Solve the System of Equations 3x-y+z=-3 -2x+y-2z=8 -4x+3y-z=6

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

Step 1

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

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

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

$- 2 x + y - 2 z = 8$

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

Add $y$ to both sides of the equation.

$z = - 3 - 3 x + y$

$- 2 x + y - 2 z = 8$

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

$z = - 3 - 3 x + y$

$- 2 x + y - 2 z = 8$

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

Step 2

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

Replace all occurrences of $z$ in $- 2 x + y - 2 z = 8$ with $- 3 - 3 x + y$ .

$- 2 x + y - 2 (- 3 - 3 x + y) = 8$

$z = - 3 - 3 x + y$

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

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$- 2 x + y - 2 \cdot - 3 - 2 (- 3 x) - 2 y = 8$

$z = - 3 - 3 x + y$

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

Simplify.

Multiply $- 2$ by $- 3$ .

$- 2 x + y + 6 - 2 (- 3 x) - 2 y = 8$

$z = - 3 - 3 x + y$

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

Multiply $- 3$ by $- 2$ .

$- 2 x + y + 6 + 6 x - 2 y = 8$

$z = - 3 - 3 x + y$

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

$- 2 x + y + 6 + 6 x - 2 y = 8$

$z = - 3 - 3 x + y$

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

$- 2 x + y + 6 + 6 x - 2 y = 8$

$z = - 3 - 3 x + y$

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

Simplify by adding terms.

Add $- 2 x$ and $6 x$ .

$4 x + y + 6 - 2 y = 8$

$z = - 3 - 3 x + y$

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

Subtract $2 y$ from $y$ .

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

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

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

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

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

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

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

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

Replace all occurrences of $z$ in $- 4 x + 3 y - z = 6$ with $- 3 - 3 x + y$ .

$- 4 x + 3 y - (- 3 - 3 x + y) = 6$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$- 4 x + 3 y + 3 - (- 3 x) - y = 6$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

Simplify.

Multiply $- 1$ by $- 3$ .

$- 4 x + 3 y + 3 - (- 3 x) - y = 6$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

Multiply $- 3$ by $- 1$ .

$- 4 x + 3 y + 3 + 3 x - y = 6$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

$- 4 x + 3 y + 3 + 3 x - y = 6$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

$- 4 x + 3 y + 3 + 3 x - y = 6$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

Simplify by adding terms.

Add $- 4 x$ and $3 x$ .

$- x + 3 y + 3 - y = 6$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

Subtract $y$ from $3 y$ .

$- x + 2 y + 3 = 6$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

$- x + 2 y + 3 = 6$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

$- x + 2 y + 3 = 6$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

$- x + 2 y + 3 = 6$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

$- x + 2 y + 3 = 6$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

Step 3

Solve for $x$ in $- x + 2 y + 3 = 6$ .

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

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

$- x + 3 = 6 - 2 y$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

Subtract $3$ from both sides of the equation.

$- x = 6 - 2 y - 3$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

Subtract $3$ from $6$ .

$- x = - 2 y + 3$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

$- x = - 2 y + 3$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

Divide each term in $- x = - 2 y + 3$ by $- 1$ and simplify.

Divide each term in $- x = - 2 y + 3$ by $- 1$ .

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

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

Simplify the left side.

Dividing two negative values results in a positive value.

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

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

Divide $x$ by $1$ .

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

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

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

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{- 2 y}{- 1}$ .

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

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

Rewrite $- 1 \cdot (- 2 y)$ as $- (- 2 y)$ .

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

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

Multiply $- 2$ by $- 1$ .

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

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

Divide $3$ by $- 1$ .

$x = 2 y - 3$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

$x = 2 y - 3$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

$x = 2 y - 3$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

$x = 2 y - 3$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

$x = 2 y - 3$

$4 x - y + 6 = 8$

$z = - 3 - 3 x + y$

Step 4

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

Replace all occurrences of $x$ in $4 x - y + 6 = 8$ with $2 y - 3$ .

$4 (2 y - 3) - y + 6 = 8$

$x = 2 y - 3$

$z = - 3 - 3 x + y$

Simplify the left side.

Simplify $4 (2 y - 3) - y + 6$ .

Simplify each term.

Apply the distributive property.

$4 (2 y) + 4 \cdot - 3 - y + 6 = 8$

$x = 2 y - 3$

$z = - 3 - 3 x + y$

Multiply $2$ by $4$ .

$8 y + 4 \cdot - 3 - y + 6 = 8$

$x = 2 y - 3$

$z = - 3 - 3 x + y$

Multiply $4$ by $- 3$ .

