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

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

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

Add $x$ to both sides of the equation.

$y - z = 2 + x$

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

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

Add $z$ to both sides of the equation.

$y = 2 + x + z$

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

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

$y = 2 + x + z$

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

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

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

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

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

$y = 2 + x + z$

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

Simplify the left side.

Simplify $- 3 x - (2 + x + z) + 2 z$ .

Simplify each term.

Apply the distributive property.

$- 3 x - 1 \cdot 2 - x - z + 2 z = 1$

$y = 2 + x + z$

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

Multiply $- 1$ by $2$ .

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

$y = 2 + x + z$

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

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

$y = 2 + x + z$

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

Simplify by adding terms.

Subtract $x$ from $- 3 x$ .

$- 4 x - 2 - z + 2 z = 1$

$y = 2 + x + z$

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

Add $- z$ and $2 z$ .

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

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

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

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

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

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

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

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

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

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

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$- x - 3 \cdot 2 - 3 x - 3 z + 4 z = - 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

Multiply $- 3$ by $2$ .

$- x - 6 - 3 x - 3 z + 4 z = - 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

$- x - 6 - 3 x - 3 z + 4 z = - 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

Simplify by adding terms.

Subtract $3 x$ from $- x$ .

$- 4 x - 6 - 3 z + 4 z = - 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

Add $- 3 z$ and $4 z$ .

$- 4 x - 6 + z = - 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

$- 4 x - 6 + z = - 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

$- 4 x - 6 + z = - 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

$- 4 x - 6 + z = - 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

$- 4 x - 6 + z = - 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

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

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

$- 6 + z = - 3 + 4 x$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

Add $6$ to both sides of the equation.

$z = - 3 + 4 x + 6$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

Add $- 3$ and $6$ .

$z = 4 x + 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

$z = 4 x + 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

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

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

$- 4 x - 2 + 4 x + 3 = 1$

$z = 4 x + 3$

$y = 2 + x + z$

Simplify the left side.

Simplify $- 4 x - 2 + 4 x + 3$ .

Remove parentheses.

$- 4 x - 2 + 4 x + 3 = 1$

$z = 4 x + 3$

$y = 2 + x + z$

Combine the opposite terms in $- 4 x - 2 + 4 x + 3$ .

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

$0 - 2 + 3 = 1$

$z = 4 x + 3$

$y = 2 + x + z$

Subtract $2$ from $0$ .

$- 2 + 3 = 1$

$z = 4 x + 3$

$y = 2 + x + z$

$- 2 + 3 = 1$

$z = 4 x + 3$

$y = 2 + x + z$

Add $- 2$ and $3$ .

$1 = 1$

$z = 4 x + 3$

$y = 2 + x + z$

$1 = 1$

$z = 4 x + 3$

$y = 2 + x + z$

$1 = 1$

$z = 4 x + 3$

$y = 2 + x + z$

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

$y = 2 + x + 4 x + 3$

$1 = 1$

$z = 4 x + 3$

Simplify $y = 2 + x + 4 x + 3$ .

Simplify the left side.

Remove parentheses.

$y = 2 + x + 4 x + 3$

$1 = 1$

$z = 4 x + 3$

$y = 2 + x + 4 x + 3$

$1 = 1$

$z = 4 x + 3$

Simplify the right side.

Simplify $2 + x + 4 x + 3$ .

Add $2$ and $3$ .

$y = x + 4 x + 5$

$1 = 1$

$z = 4 x + 3$

Add $x$ and $4 x$ .

$y = 5 x + 5$

$1 = 1$

$z = 4 x + 3$

$y = 5 x + 5$

$1 = 1$

$z = 4 x + 3$

$y = 5 x + 5$

$1 = 1$

$z = 4 x + 3$

$y = 5 x + 5$

$1 = 1$

$z = 4 x + 3$

$y = 5 x + 5$

$1 = 1$

$z = 4 x + 3$

Remove any equations from the system that are always true.

$y = 5 x + 5$

$z = 4 x + 3$

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

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