Solve the System of Equations 3x-3y+6z=-21 -4x-y+z=15 -2x-y+z=7

$3 x - 3 y + 6 z = - 21$ $- 4 x - y + z = 15$ $- 2 x - y + z = 7$

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

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

$- y + z = 15 + 4 x$

$3 x - 3 y + 6 z = - 21$

$- 2 x - y + z = 7$

Add $y$ to both sides of the equation.

$z = 15 + 4 x + y$

$3 x - 3 y + 6 z = - 21$

$- 2 x - y + z = 7$

$z = 15 + 4 x + y$

$3 x - 3 y + 6 z = - 21$

$- 2 x - y + z = 7$

Replace all occurrences of $z$ with $15 + 4 x + y$ in each equation.

Replace all occurrences of $z$ in $3 x - 3 y + 6 z = - 21$ with $15 + 4 x + y$ .

$3 x - 3 y + 6 (15 + 4 x + y) = - 21$

$z = 15 + 4 x + y$

$- 2 x - y + z = 7$

Simplify the left side.

Simplify $3 x - 3 y + 6 (15 + 4 x + y)$ .

Simplify each term.

Apply the distributive property.

$3 x - 3 y + 6 \cdot 15 + 6 (4 x) + 6 y = - 21$

$z = 15 + 4 x + y$

$- 2 x - y + z = 7$

Simplify.

Multiply $6$ by $15$ .

$3 x - 3 y + 90 + 6 (4 x) + 6 y = - 21$

$z = 15 + 4 x + y$

$- 2 x - y + z = 7$

Multiply $4$ by $6$ .

$3 x - 3 y + 90 + 24 x + 6 y = - 21$

$z = 15 + 4 x + y$

$- 2 x - y + z = 7$

$3 x - 3 y + 90 + 24 x + 6 y = - 21$

$z = 15 + 4 x + y$

$- 2 x - y + z = 7$

$3 x - 3 y + 90 + 24 x + 6 y = - 21$

$z = 15 + 4 x + y$

$- 2 x - y + z = 7$

Simplify by adding terms.

Add $3 x$ and $24 x$ .

$27 x - 3 y + 90 + 6 y = - 21$

$z = 15 + 4 x + y$

$- 2 x - y + z = 7$

Add $- 3 y$ and $6 y$ .

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

$- 2 x - y + z = 7$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

$- 2 x - y + z = 7$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

$- 2 x - y + z = 7$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

$- 2 x - y + z = 7$

Replace all occurrences of $z$ in $- 2 x - y + z = 7$ with $15 + 4 x + y$ .

$- 2 x - y + 15 + 4 x + y = 7$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

Simplify the left side.

Simplify $- 2 x - y + 15 + 4 x + y$ .

Remove parentheses.

$- 2 x - y + 15 + 4 x + y = 7$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

Combine the opposite terms in $- 2 x - y + 15 + 4 x + y$ .

Add $- y$ and $y$ .

$- 2 x + 15 + 4 x + 0 = 7$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

Add $- 2 x + 15 + 4 x$ and $0$ .

$- 2 x + 15 + 4 x = 7$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

$- 2 x + 15 + 4 x = 7$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

Add $- 2 x$ and $4 x$ .

$2 x + 15 = 7$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

$2 x + 15 = 7$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

$2 x + 15 = 7$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

$2 x + 15 = 7$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

Solve for $x$ in $2 x + 15 = 7$ .

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

Subtract $15$ from both sides of the equation.

$2 x = 7 - 15$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

Subtract $15$ from $7$ .

$2 x = - 8$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

$2 x = - 8$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

Divide each term in $2 x = - 8$ by $2$ and simplify.

Divide each term in $2 x = - 8$ by $2$ .

$\frac{2 x}{2} = \frac{- 8}{2}$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 8}{2}$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

Divide $x$ by $1$ .

