Solve the System of Equations x+y+z=6 2x-y+z=3 3x-z=0

$x + y + z = 6$ $2 x - y + z = 3$ $3 x - z = 0$

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

Subtract $y$ from both sides of the equation.

$x + z = 6 - y$

$2 x - y + z = 3$

$3 x - z = 0$

Subtract $z$ from both sides of the equation.

$x = 6 - y - z$

$2 x - y + z = 3$

$3 x - z = 0$

$x = 6 - y - z$

$2 x - y + z = 3$

$3 x - z = 0$

Replace all occurrences of $x$ with $6 - y - z$ in each equation.

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

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

$x = 6 - y - z$

$3 x - z = 0$

Simplify the left side.

Simplify $2 (6 - y - z) - y + z$ .

Simplify each term.

Apply the distributive property.

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

$x = 6 - y - z$

$3 x - z = 0$

Simplify.

Multiply $2$ by $6$ .

$12 + 2 (- y) + 2 (- z) - y + z = 3$

$x = 6 - y - z$

$3 x - z = 0$

Multiply $- 1$ by $2$ .

$12 - 2 y + 2 (- z) - y + z = 3$

$x = 6 - y - z$

$3 x - z = 0$

Multiply $- 1$ by $2$ .

$12 - 2 y - 2 z - y + z = 3$

$x = 6 - y - z$

$3 x - z = 0$

$12 - 2 y - 2 z - y + z = 3$

$x = 6 - y - z$

$3 x - z = 0$

$12 - 2 y - 2 z - y + z = 3$

$x = 6 - y - z$

$3 x - z = 0$

Simplify by adding terms.

Subtract $y$ from $- 2 y$ .

$12 - 3 y - 2 z + z = 3$

$x = 6 - y - z$

$3 x - z = 0$

Add $- 2 z$ and $z$ .

$12 - 3 y - z = 3$

$x = 6 - y - z$

$3 x - z = 0$

$12 - 3 y - z = 3$

$x = 6 - y - z$

$3 x - z = 0$

$12 - 3 y - z = 3$

$x = 6 - y - z$

$3 x - z = 0$

$12 - 3 y - z = 3$

$x = 6 - y - z$

$3 x - z = 0$

Replace all occurrences of $x$ in $3 x - z = 0$ with $6 - y - z$ .

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

$12 - 3 y - z = 3$

$x = 6 - y - z$

Simplify the left side.

Simplify $3 (6 - y - z) - z$ .

Simplify each term.

Apply the distributive property.

$3 \cdot 6 + 3 (- y) + 3 (- z) - z = 0$

$12 - 3 y - z = 3$

$x = 6 - y - z$

Simplify.

Multiply $3$ by $6$ .

$18 + 3 (- y) + 3 (- z) - z = 0$

$12 - 3 y - z = 3$

$x = 6 - y - z$

Multiply $- 1$ by $3$ .

$18 - 3 y + 3 (- z) - z = 0$

$12 - 3 y - z = 3$

$x = 6 - y - z$

Multiply $- 1$ by $3$ .

$18 - 3 y - 3 z - z = 0$

$12 - 3 y - z = 3$

$x = 6 - y - z$

$18 - 3 y - 3 z - z = 0$

$12 - 3 y - z = 3$

$x = 6 - y - z$

$18 - 3 y - 3 z - z = 0$

$12 - 3 y - z = 3$

$x = 6 - y - z$

Subtract $z$ from $- 3 z$ .

$18 - 3 y - 4 z = 0$

$12 - 3 y - z = 3$

$x = 6 - y - z$

$18 - 3 y - 4 z = 0$

$12 - 3 y - z = 3$

$x = 6 - y - z$

$18 - 3 y - 4 z = 0$

$12 - 3 y - z = 3$

$x = 6 - y - z$

$18 - 3 y - 4 z = 0$

$12 - 3 y - z = 3$

$x = 6 - y - z$

Solve for $y$ in $18 - 3 y - 4 z = 0$ .

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

Subtract $18$ from both sides of the equation.

$- 3 y - 4 z = - 18$

$12 - 3 y - z = 3$

$x = 6 - y - z$

Add $4 z$ to both sides of the equation.

$- 3 y = - 18 + 4 z$

$12 - 3 y - z = 3$

$x = 6 - y - z$

$- 3 y = - 18 + 4 z$

$12 - 3 y - z = 3$

$x = 6 - y - z$

Divide each term in $- 3 y = - 18 + 4 z$ by $- 3$ and simplify.

