Solve by Substitution x-2y-z=2 , -x+6y-3z=-6 , 2x-11y+5z=11

$x - 2 y - z = 2$ , $- x + 6 y - 3 z = - 6$ , $2 x - 11 y + 5 z = 11$

Step 1

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

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

$x - z = 2 + 2 y$

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

$2 x - 11 y + 5 z = 11$

Add $z$ to both sides of the equation.

$x = 2 + 2 y + z$

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

$2 x - 11 y + 5 z = 11$

$x = 2 + 2 y + z$

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

$2 x - 11 y + 5 z = 11$

Step 2

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

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

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

$x = 2 + 2 y + z$

$2 x - 11 y + 5 z = 11$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$- 1 \cdot 2 - (2 y) - z + 6 y - 3 z = - 6$

$x = 2 + 2 y + z$

$2 x - 11 y + 5 z = 11$

Simplify.

Multiply $- 1$ by $2$ .

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

$x = 2 + 2 y + z$

$2 x - 11 y + 5 z = 11$

Multiply $2$ by $- 1$ .

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

$x = 2 + 2 y + z$

$2 x - 11 y + 5 z = 11$

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

$x = 2 + 2 y + z$

$2 x - 11 y + 5 z = 11$

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

$x = 2 + 2 y + z$

$2 x - 11 y + 5 z = 11$

Simplify by adding terms.

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

$- 2 + 4 y - z - 3 z = - 6$

$x = 2 + 2 y + z$

$2 x - 11 y + 5 z = 11$

Subtract $3 z$ from $- z$ .

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$2 x - 11 y + 5 z = 11$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$2 x - 11 y + 5 z = 11$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$2 x - 11 y + 5 z = 11$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$2 x - 11 y + 5 z = 11$

Replace all occurrences of $x$ in $2 x - 11 y + 5 z = 11$ with $2 + 2 y + z$ .

$2 (2 + 2 y + z) - 11 y + 5 z = 11$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

Simplify the left side.

Simplify $2 (2 + 2 y + z) - 11 y + 5 z$ .

Simplify each term.

Apply the distributive property.

$2 \cdot 2 + 2 (2 y) + 2 z - 11 y + 5 z = 11$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

Simplify.

Multiply $2$ by $2$ .

$4 + 2 (2 y) + 2 z - 11 y + 5 z = 11$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

Multiply $2$ by $2$ .

$4 + 4 y + 2 z - 11 y + 5 z = 11$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$4 + 4 y + 2 z - 11 y + 5 z = 11$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$4 + 4 y + 2 z - 11 y + 5 z = 11$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

Simplify by adding terms.

Subtract $11 y$ from $4 y$ .

$4 - 7 y + 2 z + 5 z = 11$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

Add $2 z$ and $5 z$ .

$4 - 7 y + 7 z = 11$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$4 - 7 y + 7 z = 11$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$4 - 7 y + 7 z = 11$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$4 - 7 y + 7 z = 11$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$4 - 7 y + 7 z = 11$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

Step 3

Solve for $y$ in $4 - 7 y + 7 z = 11$ .

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

Subtract $4$ from both sides of the equation.

$- 7 y + 7 z = 11 - 4$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

Subtract $7 z$ from both sides of the equation.

$- 7 y = 11 - 4 - 7 z$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

Subtract $4$ from $11$ .

$- 7 y = 7 - 7 z$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$- 7 y = 7 - 7 z$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

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

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

$\frac{- 7 y}{- 7} = \frac{7}{- 7} + \frac{- 7 z}{- 7}$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 y}{- 7} = \frac{7}{- 7} + \frac{- 7 z}{- 7}$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

Divide $y$ by $1$ .

$y = \frac{7}{- 7} + \frac{- 7 z}{- 7}$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$y = \frac{7}{- 7} + \frac{- 7 z}{- 7}$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$y = \frac{7}{- 7} + \frac{- 7 z}{- 7}$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

Simplify the right side.

Simplify each term.

