Precalculus Examples

$3 x - 3 y + 2 z = 9$ , $3 x + y - 3 z = - 8$ , $11 x - y - 2 z = 0$

Step 1

Choose two equations and eliminate one variable. In this case, eliminate $x$ .

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

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

Step 2

Eliminate $x$ from the system.

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Step 2.1

Multiply each equation by the value that makes the coefficients of $x$ opposite.

$(- 1) \cdot (3 x - 3 y + 2 z) = (- 1) (9)$

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

Step 2.2

Simplify.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify $(- 1) \cdot (3 x - 3 y + 2 z)$ .

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Step 2.2.1.1.1

Apply the distributive property.

$- 1 (3 x) - 1 (- 3 y) - 1 (2 z) = (- 1) (9)$

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

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $3$ by $- 1$ .

$- 3 x - 1 (- 3 y) - 1 (2 z) = (- 1) (9)$

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

Step 2.2.1.1.2.2

Multiply $- 3$ by $- 1$ .

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

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

Step 2.2.1.1.2.3

Multiply $2$ by $- 1$ .

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

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

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

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

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

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

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

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

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $- 1$ by $9$ .

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

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

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

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

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

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

Step 2.3

Add the two equations together to eliminate $x$ from the system.

	$-$	$3$	$x$	$+$	$3$	$y$	$-$	$2$	$z$	$=$	$-$	$9$
$+$		$3$	$x$	$+$		$y$	$-$	$3$	$z$	$=$	$-$	$8$
					$4$	$y$	$-$	$5$	$z$	$=$	$-$	$1$	$7$

Step 2.4

The resultant equation has $x$ eliminated.

$4 y - 5 z = - 17$

Step 3

Choose another two equations and eliminate $x$ .

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

$11 x - y - 2 z = 0$

Step 4

Eliminate $x$ from the system.

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Step 4.1

Multiply each equation by the value that makes the coefficients of $x$ opposite.

$(- 11) \cdot (3 x + y - 3 z) = (- 11) (- 8)$

$(3) \cdot (11 x - y - 2 z) = (3) (0)$

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify $(- 11) \cdot (3 x + y - 3 z)$ .

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Step 4.2.1.1.1

Apply the distributive property.

$- 11 (3 x) - 11 y - 11 (- 3 z) = (- 11) (- 8)$

$(3) \cdot (11 x - y - 2 z) = (3) (0)$

Step 4.2.1.1.2

Simplify.

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Step 4.2.1.1.2.1

Multiply $3$ by $- 11$ .

$- 33 x - 11 y - 11 (- 3 z) = (- 11) (- 8)$

$(3) \cdot (11 x - y - 2 z) = (3) (0)$

Step 4.2.1.1.2.2

Multiply $- 3$ by $- 11$ .

$- 33 x - 11 y + 33 z = (- 11) (- 8)$

$(3) \cdot (11 x - y - 2 z) = (3) (0)$

$- 33 x - 11 y + 33 z = (- 11) (- 8)$

$(3) \cdot (11 x - y - 2 z) = (3) (0)$

$- 33 x - 11 y + 33 z = (- 11) (- 8)$

$(3) \cdot (11 x - y - 2 z) = (3) (0)$

$- 33 x - 11 y + 33 z = (- 11) (- 8)$

$(3) \cdot (11 x - y - 2 z) = (3) (0)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Multiply $- 11$ by $- 8$ .

$- 33 x - 11 y + 33 z = 88$

$(3) \cdot (11 x - y - 2 z) = (3) (0)$

$- 33 x - 11 y + 33 z = 88$

$(3) \cdot (11 x - y - 2 z) = (3) (0)$

Step 4.2.3

Simplify the left side.

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Step 4.2.3.1

Simplify $(3) \cdot (11 x - y - 2 z)$ .

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Step 4.2.3.1.1

Apply the distributive property.

$- 33 x - 11 y + 33 z = 88$

$3 (11 x) + 3 (- y) + 3 (- 2 z) = (3) (0)$

Step 4.2.3.1.2

Simplify.

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Step 4.2.3.1.2.1

Multiply $11$ by $3$ .

$- 33 x - 11 y + 33 z = 88$

$33 x + 3 (- y) + 3 (- 2 z) = (3) (0)$

Step 4.2.3.1.2.2

Multiply $- 1$ by $3$ .

$- 33 x - 11 y + 33 z = 88$

$33 x - 3 y + 3 (- 2 z) = (3) (0)$

Step 4.2.3.1.2.3

Multiply $- 2$ by $3$ .

