Solve the System of Equations -x+2y+z=1 -2x-y-z=3 -3x-4y-3z=5

$- x + 2 y + z = 1$ $- 2 x - y - z = 3$ $- 3 x - 4 y - 3 z = 5$

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

Add $x$ to both sides of the equation.

$2 y + z = 1 + x$

$- 2 x - y - z = 3$

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

Subtract $2 y$ from both sides of the equation.

$z = 1 + x - 2 y$

$- 2 x - y - z = 3$

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

$z = 1 + x - 2 y$

$- 2 x - y - z = 3$

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

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

Replace all occurrences of $z$ in $- 2 x - y - z = 3$ with $1 + x - 2 y$ .

$- 2 x - y - (1 + x - 2 y) = 3$

$z = 1 + x - 2 y$

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

Simplify the left side.

Simplify $- 2 x - y - (1 + x - 2 y)$ .

Simplify each term.

Apply the distributive property.

$- 2 x - y - 1 \cdot 1 - x - (- 2 y) = 3$

$z = 1 + x - 2 y$

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

Simplify.

Multiply $- 1$ by $1$ .

$- 2 x - y - 1 - x - (- 2 y) = 3$

$z = 1 + x - 2 y$

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

Multiply $- 2$ by $- 1$ .

$- 2 x - y - 1 - x + 2 y = 3$

$z = 1 + x - 2 y$

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

$- 2 x - y - 1 - x + 2 y = 3$

$z = 1 + x - 2 y$

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

$- 2 x - y - 1 - x + 2 y = 3$

$z = 1 + x - 2 y$

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

Simplify by adding terms.

Subtract $x$ from $- 2 x$ .

$- 3 x - y - 1 + 2 y = 3$

$z = 1 + x - 2 y$

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

Add $- y$ and $2 y$ .

$- 3 x + y - 1 = 3$

$z = 1 + x - 2 y$

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

$- 3 x + y - 1 = 3$

$z = 1 + x - 2 y$

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

$- 3 x + y - 1 = 3$

$z = 1 + x - 2 y$

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

$- 3 x + y - 1 = 3$

$z = 1 + x - 2 y$

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

Replace all occurrences of $z$ in $- 3 x - 4 y - 3 z = 5$ with $1 + x - 2 y$ .

$- 3 x - 4 y - 3 (1 + x - 2 y) = 5$

$- 3 x + y - 1 = 3$

$z = 1 + x - 2 y$

Simplify the left side.

Simplify $- 3 x - 4 y - 3 (1 + x - 2 y)$ .

Simplify each term.

Apply the distributive property.

$- 3 x - 4 y - 3 \cdot 1 - 3 x - 3 (- 2 y) = 5$

$- 3 x + y - 1 = 3$

$z = 1 + x - 2 y$

Simplify.

Multiply $- 3$ by $1$ .

$- 3 x - 4 y - 3 - 3 x - 3 (- 2 y) = 5$

$- 3 x + y - 1 = 3$

$z = 1 + x - 2 y$

Multiply $- 2$ by $- 3$ .

$- 3 x - 4 y - 3 - 3 x + 6 y = 5$

$- 3 x + y - 1 = 3$

$z = 1 + x - 2 y$

$- 3 x - 4 y - 3 - 3 x + 6 y = 5$

$- 3 x + y - 1 = 3$

$z = 1 + x - 2 y$

$- 3 x - 4 y - 3 - 3 x + 6 y = 5$

$- 3 x + y - 1 = 3$

$z = 1 + x - 2 y$

Simplify by adding terms.

Subtract $3 x$ from $- 3 x$ .

$- 6 x - 4 y - 3 + 6 y = 5$

$- 3 x + y - 1 = 3$

$z = 1 + x - 2 y$

Add $- 4 y$ and $6 y$ .

$- 6 x + 2 y - 3 = 5$

$- 3 x + y - 1 = 3$

$z = 1 + x - 2 y$

$- 6 x + 2 y - 3 = 5$

$- 3 x + y - 1 = 3$

$z = 1 + x - 2 y$

$- 6 x + 2 y - 3 = 5$

$- 3 x + y - 1 = 3$

$z = 1 + x - 2 y$

$- 6 x + 2 y - 3 = 5$

$- 3 x + y - 1 = 3$

$z = 1 + x - 2 y$

$- 6 x + 2 y - 3 = 5$

$- 3 x + y - 1 = 3$

$z = 1 + x - 2 y$

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

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

$y - 1 = 3 + 3 x$

$- 6 x + 2 y - 3 = 5$

$z = 1 + x - 2 y$

Add $1$ to both sides of the equation.