$8 y - 12 - y + 6 = 8$

$x = 2 y - 3$

$z = - 3 - 3 x + y$

$8 y - 12 - y + 6 = 8$

$x = 2 y - 3$

$z = - 3 - 3 x + y$

Simplify by adding terms.

Subtract $y$ from $8 y$ .

$7 y - 12 + 6 = 8$

$x = 2 y - 3$

$z = - 3 - 3 x + y$

Add $- 12$ and $6$ .

$7 y - 6 = 8$

$x = 2 y - 3$

$z = - 3 - 3 x + y$

$7 y - 6 = 8$

$x = 2 y - 3$

$z = - 3 - 3 x + y$

$7 y - 6 = 8$

$x = 2 y - 3$

$z = - 3 - 3 x + y$

$7 y - 6 = 8$

$x = 2 y - 3$

$z = - 3 - 3 x + y$

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

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

$7 y - 6 = 8$

$x = 2 y - 3$

Simplify the right side.

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

Simplify each term.

Apply the distributive property.

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

$7 y - 6 = 8$

$x = 2 y - 3$

Multiply $2$ by $- 3$ .

$z = - 3 - 6 y - 3 \cdot - 3 + y$

$7 y - 6 = 8$

$x = 2 y - 3$

Multiply $- 3$ by $- 3$ .

$z = - 3 - 6 y + 9 + y$

$7 y - 6 = 8$

$x = 2 y - 3$

$z = - 3 - 6 y + 9 + y$

$7 y - 6 = 8$

$x = 2 y - 3$

Simplify by adding terms.

Add $- 3$ and $9$ .

$z = - 6 y + 6 + y$

$7 y - 6 = 8$

$x = 2 y - 3$

Add $- 6 y$ and $y$ .

$z = - 5 y + 6$

$7 y - 6 = 8$

$x = 2 y - 3$

$z = - 5 y + 6$

$7 y - 6 = 8$

$x = 2 y - 3$

$z = - 5 y + 6$

$7 y - 6 = 8$

$x = 2 y - 3$

$z = - 5 y + 6$

$7 y - 6 = 8$

$x = 2 y - 3$

$z = - 5 y + 6$

$7 y - 6 = 8$

$x = 2 y - 3$

Step 5

Solve for $y$ in $7 y - 6 = 8$ .

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

Add $6$ to both sides of the equation.

$7 y = 8 + 6$

$z = - 5 y + 6$

$x = 2 y - 3$

Add $8$ and $6$ .

$7 y = 14$

$z = - 5 y + 6$

$x = 2 y - 3$

$7 y = 14$

$z = - 5 y + 6$

$x = 2 y - 3$

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

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

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

$z = - 5 y + 6$

$x = 2 y - 3$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

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

$z = - 5 y + 6$

$x = 2 y - 3$

Divide $y$ by $1$ .

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

$z = - 5 y + 6$

$x = 2 y - 3$

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

$z = - 5 y + 6$

$x = 2 y - 3$

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

$z = - 5 y + 6$

$x = 2 y - 3$

Simplify the right side.

Divide $14$ by $7$ .

$y = 2$

$z = - 5 y + 6$

$x = 2 y - 3$

$y = 2$

$z = - 5 y + 6$

$x = 2 y - 3$

$y = 2$

$z = - 5 y + 6$

$x = 2 y - 3$

$y = 2$

$z = - 5 y + 6$

$x = 2 y - 3$

Step 6

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

Replace all occurrences of $y$ in $z = - 5 y + 6$ with $2$ .

$z = - 5 \cdot 2 + 6$

$y = 2$

$x = 2 y - 3$

Simplify the right side.

Simplify $- 5 \cdot 2 + 6$ .

Multiply $- 5$ by $2$ .

$z = - 10 + 6$

$y = 2$

$x = 2 y - 3$

Add $- 10$ and $6$ .

$z = - 4$

$y = 2$

$x = 2 y - 3$

$z = - 4$

$y = 2$

$x = 2 y - 3$

$z = - 4$

$y = 2$

$x = 2 y - 3$

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

$x = 2 (2) - 3$

$z = - 4$

$y = 2$

Simplify the right side.

Simplify $2 (2) - 3$ .

Multiply $2$ by $2$ .

$x = 4 - 3$

$z = - 4$

$y = 2$

Subtract $3$ from $4$ .

$x = 1$

$z = - 4$

$y = 2$

$x = 1$

$z = - 4$

$y = 2$

$x = 1$

$z = - 4$

$y = 2$

$x = 1$

$z = - 4$

$y = 2$

Step 7

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

$(1, 2, - 4)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(1, 2, - 4)$

Equation Form:

$x = 1, y = 2, z = - 4$

Solve the System of Equations 3x-y+z=-3 -2x+y-2z=8 -4x+3y-z=6

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