Solve the System of Equations x+y+z=12 2x-y+z=7 x+2y-z=6

$x + y + z = 12$ $2 x - y + z = 7$ $x + 2 y - z = 6$

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

Subtract $y$ from both sides of the equation.

$x + z = 12 - y$

$2 x - y + z = 7$

$x + 2 y - z = 6$

Subtract $z$ from both sides of the equation.

$x = 12 - y - z$

$2 x - y + z = 7$

$x + 2 y - z = 6$

$x = 12 - y - z$

$2 x - y + z = 7$

$x + 2 y - z = 6$

Replace all occurrences of $x$ with $12 - y - z$ in each equation.

Replace all occurrences of $x$ in $2 x - y + z = 7$ with $12 - y - z$ .

$2 (12 - y - z) - y + z = 7$

$x = 12 - y - z$

$x + 2 y - z = 6$

Simplify the left side.

Simplify $2 (12 - y - z) - y + z$ .

Simplify each term.

Apply the distributive property.

$2 \cdot 12 + 2 (- y) + 2 (- z) - y + z = 7$

$x = 12 - y - z$

$x + 2 y - z = 6$

Simplify.

Multiply $2$ by $12$ .

$24 + 2 (- y) + 2 (- z) - y + z = 7$

$x = 12 - y - z$

$x + 2 y - z = 6$

Multiply $- 1$ by $2$ .

$24 - 2 y + 2 (- z) - y + z = 7$

$x = 12 - y - z$

$x + 2 y - z = 6$

Multiply $- 1$ by $2$ .

$24 - 2 y - 2 z - y + z = 7$

$x = 12 - y - z$

$x + 2 y - z = 6$

$24 - 2 y - 2 z - y + z = 7$

$x = 12 - y - z$

$x + 2 y - z = 6$

$24 - 2 y - 2 z - y + z = 7$

$x = 12 - y - z$

$x + 2 y - z = 6$

Simplify by adding terms.

Subtract $y$ from $- 2 y$ .

$24 - 3 y - 2 z + z = 7$

$x = 12 - y - z$

$x + 2 y - z = 6$

Add $- 2 z$ and $z$ .

$24 - 3 y - z = 7$

$x = 12 - y - z$

$x + 2 y - z = 6$

$24 - 3 y - z = 7$

$x = 12 - y - z$

$x + 2 y - z = 6$

$24 - 3 y - z = 7$

$x = 12 - y - z$

$x + 2 y - z = 6$

$24 - 3 y - z = 7$

$x = 12 - y - z$

$x + 2 y - z = 6$

Replace all occurrences of $x$ in $x + 2 y - z = 6$ with $12 - y - z$ .

$(12 - y - z) + 2 y - z = 6$

$24 - 3 y - z = 7$

$x = 12 - y - z$

Simplify the left side.

Simplify $(12 - y - z) + 2 y - z$ .

Add $- y$ and $2 y$ .

$12 + y - z - z = 6$

$24 - 3 y - z = 7$

$x = 12 - y - z$

Subtract $z$ from $- z$ .

$12 + y - 2 z = 6$

$24 - 3 y - z = 7$

$x = 12 - y - z$

$12 + y - 2 z = 6$

$24 - 3 y - z = 7$

$x = 12 - y - z$

$12 + y - 2 z = 6$

$24 - 3 y - z = 7$

$x = 12 - y - z$

$12 + y - 2 z = 6$

$24 - 3 y - z = 7$

$x = 12 - y - z$

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

Subtract $12$ from both sides of the equation.

$y - 2 z = 6 - 12$

$24 - 3 y - z = 7$

$x = 12 - y - z$

Add $2 z$ to both sides of the equation.

$y = 6 - 12 + 2 z$

$24 - 3 y - z = 7$

$x = 12 - y - z$

Subtract $12$ from $6$ .

$y = - 6 + 2 z$

$24 - 3 y - z = 7$

$x = 12 - y - z$

$y = - 6 + 2 z$

$24 - 3 y - z = 7$

$x = 12 - y - z$

Replace all occurrences of $y$ with $- 6 + 2 z$ in each equation.

Replace all occurrences of $y$ in $24 - 3 y - z = 7$ with $- 6 + 2 z$ .

$24 - 3 (- 6 + 2 z) - z = 7$

$y = - 6 + 2 z$

$x = 12 - y - z$

Simplify the left side.

Simplify $24 - 3 (- 6 + 2 z) - z$ .

Simplify each term.

Apply the distributive property.

$24 - 3 \cdot - 6 - 3 (2 z) - z = 7$

$y = - 6 + 2 z$

$x = 12 - y - z$

Multiply $- 3$ by $- 6$ .

$24 + 18 - 3 (2 z) - z = 7$

$y = - 6 + 2 z$

$x = 12 - y - z$

Multiply $2$ by $- 3$ .

$24 + 18 - 6 z - z = 7$

$y = - 6 + 2 z$

$x = 12 - y - z$

$24 + 18 - 6 z - z = 7$

$y = - 6 + 2 z$

$x = 12 - y - z$

Simplify by adding terms.

Add $24$ and $18$ .

$42 - 6 z - z = 7$

$y = - 6 + 2 z$

$x = 12 - y - z$

Subtract $z$ from $- 6 z$ .