$x = \frac{- 8}{2}$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

$x = \frac{- 8}{2}$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

$x = \frac{- 8}{2}$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

Simplify the right side.

Divide $- 8$ by $2$ .

$x = - 4$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

$x = - 4$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

$x = - 4$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

$x = - 4$

$27 x + 3 y + 90 = - 21$

$z = 15 + 4 x + y$

Replace all occurrences of $x$ with $- 4$ in each equation.

Replace all occurrences of $x$ in $27 x + 3 y + 90 = - 21$ with $- 4$ .

$27 (- 4) + 3 y + 90 = - 21$

$x = - 4$

$z = 15 + 4 x + y$

Simplify the left side.

Simplify $27 (- 4) + 3 y + 90$ .

Multiply $27$ by $- 4$ .

$- 108 + 3 y + 90 = - 21$

$x = - 4$

$z = 15 + 4 x + y$

Add $- 108$ and $90$ .

$3 y - 18 = - 21$

$x = - 4$

$z = 15 + 4 x + y$

$3 y - 18 = - 21$

$x = - 4$

$z = 15 + 4 x + y$

$3 y - 18 = - 21$

$x = - 4$

$z = 15 + 4 x + y$

Replace all occurrences of $x$ in $z = 15 + 4 x + y$ with $- 4$ .

$z = 15 + 4 (- 4) + y$

$3 y - 18 = - 21$

$x = - 4$

Simplify the right side.

Simplify $15 + 4 (- 4) + y$ .

Multiply $4$ by $- 4$ .

$z = 15 - 16 + y$

$3 y - 18 = - 21$

$x = - 4$

Subtract $16$ from $15$ .

$z = - 1 + y$

$3 y - 18 = - 21$

$x = - 4$

$z = - 1 + y$

$3 y - 18 = - 21$

$x = - 4$

$z = - 1 + y$

$3 y - 18 = - 21$

$x = - 4$

$z = - 1 + y$

$3 y - 18 = - 21$

$x = - 4$

Solve for $y$ in $3 y - 18 = - 21$ .

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

Add $18$ to both sides of the equation.

$3 y = - 21 + 18$

$z = - 1 + y$

$x = - 4$

Add $- 21$ and $18$ .

$3 y = - 3$

$z = - 1 + y$

$x = - 4$

$3 y = - 3$

$z = - 1 + y$

$x = - 4$

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

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

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

$z = - 1 + y$

$x = - 4$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

$z = - 1 + y$

$x = - 4$

Divide $y$ by $1$ .

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

$z = - 1 + y$

$x = - 4$

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

$z = - 1 + y$

$x = - 4$

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

$z = - 1 + y$

$x = - 4$

Simplify the right side.

Divide $- 3$ by $3$ .

$y = - 1$

$z = - 1 + y$

$x = - 4$

$y = - 1$

$z = - 1 + y$

$x = - 4$

$y = - 1$

$z = - 1 + y$

$x = - 4$

$y = - 1$

$z = - 1 + y$

$x = - 4$

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

Replace all occurrences of $y$ in $z = - 1 + y$ with $- 1$ .

$z = - 1 - 1$

$y = - 1$

$x = - 4$

Simplify $z = - 1 - 1$ .

Simplify the left side.

Remove parentheses.

$z = - 1 - 1$

$y = - 1$

$x = - 4$

$z = - 1 - 1$

$y = - 1$

$x = - 4$

Simplify the right side.

Subtract $1$ from $- 1$ .

$z = - 2$

$y = - 1$

$x = - 4$

$z = - 2$

$y = - 1$

$x = - 4$

$z = - 2$

$y = - 1$

$x = - 4$

$z = - 2$

$y = - 1$

$x = - 4$

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

$(- 4, - 1, - 2)$

The result can be shown in multiple forms.

Point Form:

$(- 4, - 1, - 2)$

Equation Form:

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

Solve the System of Equations 3x-3y+6z=-21 -4x-y+z=15 -2x-y+z=7

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