Solve the System of Equations 3x+2y-z=3 -x+4y-z=-7 -9x-3y+z=-6

$3 x + 2 y - z = 3$ $- x + 4 y - z = - 7$ $- 9 x - 3 y + z = - 6$

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

Add $9 x$ to both sides of the equation.

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

$3 x + 2 y - z = 3$

$- x + 4 y - z = - 7$

Add $3 y$ to both sides of the equation.

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

$3 x + 2 y - z = 3$

$- x + 4 y - z = - 7$

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

$3 x + 2 y - z = 3$

$- x + 4 y - z = - 7$

Replace all occurrences of $z$ with $- 6 + 9 x + 3 y$ in each equation.

Replace all occurrences of $z$ in $3 x + 2 y - z = 3$ with $- 6 + 9 x + 3 y$ .

$3 x + 2 y - (- 6 + 9 x + 3 y) = 3$

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

$- x + 4 y - z = - 7$

Simplify the left side.

Simplify $3 x + 2 y - (- 6 + 9 x + 3 y)$ .

Simplify each term.

Apply the distributive property.

$3 x + 2 y + 6 - (9 x) - (3 y) = 3$

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

$- x + 4 y - z = - 7$

Simplify.

Multiply $- 1$ by $- 6$ .

$3 x + 2 y + 6 - (9 x) - (3 y) = 3$

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

$- x + 4 y - z = - 7$

Multiply $9$ by $- 1$ .

$3 x + 2 y + 6 - 9 x - (3 y) = 3$

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

$- x + 4 y - z = - 7$

Multiply $3$ by $- 1$ .

$3 x + 2 y + 6 - 9 x - 3 y = 3$

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

$- x + 4 y - z = - 7$

$3 x + 2 y + 6 - 9 x - 3 y = 3$

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

$- x + 4 y - z = - 7$

$3 x + 2 y + 6 - 9 x - 3 y = 3$

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

$- x + 4 y - z = - 7$

Simplify by adding terms.

Subtract $9 x$ from $3 x$ .

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

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

$- x + 4 y - z = - 7$

Subtract $3 y$ from $2 y$ .

$- 6 x - y + 6 = 3$

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

$- x + 4 y - z = - 7$

$- 6 x - y + 6 = 3$

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

$- x + 4 y - z = - 7$

$- 6 x - y + 6 = 3$

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

$- x + 4 y - z = - 7$

$- 6 x - y + 6 = 3$

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

$- x + 4 y - z = - 7$

Replace all occurrences of $z$ in $- x + 4 y - z = - 7$ with $- 6 + 9 x + 3 y$ .

$- x + 4 y - (- 6 + 9 x + 3 y) = - 7$

$- 6 x - y + 6 = 3$

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

Simplify the left side.

Simplify $- x + 4 y - (- 6 + 9 x + 3 y)$ .

Simplify each term.

Apply the distributive property.

$- x + 4 y + 6 - (9 x) - (3 y) = - 7$

$- 6 x - y + 6 = 3$

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

Simplify.

Multiply $- 1$ by $- 6$ .

$- x + 4 y + 6 - (9 x) - (3 y) = - 7$

$- 6 x - y + 6 = 3$

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

Multiply $9$ by $- 1$ .

$- x + 4 y + 6 - 9 x - (3 y) = - 7$

$- 6 x - y + 6 = 3$

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

Multiply $3$ by $- 1$ .

$- x + 4 y + 6 - 9 x - 3 y = - 7$

$- 6 x - y + 6 = 3$

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

$- x + 4 y + 6 - 9 x - 3 y = - 7$

$- 6 x - y + 6 = 3$

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

$- x + 4 y + 6 - 9 x - 3 y = - 7$

$- 6 x - y + 6 = 3$

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

Simplify by adding terms.

Subtract $9 x$ from $- x$ .

$- 10 x + 4 y + 6 - 3 y = - 7$

$- 6 x - y + 6 = 3$

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

Subtract $3 y$ from $4 y$ .

$- 10 x + y + 6 = - 7$

$- 6 x - y + 6 = 3$

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

$- 10 x + y + 6 = - 7$

$- 6 x - y + 6 = 3$

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

$- 10 x + y + 6 = - 7$

$- 6 x - y + 6 = 3$

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

$- 10 x + y + 6 = - 7$

$- 6 x - y + 6 = 3$

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

$- 10 x + y + 6 = - 7$

$- 6 x - y + 6 = 3$

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

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

Add $10 x$ to both sides of the equation.

$y + 6 = - 7 + 10 x$

$- 6 x - y + 6 = 3$

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

Subtract $6$ from both sides of the equation.

$y = - 7 + 10 x - 6$

$- 6 x - y + 6 = 3$

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

Subtract $6$ from $- 7$ .

$y = 10 x - 13$

$- 6 x - y + 6 = 3$

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

$y = 10 x - 13$

$- 6 x - y + 6 = 3$

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

Replace all occurrences of $y$ with $10 x - 13$ in each equation.

Replace all occurrences of $y$ in $- 6 x - y + 6 = 3$ with $10 x - 13$ .

$- 6 x - (10 x - 13) + 6 = 3$

$y = 10 x - 13$

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

Simplify the left side.

Simplify $- 6 x - (10 x - 13) + 6$ .

Simplify each term.

Apply the distributive property.

$- 6 x - (10 x) + 13 + 6 = 3$

$y = 10 x - 13$

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

Multiply $10$ by $- 1$ .

$- 6 x - 10 x + 13 + 6 = 3$

$y = 10 x - 13$

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

Multiply $- 1$ by $- 13$ .

