4.2.1.2

Add $- 12$ and $3$ .

$\frac{3 z}{2} - 9 + 2 z = - 2$

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

$y = - 3 - 2 x + z$

Step 4.2.1.3

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

$\frac{3 z}{2} + 2 z \cdot \frac{2}{2} - 9 = - 2$

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

$y = - 3 - 2 x + z$

Step 4.2.1.4

Simplify terms.

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Step 4.2.1.4.1

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

$\frac{3 z}{2} + \frac{2 z \cdot 2}{2} - 9 = - 2$

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

$y = - 3 - 2 x + z$

Step 4.2.1.4.2

Combine the numerators over the common denominator.

$\frac{3 z + 2 z \cdot 2}{2} - 9 = - 2$

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

$y = - 3 - 2 x + z$

$\frac{3 z + 2 z \cdot 2}{2} - 9 = - 2$

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

$y = - 3 - 2 x + z$

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 $3 z + 2 z \cdot 2$ .

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Step 4.2.1.5.1.1.1

Factor $z$ out of $3 z$ .

$\frac{z \cdot 3 + 2 z \cdot 2}{2} - 9 = - 2$

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

$y = - 3 - 2 x + z$

Step 4.2.1.5.1.1.2

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

$\frac{z \cdot 3 + z (2 \cdot 2)}{2} - 9 = - 2$

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

$y = - 3 - 2 x + z$

Step 4.2.1.5.1.1.3

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

$\frac{z (3 + 2 \cdot 2)}{2} - 9 = - 2$

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

$y = - 3 - 2 x + z$

$\frac{z (3 + 2 \cdot 2)}{2} - 9 = - 2$

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

$y = - 3 - 2 x + z$

Step 4.2.1.5.1.2

Multiply $2$ by $2$ .

$\frac{z (3 + 4)}{2} - 9 = - 2$

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

$y = - 3 - 2 x + z$

Step 4.2.1.5.1.3

Add $3$ and $4$ .

$\frac{z \cdot 7}{2} - 9 = - 2$

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

$y = - 3 - 2 x + z$

$\frac{z \cdot 7}{2} - 9 = - 2$

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

$y = - 3 - 2 x + z$

Step 4.2.1.5.2

Move $7$ to the left of $z$ .

$\frac{7 z}{2} - 9 = - 2$

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

$y = - 3 - 2 x + z$

$\frac{7 z}{2} - 9 = - 2$

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

$y = - 3 - 2 x + z$

$\frac{7 z}{2} - 9 = - 2$

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

$y = - 3 - 2 x + z$

$\frac{7 z}{2} - 9 = - 2$

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

$y = - 3 - 2 x + z$

Step 4.3

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

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

$\frac{7 z}{2} - 9 = - 2$

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

Step 4.4

Simplify the right side.

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Step 4.4.1

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

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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.

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

$\frac{7 z}{2} - 9 = - 2$

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

Step 4.4.1.1.2

Multiply $- 2$ by $- 4$ .

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

$\frac{7 z}{2} - 9 = - 2$

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

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 $- 2$ .

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

$\frac{7 z}{2} - 9 = - 2$

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

Step 4.4.1.1.3.2

Cancel the common factor.

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

$\frac{7 z}{2} - 9 = - 2$

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

Step 4.4.1.1.3.3

Rewrite the expression.

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

$\frac{7 z}{2} - 9 = - 2$

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

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

$\frac{7 z}{2} - 9 = - 2$

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

Step 4.4.1.1.4

Rewrite $- 1 z$ as $- z$ .

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

$\frac{7 z}{2} - 9 = - 2$

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

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

$\frac{7 z}{2} - 9 = - 2$

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

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

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

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Step 4.4.1.2.1.1

Add $- z$ and $z$ .

$y = - 3 + 8 + 0$

$\frac{7 z}{2} - 9 = - 2$

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

Step 4.4.1.2.1.2

Add $- 3 + 8$ and $0$ .

$y = - 3 + 8$

$\frac{7 z}{2} - 9 = - 2$

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

$y = - 3 + 8$

$\frac{7 z}{2} - 9 = - 2$

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

Step 4.4.1.2.2

Add $- 3$ and $8$ .

$y = 5$

$\frac{7 z}{2} - 9 = - 2$

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

$y = 5$

$\frac{7 z}{2} - 9 = - 2$

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

$y = 5$

$\frac{7 z}{2} - 9 = - 2$

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

$y = 5$

$\frac{7 z}{2} - 9 = - 2$

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

$y = 5$

$\frac{7 z}{2} - 9 = - 2$

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

Step 5

Solve for $z$ in $\frac{7 z}{2} - 9 = - 2$ .

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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 $9$ to both sides of the equation.

$\frac{7 z}{2} = - 2 + 9$

$y = 5$

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

Step 5.1.2

Add $- 2$ and $9$ .

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

$y = 5$

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

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

$y = 5$

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

Step 5.2

Multiply both sides of the equation by $\frac{2}{7}$ .

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

$y = 5$

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

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

Simplify $\frac{2}{7} \cdot \frac{7 z}{2}$ .

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Step 5.3.1.1.1

Cancel the common factor of $2$ .

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Step 5.3.1.1.1.1

Cancel the common factor.

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

$y = 5$

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

Step 5.3.1.1.1.2

Rewrite the expression.

$\frac{1}{7} \cdot (7 z) = \frac{2}{7} \cdot 7$

$y = 5$

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

$\frac{1}{7} \cdot (7 z) = \frac{2}{7} \cdot 7$

$y = 5$

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

Step 5.3.1.1.2

Cancel the common factor of $7$ .

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Step 5.3.1.1.2.1

Factor $7$ out of $7 z$ .

$\frac{1}{7} \cdot (7 (z)) = \frac{2}{7} \cdot 7$

$y = 5$

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

Step 5.3.1.1.2.2

Cancel the common factor.

$\frac{1}{7} \cdot (7 z) = \frac{2}{7} \cdot 7$

$y = 5$

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

Step 5.3.1.1.2.3

Rewrite the expression.

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

$y = 5$

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

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

$y = 5$

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

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

$y = 5$

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

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

$y = 5$

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

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Cancel the common factor of $7$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

$y = 5$

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

Step 5.3.2.1.2

Rewrite the expression.

$z = 2$

$y = 5$

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

$z = 2$

$y = 5$

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

$z = 2$

$y = 5$

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

$z = 2$

$y = 5$

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

$z = 2$

$y = 5$

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

Step 6

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

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Step 6.1

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

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

$z = 2$

$y = 5$

Step 6.2

Simplify the right side.

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Step 6.2.1

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

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Step 6.2.1.1

Divide $2$ by $2$ .

$x = - 4 + 1$

$z = 2$

$y = 5$

Step 6.2.1.2

Add $- 4$ and $1$ .

$x = - 3$

$z = 2$

$y = 5$

$x = - 3$

$z = 2$

$y = 5$

$x = - 3$

$z = 2$

$y = 5$

$x = - 3$

$z = 2$

$y = 5$

Step 7

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

$(- 3, 5, 2)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 3, 5, 2)$

Equation Form:

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