$- 33 x - 11 y + 33 z = 88$

$33 x - 3 y - 6 z = (3) (0)$

$- 33 x - 11 y + 33 z = 88$

$33 x - 3 y - 6 z = (3) (0)$

$- 33 x - 11 y + 33 z = 88$

$33 x - 3 y - 6 z = (3) (0)$

$- 33 x - 11 y + 33 z = 88$

$33 x - 3 y - 6 z = (3) (0)$

Step 4.2.4

Simplify the right side.

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Step 4.2.4.1

Multiply $3$ by $0$ .

$- 33 x - 11 y + 33 z = 88$

$33 x - 3 y - 6 z = 0$

$- 33 x - 11 y + 33 z = 88$

$33 x - 3 y - 6 z = 0$

$- 33 x - 11 y + 33 z = 88$

$33 x - 3 y - 6 z = 0$

Step 4.3

Add the two equations together to eliminate $x$ from the system.

	$-$	$3$	$3$	$x$	$-$	$1$	$1$	$y$	$+$	$3$	$3$	$z$	$=$	$8$	$8$
$+$		$3$	$3$	$x$		$-$	$3$	$y$		$-$	$6$	$z$	$=$		$0$
					$-$	$1$	$4$	$y$	$+$	$2$	$7$	$z$	$=$	$8$	$8$

Step 4.4

The resultant equation has $x$ eliminated.

$- 14 y + 27 z = 88$

Step 5

Take the resultant equations and eliminate another variable. In this case, eliminate $y$ .

$4 y - 5 z = - 17$

$- 14 y + 27 z = 88$

Step 6

Eliminate $y$ from the system.

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Step 6.1

Multiply each equation by the value that makes the coefficients of $y$ opposite.

$(7) \cdot (4 y - 5 z) = (7) (- 17)$

$(2) \cdot (- 14 y + 27 z) = (2) (88)$

Step 6.2

Simplify.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Simplify $(7) \cdot (4 y - 5 z)$ .

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Step 6.2.1.1.1

Apply the distributive property.

$7 (4 y) + 7 (- 5 z) = (7) (- 17)$

$(2) \cdot (- 14 y + 27 z) = (2) (88)$

Step 6.2.1.1.2

Multiply.

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Step 6.2.1.1.2.1

Multiply $4$ by $7$ .

$28 y + 7 (- 5 z) = (7) (- 17)$

$(2) \cdot (- 14 y + 27 z) = (2) (88)$

Step 6.2.1.1.2.2

Multiply $- 5$ by $7$ .

$28 y - 35 z = (7) (- 17)$

$(2) \cdot (- 14 y + 27 z) = (2) (88)$

$28 y - 35 z = (7) (- 17)$

$(2) \cdot (- 14 y + 27 z) = (2) (88)$

$28 y - 35 z = (7) (- 17)$

$(2) \cdot (- 14 y + 27 z) = (2) (88)$

$28 y - 35 z = (7) (- 17)$

$(2) \cdot (- 14 y + 27 z) = (2) (88)$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Multiply $7$ by $- 17$ .

$28 y - 35 z = - 119$

$(2) \cdot (- 14 y + 27 z) = (2) (88)$

$28 y - 35 z = - 119$

$(2) \cdot (- 14 y + 27 z) = (2) (88)$

Step 6.2.3

Simplify the left side.

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Step 6.2.3.1

Simplify $(2) \cdot (- 14 y + 27 z)$ .

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Step 6.2.3.1.1

Apply the distributive property.

$28 y - 35 z = - 119$

$2 (- 14 y) + 2 (27 z) = (2) (88)$

Step 6.2.3.1.2

Multiply.

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Step 6.2.3.1.2.1

Multiply $- 14$ by $2$ .

$28 y - 35 z = - 119$

$- 28 y + 2 (27 z) = (2) (88)$

Step 6.2.3.1.2.2

Multiply $27$ by $2$ .

$28 y - 35 z = - 119$

$- 28 y + 54 z = (2) (88)$

$28 y - 35 z = - 119$

$- 28 y + 54 z = (2) (88)$

$28 y - 35 z = - 119$

$- 28 y + 54 z = (2) (88)$

$28 y - 35 z = - 119$

$- 28 y + 54 z = (2) (88)$

Step 6.2.4

Simplify the right side.

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Step 6.2.4.1

Multiply $2$ by $88$ .

$28 y - 35 z = - 119$

$- 28 y + 54 z = 176$

$28 y - 35 z = - 119$

$- 28 y + 54 z = 176$

$28 y - 35 z = - 119$

$- 28 y + 54 z = 176$

Step 6.3

Add the two equations together to eliminate $y$ from the system.