$42 - 7 z = 7$

$y = - 6 + 2 z$

$x = 12 - y - z$

$42 - 7 z = 7$

$y = - 6 + 2 z$

$x = 12 - y - z$

$42 - 7 z = 7$

$y = - 6 + 2 z$

$x = 12 - y - z$

$42 - 7 z = 7$

$y = - 6 + 2 z$

$x = 12 - y - z$

Replace all occurrences of $y$ in $x = 12 - y - z$ with $- 6 + 2 z$ .

$x = 12 - (- 6 + 2 z) - z$

$42 - 7 z = 7$

$y = - 6 + 2 z$

Simplify the right side.

Simplify $12 - (- 6 + 2 z) - z$ .

Simplify each term.

Apply the distributive property.

$x = 12 + 6 - (2 z) - z$

$42 - 7 z = 7$

$y = - 6 + 2 z$

Multiply $- 1$ by $- 6$ .

$x = 12 + 6 - (2 z) - z$

$42 - 7 z = 7$

$y = - 6 + 2 z$

Multiply $2$ by $- 1$ .

$x = 12 + 6 - 2 z - z$

$42 - 7 z = 7$

$y = - 6 + 2 z$

$x = 12 + 6 - 2 z - z$

$42 - 7 z = 7$

$y = - 6 + 2 z$

Simplify by adding terms.

Add $12$ and $6$ .

$x = 18 - 2 z - z$

$42 - 7 z = 7$

$y = - 6 + 2 z$

Subtract $z$ from $- 2 z$ .

$x = 18 - 3 z$

$42 - 7 z = 7$

$y = - 6 + 2 z$

$x = 18 - 3 z$

$42 - 7 z = 7$

$y = - 6 + 2 z$

$x = 18 - 3 z$

$42 - 7 z = 7$

$y = - 6 + 2 z$

$x = 18 - 3 z$

$42 - 7 z = 7$

$y = - 6 + 2 z$

$x = 18 - 3 z$

$42 - 7 z = 7$

$y = - 6 + 2 z$

Solve for $z$ in $42 - 7 z = 7$ .

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

Subtract $42$ from both sides of the equation.

$- 7 z = 7 - 42$

$x = 18 - 3 z$

$y = - 6 + 2 z$

Subtract $42$ from $7$ .

$- 7 z = - 35$

$x = 18 - 3 z$

$y = - 6 + 2 z$

$- 7 z = - 35$

$x = 18 - 3 z$

$y = - 6 + 2 z$

Divide each term in $- 7 z = - 35$ by $- 7$ and simplify.

Divide each term in $- 7 z = - 35$ by $- 7$ .

$\frac{- 7 z}{- 7} = \frac{- 35}{- 7}$

$x = 18 - 3 z$

$y = - 6 + 2 z$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 z}{- 7} = \frac{- 35}{- 7}$

$x = 18 - 3 z$

$y = - 6 + 2 z$

Divide $z$ by $1$ .

$z = \frac{- 35}{- 7}$

$x = 18 - 3 z$

$y = - 6 + 2 z$

$z = \frac{- 35}{- 7}$

$x = 18 - 3 z$

$y = - 6 + 2 z$

$z = \frac{- 35}{- 7}$

$x = 18 - 3 z$

$y = - 6 + 2 z$

Simplify the right side.

Divide $- 35$ by $- 7$ .

$z = 5$

$x = 18 - 3 z$

$y = - 6 + 2 z$

$z = 5$

$x = 18 - 3 z$

$y = - 6 + 2 z$

$z = 5$

$x = 18 - 3 z$

$y = - 6 + 2 z$

$z = 5$

$x = 18 - 3 z$

$y = - 6 + 2 z$

Replace all occurrences of $z$ with $5$ in each equation.

Replace all occurrences of $z$ in $x = 18 - 3 z$ with $5$ .

$x = 18 - 3 \cdot 5$

$z = 5$

$y = - 6 + 2 z$

Simplify the right side.

Simplify $18 - 3 \cdot 5$ .

Multiply $- 3$ by $5$ .

$x = 18 - 15$

$z = 5$

$y = - 6 + 2 z$

Subtract $15$ from $18$ .

$x = 3$

$z = 5$

$y = - 6 + 2 z$

$x = 3$

$z = 5$

$y = - 6 + 2 z$

$x = 3$

$z = 5$

$y = - 6 + 2 z$

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

$y = - 6 + 2 (5)$

$x = 3$

$z = 5$

Simplify the right side.

Simplify $- 6 + 2 (5)$ .

Multiply $2$ by $5$ .

$y = - 6 + 10$

$x = 3$

$z = 5$

Add $- 6$ and $10$ .

$y = 4$

$x = 3$

$z = 5$

$y = 4$

$x = 3$

$z = 5$

$y = 4$

$x = 3$

$z = 5$

$y = 4$

$x = 3$

$z = 5$

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

$(3, 4, 5)$

The result can be shown in multiple forms.

Point Form:

$(3, 4, 5)$

Equation Form:

$x = 3, y = 4, z = 5$

Solve the System of Equations x+y+z=12 2x-y+z=7 x+2y-z=6

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