Divide $7$ by $- 7$ .

$y = - 1 + \frac{- 7 z}{- 7}$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

Cancel the common factor of $- 7$ .

Cancel the common factor.

$y = - 1 + \frac{- 7 z}{- 7}$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

Divide $z$ by $1$ .

$y = - 1 + z$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$y = - 1 + z$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$y = - 1 + z$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$y = - 1 + z$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$y = - 1 + z$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

$y = - 1 + z$

$- 2 + 4 y - 4 z = - 6$

$x = 2 + 2 y + z$

Step 4

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

Replace all occurrences of $y$ in $- 2 + 4 y - 4 z = - 6$ with $- 1 + z$ .

$- 2 + 4 (- 1 + z) - 4 z = - 6$

$y = - 1 + z$

$x = 2 + 2 y + z$

Simplify the left side.

Simplify $- 2 + 4 (- 1 + z) - 4 z$ .

Simplify each term.

Apply the distributive property.

$- 2 + 4 \cdot - 1 + 4 z - 4 z = - 6$

$y = - 1 + z$

$x = 2 + 2 y + z$

Multiply $4$ by $- 1$ .

$- 2 - 4 + 4 z - 4 z = - 6$

$y = - 1 + z$

$x = 2 + 2 y + z$

$- 2 - 4 + 4 z - 4 z = - 6$

$y = - 1 + z$

$x = 2 + 2 y + z$

Simplify by adding terms.

Combine the opposite terms in $- 2 - 4 + 4 z - 4 z$ .

Subtract $4 z$ from $4 z$ .

$- 2 - 4 + 0 = - 6$

$y = - 1 + z$

$x = 2 + 2 y + z$

Add $- 2 - 4$ and $0$ .

$- 2 - 4 = - 6$

$y = - 1 + z$

$x = 2 + 2 y + z$

$- 2 - 4 = - 6$

$y = - 1 + z$

$x = 2 + 2 y + z$

Subtract $4$ from $- 2$ .

$- 6 = - 6$

$y = - 1 + z$

$x = 2 + 2 y + z$

$- 6 = - 6$

$y = - 1 + z$

$x = 2 + 2 y + z$

$- 6 = - 6$

$y = - 1 + z$

$x = 2 + 2 y + z$

$- 6 = - 6$

$y = - 1 + z$

$x = 2 + 2 y + z$

Replace all occurrences of $y$ in $x = 2 + 2 y + z$ with $- 1 + z$ .

$x = 2 + 2 (- 1 + z) + z$

$- 6 = - 6$

$y = - 1 + z$

Simplify the right side.

Simplify $2 + 2 (- 1 + z) + z$ .

Simplify each term.

Apply the distributive property.

$x = 2 + 2 \cdot - 1 + 2 z + z$

$- 6 = - 6$

$y = - 1 + z$

Multiply $2$ by $- 1$ .

$x = 2 - 2 + 2 z + z$

$- 6 = - 6$

$y = - 1 + z$

$x = 2 - 2 + 2 z + z$

$- 6 = - 6$

$y = - 1 + z$

Simplify by adding terms.

Subtract $2$ from $2$ .

$x = 0 + 2 z + z$

$- 6 = - 6$

$y = - 1 + z$

Add $0$ and $2 z$ .

$x = 2 z + z$

$- 6 = - 6$

$y = - 1 + z$

Add $2 z$ and $z$ .

$x = 3 z$

$- 6 = - 6$

$y = - 1 + z$

$x = 3 z$

$- 6 = - 6$

$y = - 1 + z$

$x = 3 z$

$- 6 = - 6$

$y = - 1 + z$

$x = 3 z$

$- 6 = - 6$

$y = - 1 + z$

$x = 3 z$

$- 6 = - 6$

$y = - 1 + z$

Step 5

Remove any equations from the system that are always true.

$x = 3 z$

$y = - 1 + z$

Solve by Substitution x-2y-z=2 , -x+6y-3z=-6 , 2x-11y+5z=11

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