		$2$	$8$	$y$	$-$	$3$	$5$	$z$	$=$	$-$	$1$	$1$	$9$
$+$	$-$	$2$	$8$	$y$	$+$	$5$	$4$	$z$	$=$		$1$	$7$	$6$
						$1$	$9$	$z$	$=$			$5$	$7$

Step 6.4

The resultant equation has $y$ eliminated.

$19 z = 57$

Step 6.5

Divide each term in $19 z = 57$ by $19$ and simplify.

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Step 6.5.1

Divide each term in $19 z = 57$ by $19$ .

$\frac{19 z}{19} = \frac{57}{19}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $19$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{19 z}{19} = \frac{57}{19}$

Step 6.5.2.1.2

Divide $z$ by $1$ .

$z = \frac{57}{19}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Divide $57$ by $19$ .

$z = 3$

Step 7

Substitute the value of $z$ into an equation with $x$ eliminated already and solve for the remaining variable.

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Step 7.1

Substitute the value of $z$ into an equation with $x$ eliminated already.

$4 y - 5 \cdot 3 = - 17$

Step 7.2

Solve for $y$ .

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Step 7.2.1

Multiply $- 5$ by $3$ .

$4 y - 15 = - 17$

Step 7.2.2

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

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Step 7.2.2.1

Add $15$ to both sides of the equation.

$4 y = - 17 + 15$

Step 7.2.2.2

Add $- 17$ and $15$ .

$4 y = - 2$

Step 7.2.3

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

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Step 7.2.3.1

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

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

Step 7.2.3.2

Simplify the left side.

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Step 7.2.3.2.1

Cancel the common factor of $4$ .

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Step 7.2.3.2.1.1

Cancel the common factor.

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

Step 7.2.3.2.1.2

Divide $y$ by $1$ .

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

Step 7.2.3.3

Simplify the right side.

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Step 7.2.3.3.1

Cancel the common factor of $- 2$ and $4$ .

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Step 7.2.3.3.1.1

Factor $2$ out of $- 2$ .

$y = \frac{2 (- 1)}{4}$

Step 7.2.3.3.1.2

Cancel the common factors.

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Step 7.2.3.3.1.2.1

Factor $2$ out of $4$ .

$y = \frac{2 \cdot - 1}{2 \cdot 2}$

Step 7.2.3.3.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot - 1}{2 \cdot 2}$

Step 7.2.3.3.1.2.3

Rewrite the expression.

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

Step 7.2.3.3.2

Move the negative in front of the fraction.

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

Step 8

Substitute the value of each known variable into one of the initial equations and solve for the last variable.

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Step 8.1

Substitute the value of each known variable into one of the initial equations.

$3 x - 3 (- \frac{1}{2}) + 2 (3) = 9$

Step 8.2

Solve for $x$ .

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Step 8.2.1

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

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Step 8.2.1.1

Simplify each term.

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Step 8.2.1.1.1

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

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Step 8.2.1.1.1.1

Multiply $- 1$ by $- 3$ .

$3 x + 3 (\frac{1}{2}) + 2 (3) = 9$

Step 8.2.1.1.1.2

Combine $3$ and $\frac{1}{2}$ .

$3 x + \frac{3}{2} + 2 (3) = 9$

Step 8.2.1.1.2

Multiply $2$ by $3$ .

$3 x + \frac{3}{2} + 6 = 9$

Step 8.2.1.2

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

$3 x + \frac{3}{2} + 6 \cdot \frac{2}{2} = 9$

Step 8.2.1.3

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

$3 x + \frac{3}{2} + \frac{6 \cdot 2}{2} = 9$

Step 8.2.1.4

Combine the numerators over the common denominator.

$3 x + \frac{3 + 6 \cdot 2}{2} = 9$

Step 8.2.1.5

Simplify the numerator.

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Step 8.2.1.5.1

Multiply $6$ by $2$ .

$3 x + \frac{3 + 12}{2} = 9$

Step 8.2.1.5.2

Add $3$ and $12$ .

$3 x + \frac{15}{2} = 9$

Step 8.2.2

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

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Step 8.2.2.1

Subtract $\frac{15}{2}$ from both sides of the equation.

$3 x = 9 - \frac{15}{2}$

Step 8.2.2.2

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 x = 9 \cdot \frac{2}{2} - \frac{15}{2}$

Step 8.2.2.3

Combine $9$ and $\frac{2}{2}$ .