$y = 3 + 3 x + 1$

$- 6 x + 2 y - 3 = 5$

$z = 1 + x - 2 y$

Add $3$ and $1$ .

$y = 3 x + 4$

$- 6 x + 2 y - 3 = 5$

$z = 1 + x - 2 y$

$y = 3 x + 4$

$- 6 x + 2 y - 3 = 5$

$z = 1 + x - 2 y$

Replace all occurrences of $y$ with $3 x + 4$ in each equation.

Replace all occurrences of $y$ in $- 6 x + 2 y - 3 = 5$ with $3 x + 4$ .

$- 6 x + 2 (3 x + 4) - 3 = 5$

$y = 3 x + 4$

$z = 1 + x - 2 y$

Simplify the left side.

Simplify $- 6 x + 2 (3 x + 4) - 3$ .

Simplify each term.

Apply the distributive property.

$- 6 x + 2 (3 x) + 2 \cdot 4 - 3 = 5$

$y = 3 x + 4$

$z = 1 + x - 2 y$

Multiply $3$ by $2$ .

$- 6 x + 6 x + 2 \cdot 4 - 3 = 5$

$y = 3 x + 4$

$z = 1 + x - 2 y$

Multiply $2$ by $4$ .

$- 6 x + 6 x + 8 - 3 = 5$

$y = 3 x + 4$

$z = 1 + x - 2 y$

$- 6 x + 6 x + 8 - 3 = 5$

$y = 3 x + 4$

$z = 1 + x - 2 y$

Simplify by adding terms.

Add $- 6 x$ and $6 x$ .

$0 + 8 - 3 = 5$

$y = 3 x + 4$

$z = 1 + x - 2 y$

Simplify by subtracting numbers.

Add $0$ and $8$ .

$8 - 3 = 5$

$y = 3 x + 4$

$z = 1 + x - 2 y$

Subtract $3$ from $8$ .

$5 = 5$

$y = 3 x + 4$

$z = 1 + x - 2 y$

$5 = 5$

$y = 3 x + 4$

$z = 1 + x - 2 y$

$5 = 5$

$y = 3 x + 4$

$z = 1 + x - 2 y$

$5 = 5$

$y = 3 x + 4$

$z = 1 + x - 2 y$

$5 = 5$

$y = 3 x + 4$

$z = 1 + x - 2 y$

Replace all occurrences of $y$ in $z = 1 + x - 2 y$ with $3 x + 4$ .

$z = 1 + x - 2 (3 x + 4)$

$5 = 5$

$y = 3 x + 4$

Simplify the right side.

Simplify $1 + x - 2 (3 x + 4)$ .

Simplify each term.

Apply the distributive property.

$z = 1 + x - 2 (3 x) - 2 \cdot 4$

$5 = 5$

$y = 3 x + 4$

Multiply $3$ by $- 2$ .

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

$5 = 5$

$y = 3 x + 4$

Multiply $- 2$ by $4$ .

$z = 1 + x - 6 x - 8$

$5 = 5$

$y = 3 x + 4$

$z = 1 + x - 6 x - 8$

$5 = 5$

$y = 3 x + 4$

Simplify by adding terms.

Subtract $8$ from $1$ .

$z = x - 6 x - 7$

$5 = 5$

$y = 3 x + 4$

Subtract $6 x$ from $x$ .

$z = - 5 x - 7$

$5 = 5$

$y = 3 x + 4$

$z = - 5 x - 7$

$5 = 5$

$y = 3 x + 4$

$z = - 5 x - 7$

$5 = 5$

$y = 3 x + 4$

$z = - 5 x - 7$

$5 = 5$

$y = 3 x + 4$

$z = - 5 x - 7$

$5 = 5$

$y = 3 x + 4$

Remove any equations from the system that are always true.

$z = - 5 x - 7$

$y = 3 x + 4$

Solve the System of Equations -x+2y+z=1 -2x-y-z=3 -3x-4y-3z=5

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