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

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

Step 1

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

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

$y - z = 1 - 2 x$

$- x - y - 2 z = 3$

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

Add $z$ to both sides of the equation.

$y = 1 - 2 x + z$

$- x - y - 2 z = 3$

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

$y = 1 - 2 x + z$

$- x - y - 2 z = 3$

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

Step 2

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

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

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

$y = 1 - 2 x + z$

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

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$y = 1 - 2 x + z$

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

Simplify.

Multiply $- 1$ by $1$ .

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

$y = 1 - 2 x + z$

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

Multiply $- 2$ by $- 1$ .

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

$y = 1 - 2 x + z$

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

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

$y = 1 - 2 x + z$

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

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

$y = 1 - 2 x + z$

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

Simplify by adding terms.

Add $- x$ and $2 x$ .

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

$y = 1 - 2 x + z$

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

Subtract $2 z$ from $- z$ .

$x - 1 - 3 z = 3$

$y = 1 - 2 x + z$

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

$x - 1 - 3 z = 3$

$y = 1 - 2 x + z$

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

$x - 1 - 3 z = 3$

$y = 1 - 2 x + z$

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

$x - 1 - 3 z = 3$

$y = 1 - 2 x + z$

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

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

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

$x - 1 - 3 z = 3$

$y = 1 - 2 x + z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$x - 1 - 3 z = 3$

$y = 1 - 2 x + z$

Simplify.

Multiply $- 3$ by $1$ .

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

$x - 1 - 3 z = 3$

$y = 1 - 2 x + z$

Multiply $- 2$ by $- 3$ .

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

$x - 1 - 3 z = 3$

$y = 1 - 2 x + z$

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

$x - 1 - 3 z = 3$

$y = 1 - 2 x + z$

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

$x - 1 - 3 z = 3$

$y = 1 - 2 x + z$

Simplify by adding terms.

Add $- 4 x$ and $6 x$ .

$2 x - 3 - 3 z - 3 z = 5$

$x - 1 - 3 z = 3$

$y = 1 - 2 x + z$

Subtract $3 z$ from $- 3 z$ .

$2 x - 3 - 6 z = 5$

$x - 1 - 3 z = 3$

$y = 1 - 2 x + z$

$2 x - 3 - 6 z = 5$

$x - 1 - 3 z = 3$

$y = 1 - 2 x + z$

$2 x - 3 - 6 z = 5$

$x - 1 - 3 z = 3$

$y = 1 - 2 x + z$

$2 x - 3 - 6 z = 5$

$x - 1 - 3 z = 3$

$y = 1 - 2 x + z$

$2 x - 3 - 6 z = 5$

$x - 1 - 3 z = 3$

$y = 1 - 2 x + z$

Step 3

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

Add $1$ to both sides of the equation.

$x - 3 z = 3 + 1$

$2 x - 3 - 6 z = 5$

$y = 1 - 2 x + z$

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

$x = 3 + 1 + 3 z$

$2 x - 3 - 6 z = 5$

$y = 1 - 2 x + z$

Add $3$ and $1$ .

$x = 4 + 3 z$

$2 x - 3 - 6 z = 5$

$y = 1 - 2 x + z$

$x = 4 + 3 z$

$2 x - 3 - 6 z = 5$

$y = 1 - 2 x + z$

Step 4

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

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

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

$x = 4 + 3 z$

$y = 1 - 2 x + z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$x = 4 + 3 z$

$y = 1 - 2 x + z$

Multiply $2$ by $4$ .

$8 + 2 (3 z) - 3 - 6 z = 5$

$x = 4 + 3 z$

$y = 1 - 2 x + z$

Multiply $3$ by $2$ .

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

$x = 4 + 3 z$

$y = 1 - 2 x + z$

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

$x = 4 + 3 z$

$y = 1 - 2 x + z$

Simplify by adding terms.

Combine the opposite terms in $8 + 6 z - 3 - 6 z$ .

Subtract $6 z$ from $6 z$ .

$8 + 0 - 3 = 5$

$x = 4 + 3 z$

$y = 1 - 2 x + z$

Add $8$ and $0$ .

$8 - 3 = 5$

$x = 4 + 3 z$

$y = 1 - 2 x + z$

$8 - 3 = 5$

$x = 4 + 3 z$

$y = 1 - 2 x + z$

Subtract $3$ from $8$ .

$5 = 5$

$x = 4 + 3 z$

$y = 1 - 2 x + z$

$5 = 5$

$x = 4 + 3 z$

$y = 1 - 2 x + z$

$5 = 5$

$x = 4 + 3 z$

$y = 1 - 2 x + z$

$5 = 5$

$x = 4 + 3 z$

$y = 1 - 2 x + z$

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

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

$5 = 5$

$x = 4 + 3 z$

Simplify the right side.

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

Simplify each term.

Apply the distributive property.

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

$5 = 5$

$x = 4 + 3 z$

Multiply $- 2$ by $4$ .

$y = 1 - 8 - 2 (3 z) + z$

$5 = 5$

$x = 4 + 3 z$

Multiply $3$ by $- 2$ .

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

$5 = 5$

$x = 4 + 3 z$

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

$5 = 5$

$x = 4 + 3 z$

Simplify by adding terms.

Subtract $8$ from $1$ .

$y = - 7 - 6 z + z$

$5 = 5$

$x = 4 + 3 z$

Add $- 6 z$ and $z$ .

$y = - 7 - 5 z$

$5 = 5$

$x = 4 + 3 z$

$y = - 7 - 5 z$

$5 = 5$

$x = 4 + 3 z$

$y = - 7 - 5 z$

$5 = 5$

$x = 4 + 3 z$

$y = - 7 - 5 z$

$5 = 5$

$x = 4 + 3 z$

$y = - 7 - 5 z$

$5 = 5$

$x = 4 + 3 z$

Step 5

Remove any equations from the system that are always true.

$y = - 7 - 5 z$

$x = 4 + 3 z$

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

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