$3 x = \frac{9 \cdot 2}{2} - \frac{15}{2}$

Step 8.2.2.4

Combine the numerators over the common denominator.

$3 x = \frac{9 \cdot 2 - 15}{2}$

Step 8.2.2.5

Simplify the numerator.

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Step 8.2.2.5.1

Multiply $9$ by $2$ .

$3 x = \frac{18 - 15}{2}$

Step 8.2.2.5.2

Subtract $15$ from $18$ .

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

Step 8.2.3

Divide each term in $3 x = \frac{3}{2}$ by $3$ and simplify.

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Step 8.2.3.1

Divide each term in $3 x = \frac{3}{2}$ by $3$ .

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

Step 8.2.3.2

Simplify the left side.

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Step 8.2.3.2.1

Cancel the common factor of $3$ .

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Step 8.2.3.2.1.1

Cancel the common factor.

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

Step 8.2.3.2.1.2

Divide $x$ by $1$ .

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

Step 8.2.3.3

Simplify the right side.

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Step 8.2.3.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2.3.3.2

Cancel the common factor of $3$ .

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Step 8.2.3.3.2.1

Cancel the common factor.

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

Step 8.2.3.3.2.2

Rewrite the expression.

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

Step 9

The solution to the system of equations can be represented as a point.

$(\frac{1}{2}, - \frac{1}{2}, 3)$

Step 10

The result can be shown in multiple forms.

Point Form:

$(\frac{1}{2}, - \frac{1}{2}, 3)$

Equation Form:

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

	$-$	$3$	$x$	$+$	$3$	$y$	$-$	$2$	$z$	$=$	$-$	$9$
$+$		$3$	$x$	$+$		$y$	$-$	$3$	$z$	$=$	$-$	$8$
					$4$	$y$	$-$	$5$	$z$	$=$	$-$	$1$	$7$

	$-$	$3$	$3$	$x$	$-$	$1$	$1$	$y$	$+$	$3$	$3$	$z$	$=$	$8$	$8$
$+$		$3$	$3$	$x$		$-$	$3$	$y$		$-$	$6$	$z$	$=$		$0$
					$-$	$1$	$4$	$y$	$+$	$2$	$7$	$z$	$=$	$8$	$8$

		$2$	$8$	$y$	$-$	$3$	$5$	$z$	$=$	$-$	$1$	$1$	$9$
$+$	$-$	$2$	$8$	$y$	$+$	$5$	$4$	$z$	$=$		$1$	$7$	$6$
						$1$	$9$	$z$	$=$			$5$	$7$

	$-$	$3$	$x$	$+$	$3$	$y$	$-$	$2$	$z$	$=$	$-$	$9$
$+$		$3$	$x$	$+$		$y$	$-$	$3$	$z$	$=$	$-$	$8$
					$4$	$y$	$-$	$5$	$z$	$=$	$-$	$1$	$7$

	$-$	$3$	$3$	$x$	$-$	$1$	$1$	$y$	$+$	$3$	$3$	$z$	$=$	$8$	$8$
$+$		$3$	$3$	$x$		$-$	$3$	$y$		$-$	$6$	$z$	$=$		$0$
					$-$	$1$	$4$	$y$	$+$	$2$	$7$	$z$	$=$	$8$	$8$

		$2$	$8$	$y$	$-$	$3$	$5$	$z$	$=$	$-$	$1$	$1$	$9$
$+$	$-$	$2$	$8$	$y$	$+$	$5$	$4$	$z$	$=$		$1$	$7$	$6$
						$1$	$9$	$z$	$=$			$5$	$7$

	$-$	$3$	$x$	$+$	$3$	$y$	$-$	$2$	$z$	$=$	$-$	$9$
$+$		$3$	$x$	$+$		$y$	$-$	$3$	$z$	$=$	$-$	$8$
					$4$	$y$	$-$	$5$	$z$	$=$	$-$	$1$	$7$

	$-$	$3$	$3$	$x$	$-$	$1$	$1$	$y$	$+$	$3$	$3$	$z$	$=$	$8$	$8$
$+$		$3$	$3$	$x$		$-$	$3$	$y$		$-$	$6$	$z$	$=$		$0$
					$-$	$1$	$4$	$y$	$+$	$2$	$7$	$z$	$=$	$8$	$8$

		$2$	$8$	$y$	$-$	$3$	$5$	$z$	$=$	$-$	$1$	$1$	$9$
$+$	$-$	$2$	$8$	$y$	$+$	$5$	$4$	$z$	$=$		$1$	$7$	$6$
						$1$	$9$	$z$	$=$			$5$	$7$