$- 6 x - 10 x + 13 + 6 = 3$

$y = 10 x - 13$

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

$- 6 x - 10 x + 13 + 6 = 3$

$y = 10 x - 13$

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

Simplify by adding terms.

Subtract $10 x$ from $- 6 x$ .

$- 16 x + 13 + 6 = 3$

$y = 10 x - 13$

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

Add $13$ and $6$ .

$- 16 x + 19 = 3$

$y = 10 x - 13$

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

$- 16 x + 19 = 3$

$y = 10 x - 13$

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

$- 16 x + 19 = 3$

$y = 10 x - 13$

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

$- 16 x + 19 = 3$

$y = 10 x - 13$

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

Replace all occurrences of $y$ in $z = - 6 + 9 x + 3 y$ with $10 x - 13$ .

$z = - 6 + 9 x + 3 (10 x - 13)$

$- 16 x + 19 = 3$

$y = 10 x - 13$

Simplify the right side.

Simplify $- 6 + 9 x + 3 (10 x - 13)$ .

Simplify each term.

Apply the distributive property.

$z = - 6 + 9 x + 3 (10 x) + 3 \cdot - 13$

$- 16 x + 19 = 3$

$y = 10 x - 13$

Multiply $10$ by $3$ .

$z = - 6 + 9 x + 30 x + 3 \cdot - 13$

$- 16 x + 19 = 3$

$y = 10 x - 13$

Multiply $3$ by $- 13$ .

$z = - 6 + 9 x + 30 x - 39$

$- 16 x + 19 = 3$

$y = 10 x - 13$

$z = - 6 + 9 x + 30 x - 39$

$- 16 x + 19 = 3$

$y = 10 x - 13$

Simplify by adding terms.

Subtract $39$ from $- 6$ .

$z = 9 x + 30 x - 45$

$- 16 x + 19 = 3$

$y = 10 x - 13$

Add $9 x$ and $30 x$ .

$z = 39 x - 45$

$- 16 x + 19 = 3$

$y = 10 x - 13$

$z = 39 x - 45$

$- 16 x + 19 = 3$

$y = 10 x - 13$

$z = 39 x - 45$

$- 16 x + 19 = 3$

$y = 10 x - 13$

$z = 39 x - 45$

$- 16 x + 19 = 3$

$y = 10 x - 13$

$z = 39 x - 45$

$- 16 x + 19 = 3$

$y = 10 x - 13$

Solve for $x$ in $- 16 x + 19 = 3$ .

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

Subtract $19$ from both sides of the equation.

$- 16 x = 3 - 19$

$z = 39 x - 45$

$y = 10 x - 13$

Subtract $19$ from $3$ .

$- 16 x = - 16$

$z = 39 x - 45$

$y = 10 x - 13$

$- 16 x = - 16$

$z = 39 x - 45$

$y = 10 x - 13$

Divide each term in $- 16 x = - 16$ by $- 16$ and simplify.

Divide each term in $- 16 x = - 16$ by $- 16$ .

$\frac{- 16 x}{- 16} = \frac{- 16}{- 16}$

$z = 39 x - 45$

$y = 10 x - 13$

Simplify the left side.

Cancel the common factor of $- 16$ .

Cancel the common factor.

$\frac{- 16 x}{- 16} = \frac{- 16}{- 16}$

$z = 39 x - 45$

$y = 10 x - 13$

Divide $x$ by $1$ .

$x = \frac{- 16}{- 16}$

$z = 39 x - 45$

$y = 10 x - 13$

$x = \frac{- 16}{- 16}$

$z = 39 x - 45$

$y = 10 x - 13$

$x = \frac{- 16}{- 16}$

$z = 39 x - 45$

$y = 10 x - 13$

Simplify the right side.

Divide $- 16$ by $- 16$ .

$x = 1$

$z = 39 x - 45$

$y = 10 x - 13$

$x = 1$

$z = 39 x - 45$

$y = 10 x - 13$

$x = 1$

$z = 39 x - 45$

$y = 10 x - 13$

$x = 1$

$z = 39 x - 45$

$y = 10 x - 13$

Replace all occurrences of $x$ with $1$ in each equation.

Replace all occurrences of $x$ in $z = 39 x - 45$ with $1$ .

$z = 39 (1) - 45$

$x = 1$

$y = 10 x - 13$

Simplify the right side.

Simplify $39 (1) - 45$ .

Multiply $39$ by $1$ .

$z = 39 - 45$

$x = 1$

$y = 10 x - 13$

Subtract $45$ from $39$ .

$z = - 6$

$x = 1$

$y = 10 x - 13$

$z = - 6$

$x = 1$

$y = 10 x - 13$

$z = - 6$

$x = 1$

$y = 10 x - 13$

Replace all occurrences of $x$ in $y = 10 x - 13$ with $1$ .

$y = 10 (1) - 13$

$z = - 6$

$x = 1$

Simplify the right side.

Simplify $10 (1) - 13$ .

Multiply $10$ by $1$ .

$y = 10 - 13$

$z = - 6$

$x = 1$

Subtract $13$ from $10$ .

$y = - 3$

$z = - 6$

$x = 1$

$y = - 3$

$z = - 6$

$x = 1$

$y = - 3$

$z = - 6$

$x = 1$

$y = - 3$

$z = - 6$

$x = 1$

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

$(1, - 3, - 6)$

The result can be shown in multiple forms.

Point Form:

$(1, - 3, - 6)$

Equation Form:

$x = 1, y = - 3, z = - 6$

Solve the System of Equations 3x+2y-z=3 -x+4y-z=-7 -9x-3y+z=-6

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