Solve the System of Equations x+y+z=3 2x-y+2z=6 3x+2y-z=13

$x + y + z = 3$ $2 x - y + 2 z = 6$ $3 x + 2 y - z = 13$

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

Subtract $y$ from both sides of the equation.

$x + z = 3 - y$

$2 x - y + 2 z = 6$

$3 x + 2 y - z = 13$

Subtract $z$ from both sides of the equation.

$x = 3 - y - z$

$2 x - y + 2 z = 6$

$3 x + 2 y - z = 13$

$x = 3 - y - z$

$2 x - y + 2 z = 6$

$3 x + 2 y - z = 13$

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

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

$2 (3 - y - z) - y + 2 z = 6$

$x = 3 - y - z$

$3 x + 2 y - z = 13$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$x = 3 - y - z$

$3 x + 2 y - z = 13$

Simplify.

Multiply $2$ by $3$ .

$6 + 2 (- y) + 2 (- z) - y + 2 z = 6$

$x = 3 - y - z$

$3 x + 2 y - z = 13$

Multiply $- 1$ by $2$ .

$6 - 2 y + 2 (- z) - y + 2 z = 6$

$x = 3 - y - z$

$3 x + 2 y - z = 13$

Multiply $- 1$ by $2$ .

$6 - 2 y - 2 z - y + 2 z = 6$

$x = 3 - y - z$

$3 x + 2 y - z = 13$

$6 - 2 y - 2 z - y + 2 z = 6$

$x = 3 - y - z$

$3 x + 2 y - z = 13$

$6 - 2 y - 2 z - y + 2 z = 6$

$x = 3 - y - z$

$3 x + 2 y - z = 13$

Simplify by adding terms.

Combine the opposite terms in $6 - 2 y - 2 z - y + 2 z$ .

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

$6 - 2 y - y + 0 = 6$

$x = 3 - y - z$

$3 x + 2 y - z = 13$

Add $6 - 2 y - y$ and $0$ .

$6 - 2 y - y = 6$

$x = 3 - y - z$

$3 x + 2 y - z = 13$

$6 - 2 y - y = 6$

$x = 3 - y - z$

$3 x + 2 y - z = 13$

Subtract $y$ from $- 2 y$ .

$6 - 3 y = 6$

$x = 3 - y - z$

$3 x + 2 y - z = 13$

$6 - 3 y = 6$

$x = 3 - y - z$

$3 x + 2 y - z = 13$

$6 - 3 y = 6$

$x = 3 - y - z$

$3 x + 2 y - z = 13$

$6 - 3 y = 6$

$x = 3 - y - z$

$3 x + 2 y - z = 13$

Replace all occurrences of $x$ in $3 x + 2 y - z = 13$ with $3 - y - z$ .

$3 (3 - y - z) + 2 y - z = 13$

$6 - 3 y = 6$

$x = 3 - y - z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$3 \cdot 3 + 3 (- y) + 3 (- z) + 2 y - z = 13$

$6 - 3 y = 6$

$x = 3 - y - z$

Simplify.

Multiply $3$ by $3$ .

$9 + 3 (- y) + 3 (- z) + 2 y - z = 13$

$6 - 3 y = 6$

$x = 3 - y - z$

Multiply $- 1$ by $3$ .

$9 - 3 y + 3 (- z) + 2 y - z = 13$

$6 - 3 y = 6$

$x = 3 - y - z$

Multiply $- 1$ by $3$ .

$9 - 3 y - 3 z + 2 y - z = 13$

$6 - 3 y = 6$

$x = 3 - y - z$

$9 - 3 y - 3 z + 2 y - z = 13$

$6 - 3 y = 6$

$x = 3 - y - z$

$9 - 3 y - 3 z + 2 y - z = 13$

$6 - 3 y = 6$

$x = 3 - y - z$

Simplify by adding terms.

Add $- 3 y$ and $2 y$ .

$9 - y - 3 z - z = 13$

$6 - 3 y = 6$

$x = 3 - y - z$

Subtract $z$ from $- 3 z$ .

$9 - y - 4 z = 13$

$6 - 3 y = 6$

$x = 3 - y - z$

$9 - y - 4 z = 13$

$6 - 3 y = 6$

$x = 3 - y - z$

$9 - y - 4 z = 13$

$6 - 3 y = 6$

$x = 3 - y - z$

$9 - y - 4 z = 13$

$6 - 3 y = 6$

$x = 3 - y - z$

$9 - y - 4 z = 13$

$6 - 3 y = 6$

$x = 3 - y - z$

Solve for $y$ in $6 - 3 y = 6$ .

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

Subtract $6$ from both sides of the equation.

$- 3 y = 6 - 6$

$9 - y - 4 z = 13$

$x = 3 - y - z$

Subtract $6$ from $6$ .