Divide each term in $- 3 y = - 18 + 4 z$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{- 18}{- 3} + \frac{4 z}{- 3}$

$12 - 3 y - z = 3$

$x = 6 - y - z$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{- 18}{- 3} + \frac{4 z}{- 3}$

$12 - 3 y - z = 3$

$x = 6 - y - z$

Divide $y$ by $1$ .

$y = \frac{- 18}{- 3} + \frac{4 z}{- 3}$

$12 - 3 y - z = 3$

$x = 6 - y - z$

$y = \frac{- 18}{- 3} + \frac{4 z}{- 3}$

$12 - 3 y - z = 3$

$x = 6 - y - z$

$y = \frac{- 18}{- 3} + \frac{4 z}{- 3}$

$12 - 3 y - z = 3$

$x = 6 - y - z$

Simplify the right side.

Simplify each term.

Divide $- 18$ by $- 3$ .

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

$12 - 3 y - z = 3$

$x = 6 - y - z$

Move the negative in front of the fraction.

$y = 6 - \frac{4 z}{3}$

$12 - 3 y - z = 3$

$x = 6 - y - z$

$y = 6 - \frac{4 z}{3}$

$12 - 3 y - z = 3$

$x = 6 - y - z$

$y = 6 - \frac{4 z}{3}$

$12 - 3 y - z = 3$

$x = 6 - y - z$

$y = 6 - \frac{4 z}{3}$

$12 - 3 y - z = 3$

$x = 6 - y - z$

$y = 6 - \frac{4 z}{3}$

$12 - 3 y - z = 3$

$x = 6 - y - z$

Replace all occurrences of $y$ with $6 - \frac{4 z}{3}$ in each equation.

Replace all occurrences of $y$ in $12 - 3 y - z = 3$ with $6 - \frac{4 z}{3}$ .

$12 - 3 (6 - \frac{4 z}{3}) - z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 6 - y - z$

Simplify the left side.

Simplify $12 - 3 (6 - \frac{4 z}{3}) - z$ .

Simplify each term.

Apply the distributive property.

$12 - 3 \cdot 6 - 3 (- \frac{4 z}{3}) - z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 6 - y - z$

Multiply $- 3$ by $6$ .

$12 - 18 - 3 (- \frac{4 z}{3}) - z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 6 - y - z$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{4 z}{3}$ into the numerator.

$12 - 18 - 3 \frac{- 4 z}{3} - z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 6 - y - z$

Factor $3$ out of $- 3$ .

$12 - 18 + 3 (- 1) (\frac{- 4 z}{3}) - z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 6 - y - z$

Cancel the common factor.

$12 - 18 + 3 \cdot (- 1 \frac{- 4 z}{3}) - z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 6 - y - z$

Rewrite the expression.

$12 - 18 - 1 (- 4 z) - z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 6 - y - z$

$12 - 18 - 1 (- 4 z) - z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 6 - y - z$

Multiply $- 4$ by $- 1$ .

$12 - 18 + 4 z - z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 6 - y - z$

$12 - 18 + 4 z - z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 6 - y - z$

Simplify by adding terms.

Subtract $18$ from $12$ .

$- 6 + 4 z - z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 6 - y - z$

Subtract $z$ from $4 z$ .

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 6 - y - z$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 6 - y - z$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 6 - y - z$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 6 - y - z$

Replace all occurrences of $y$ in $x = 6 - y - z$ with $6 - \frac{4 z}{3}$ .

$x = 6 - (6 - \frac{4 z}{3}) - z$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

Simplify the right side.

Simplify $6 - (6 - \frac{4 z}{3}) - z$ .

Simplify each term.

Apply the distributive property.

$x = 6 - 1 \cdot 6 + \frac{4 z}{3} - z$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

Multiply $- 1$ by $6$ .

$x = 6 - 6 + \frac{4 z}{3} - z$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

Multiply $- - \frac{4 z}{3}$ .

Multiply $- 1$ by $- 1$ .

$x = 6 - 6 + 1 (\frac{4 z}{3}) - z$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

Multiply $\frac{4 z}{3}$ by $1$ .

$x = 6 - 6 + \frac{4 z}{3} - z$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 6 - 6 + \frac{4 z}{3} - z$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 6 - 6 + \frac{4 z}{3} - z$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

Simplify by subtracting numbers.

Subtract $6$ from $6$ .

$x = 0 + \frac{4 z}{3} - z$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

Add $0$ and $\frac{4 z}{3}$ .

$x = \frac{4 z}{3} - z$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

$x = \frac{4 z}{3} - z$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

To write $- z$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{4 z}{3} - z \cdot \frac{3}{3}$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

Simplify terms.

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

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

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

Combine the numerators over the common denominator.

$x = \frac{4 z - z \cdot 3}{3}$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

$x = \frac{4 z - z \cdot 3}{3}$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

Simplify the numerator.

Factor $z$ out of $4 z - z \cdot 3$ .

Factor $z$ out of $4 z$ .

$x = \frac{z \cdot 4 - z \cdot 3}{3}$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

Factor $z$ out of $- z \cdot 3$ .

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

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

Factor $z$ out of $z \cdot 4 + z (- 1 \cdot 3)$ .

$x = \frac{z (4 - 1 \cdot 3)}{3}$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

$x = \frac{z (4 - 1 \cdot 3)}{3}$

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

Multiply $- 1$ by $3$ .

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

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

Subtract $3$ from $4$ .

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

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

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

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

Multiply $z$ by $1$ .

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

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

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

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

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

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

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

$- 6 + 3 z = 3$

$y = 6 - \frac{4 z}{3}$

Solve for $z$ in $- 6 + 3 z = 3$ .

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

Add $6$ to both sides of the equation.

$3 z = 3 + 6$

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

$y = 6 - \frac{4 z}{3}$

Add $3$ and $6$ .

$3 z = 9$

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

$y = 6 - \frac{4 z}{3}$

$3 z = 9$

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

$y = 6 - \frac{4 z}{3}$

Divide each term in $3 z = 9$ by $3$ and simplify.

Divide each term in $3 z = 9$ by $3$ .

$\frac{3 z}{3} = \frac{9}{3}$

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

$y = 6 - \frac{4 z}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 z}{3} = \frac{9}{3}$

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

$y = 6 - \frac{4 z}{3}$

Divide $z$ by $1$ .

$z = \frac{9}{3}$

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

$y = 6 - \frac{4 z}{3}$

$z = \frac{9}{3}$

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

$y = 6 - \frac{4 z}{3}$

$z = \frac{9}{3}$

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

$y = 6 - \frac{4 z}{3}$

Simplify the right side.

Divide $9$ by $3$ .

$z = 3$

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

$y = 6 - \frac{4 z}{3}$

$z = 3$

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

$y = 6 - \frac{4 z}{3}$

$z = 3$

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

$y = 6 - \frac{4 z}{3}$

$z = 3$

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

$y = 6 - \frac{4 z}{3}$

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

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

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

$z = 3$

$y = 6 - \frac{4 z}{3}$

Simplify the right side.

Divide $3$ by $3$ .

$x = 1$

$z = 3$

$y = 6 - \frac{4 z}{3}$

$x = 1$

$z = 3$

$y = 6 - \frac{4 z}{3}$

Replace all occurrences of $z$ in $y = 6 - \frac{4 z}{3}$ with $3$ .

$y = 6 - \frac{4 (3)}{3}$

$x = 1$

$z = 3$

Simplify the right side.

Simplify $6 - \frac{4 (3)}{3}$ .

Simplify each term.

Cancel the common factor of $3$ .

Cancel the common factor.

$y = 6 - \frac{4 \cdot 3}{3}$

$x = 1$

$z = 3$

Divide $4$ by $1$ .

$y = 6 - 1 \cdot 4$

$x = 1$

$z = 3$

$y = 6 - 1 \cdot 4$

$x = 1$

$z = 3$

Multiply $- 1$ by $4$ .

$y = 6 - 4$

$x = 1$

$z = 3$

$y = 6 - 4$

$x = 1$

$z = 3$

Subtract $4$ from $6$ .

$y = 2$

$x = 1$

$z = 3$

$y = 2$

$x = 1$

$z = 3$

$y = 2$

$x = 1$

$z = 3$

$y = 2$

$x = 1$

$z = 3$

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

$(1, 2, 3)$

The result can be shown in multiple forms.

Point Form:

$(1, 2, 3)$

Equation Form:

$x = 1, y = 2, z = 3$

Solve the System of Equations x+y+z=6 2x-y+z=3 3x-z=0

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