$- 3 y = 0$

$9 - y - 4 z = 13$

$x = 3 - y - z$

$- 3 y = 0$

$9 - y - 4 z = 13$

$x = 3 - y - z$

Divide each term in $- 3 y = 0$ by $- 3$ and simplify.

Divide each term in $- 3 y = 0$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{0}{- 3}$

$9 - y - 4 z = 13$

$x = 3 - y - z$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{0}{- 3}$

$9 - y - 4 z = 13$

$x = 3 - y - z$

Divide $y$ by $1$ .

$y = \frac{0}{- 3}$

$9 - y - 4 z = 13$

$x = 3 - y - z$

$y = \frac{0}{- 3}$

$9 - y - 4 z = 13$

$x = 3 - y - z$

$y = \frac{0}{- 3}$

$9 - y - 4 z = 13$

$x = 3 - y - z$

Simplify the right side.

Divide $0$ by $- 3$ .

$y = 0$

$9 - y - 4 z = 13$

$x = 3 - y - z$

$y = 0$

$9 - y - 4 z = 13$

$x = 3 - y - z$

$y = 0$

$9 - y - 4 z = 13$

$x = 3 - y - z$

$y = 0$

$9 - y - 4 z = 13$

$x = 3 - y - z$

Replace all occurrences of $y$ with $0$ in each equation.

Replace all occurrences of $y$ in $9 - y - 4 z = 13$ with $0$ .

$9 - (0) - 4 z = 13$

$y = 0$

$x = 3 - y - z$

Simplify the left side.

Simplify $9 - (0) - 4 z$ .

Multiply $- 1$ by $0$ .

$9 + 0 - 4 z = 13$

$y = 0$

$x = 3 - y - z$

Add $9$ and $0$ .

$9 - 4 z = 13$

$y = 0$

$x = 3 - y - z$

$9 - 4 z = 13$

$y = 0$

$x = 3 - y - z$

$9 - 4 z = 13$

$y = 0$

$x = 3 - y - z$

Replace all occurrences of $y$ in $x = 3 - y - z$ with $0$ .

$x = 3 - (0) - z$

$9 - 4 z = 13$

$y = 0$

Simplify the right side.

Simplify $3 - (0) - z$ .

Multiply $- 1$ by $0$ .

$x = 3 + 0 - z$

$9 - 4 z = 13$

$y = 0$

Add $3$ and $0$ .

$x = 3 - z$

$9 - 4 z = 13$

$y = 0$

$x = 3 - z$

$9 - 4 z = 13$

$y = 0$

$x = 3 - z$

$9 - 4 z = 13$

$y = 0$

$x = 3 - z$

$9 - 4 z = 13$

$y = 0$

Solve for $z$ in $9 - 4 z = 13$ .

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

Subtract $9$ from both sides of the equation.

$- 4 z = 13 - 9$

$x = 3 - z$

$y = 0$

Subtract $9$ from $13$ .

$- 4 z = 4$

$x = 3 - z$

$y = 0$

$- 4 z = 4$

$x = 3 - z$

$y = 0$

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

Divide each term in $- 4 z = 4$ by $- 4$ .

$\frac{- 4 z}{- 4} = \frac{4}{- 4}$

$x = 3 - z$

$y = 0$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 z}{- 4} = \frac{4}{- 4}$

$x = 3 - z$

$y = 0$

Divide $z$ by $1$ .

$z = \frac{4}{- 4}$

$x = 3 - z$

$y = 0$

$z = \frac{4}{- 4}$

$x = 3 - z$

$y = 0$

$z = \frac{4}{- 4}$

$x = 3 - z$

$y = 0$

Simplify the right side.

Divide $4$ by $- 4$ .

$z = - 1$

$x = 3 - z$

$y = 0$

$z = - 1$

$x = 3 - z$

$y = 0$

$z = - 1$

$x = 3 - z$

$y = 0$

$z = - 1$

$x = 3 - z$

$y = 0$

Replace all occurrences of $z$ with $- 1$ in each equation.

Replace all occurrences of $z$ in $x = 3 - z$ with $- 1$ .

$x = 3 - (- 1)$

$z = - 1$

$y = 0$

Simplify the right side.

Simplify $3 - (- 1)$ .

Multiply $- 1$ by $- 1$ .

$x = 3 + 1$

$z = - 1$

$y = 0$

Add $3$ and $1$ .

$x = 4$

$z = - 1$

$y = 0$

$x = 4$

$z = - 1$

$y = 0$

$x = 4$

$z = - 1$

$y = 0$

$x = 4$

$z = - 1$

$y = 0$

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

$(4, 0, - 1)$

The result can be shown in multiple forms.

Point Form:

$(4, 0, - 1)$

Equation Form:

$x = 4, y = 0, z = - 1$

Solve the System of Equations x+y+z=3 2x-y+2z=6 3x+2y-z=13

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