Solve the System of Equations 3x-2y-z=4 , x-y-2z=0 , 4x+3y+z=2

$3 x - 2 y - z = 4$ , $x - y - 2 z = 0$ , $4 x + 3 y + z = 2$

Step 1

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

Add $y$ to both sides of the equation.

$x - 2 z = y$

$3 x - 2 y - z = 4$

$4 x + 3 y + z = 2$

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

$x = y + 2 z$

$3 x - 2 y - z = 4$

$4 x + 3 y + z = 2$

$x = y + 2 z$

$3 x - 2 y - z = 4$

$4 x + 3 y + z = 2$

Step 2

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

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

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

$x = y + 2 z$

$4 x + 3 y + z = 2$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$x = y + 2 z$

$4 x + 3 y + z = 2$

Multiply $2$ by $3$ .

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

$x = y + 2 z$

$4 x + 3 y + z = 2$

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

$x = y + 2 z$

$4 x + 3 y + z = 2$

Simplify by adding terms.

Subtract $2 y$ from $3 y$ .

$y + 6 z - z = 4$

$x = y + 2 z$

$4 x + 3 y + z = 2$

Subtract $z$ from $6 z$ .

$y + 5 z = 4$

$x = y + 2 z$

$4 x + 3 y + z = 2$

$y + 5 z = 4$

$x = y + 2 z$

$4 x + 3 y + z = 2$

$y + 5 z = 4$

$x = y + 2 z$

$4 x + 3 y + z = 2$

$y + 5 z = 4$

$x = y + 2 z$

$4 x + 3 y + z = 2$

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

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

$y + 5 z = 4$

$x = y + 2 z$

Simplify the left side.

Simplify $4 (y + 2 z) + 3 y + z$ .

Simplify each term.

Apply the distributive property.

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

$y + 5 z = 4$

$x = y + 2 z$

Multiply $2$ by $4$ .

$4 y + 8 z + 3 y + z = 2$

$y + 5 z = 4$

$x = y + 2 z$

$4 y + 8 z + 3 y + z = 2$

$y + 5 z = 4$

$x = y + 2 z$

Simplify by adding terms.

Add $4 y$ and $3 y$ .

$7 y + 8 z + z = 2$

$y + 5 z = 4$

$x = y + 2 z$

Add $8 z$ and $z$ .

$7 y + 9 z = 2$

$y + 5 z = 4$

$x = y + 2 z$

$7 y + 9 z = 2$

$y + 5 z = 4$

$x = y + 2 z$

$7 y + 9 z = 2$

$y + 5 z = 4$

$x = y + 2 z$

$7 y + 9 z = 2$

$y + 5 z = 4$

$x = y + 2 z$

$7 y + 9 z = 2$

$y + 5 z = 4$

$x = y + 2 z$

Step 3

Subtract $5 z$ from both sides of the equation.

$y = 4 - 5 z$

$7 y + 9 z = 2$

$x = y + 2 z$

Step 4

Replace all occurrences of $y$ with $4 - 5 z$ in each equation.

Replace all occurrences of $y$ in $7 y + 9 z = 2$ with $4 - 5 z$ .

$7 (4 - 5 z) + 9 z = 2$

$y = 4 - 5 z$

$x = y + 2 z$

Simplify the left side.

Simplify $7 (4 - 5 z) + 9 z$ .

Simplify each term.

Apply the distributive property.

$7 \cdot 4 + 7 (- 5 z) + 9 z = 2$

$y = 4 - 5 z$

$x = y + 2 z$

Multiply $7$ by $4$ .

$28 + 7 (- 5 z) + 9 z = 2$

$y = 4 - 5 z$

$x = y + 2 z$

Multiply $- 5$ by $7$ .

$28 - 35 z + 9 z = 2$

$y = 4 - 5 z$

$x = y + 2 z$

$28 - 35 z + 9 z = 2$

$y = 4 - 5 z$

$x = y + 2 z$

Add $- 35 z$ and $9 z$ .

$28 - 26 z = 2$

$y = 4 - 5 z$

$x = y + 2 z$

$28 - 26 z = 2$

$y = 4 - 5 z$

$x = y + 2 z$

$28 - 26 z = 2$

$y = 4 - 5 z$

$x = y + 2 z$

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

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

$28 - 26 z = 2$

$y = 4 - 5 z$

Simplify the right side.

Add $- 5 z$ and $2 z$ .

$x = 4 - 3 z$

$28 - 26 z = 2$

$y = 4 - 5 z$

$x = 4 - 3 z$

$28 - 26 z = 2$

$y = 4 - 5 z$

$x = 4 - 3 z$

$28 - 26 z = 2$

$y = 4 - 5 z$

Step 5

Solve for $z$ in $28 - 26 z = 2$ .

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

Subtract $28$ from both sides of the equation.

$- 26 z = 2 - 28$

$x = 4 - 3 z$

$y = 4 - 5 z$

Subtract $28$ from $2$ .

$- 26 z = - 26$

$x = 4 - 3 z$

$y = 4 - 5 z$

$- 26 z = - 26$

$x = 4 - 3 z$

$y = 4 - 5 z$

Divide each term in $- 26 z = - 26$ by $- 26$ and simplify.

Divide each term in $- 26 z = - 26$ by $- 26$ .

$\frac{- 26 z}{- 26} = \frac{- 26}{- 26}$

$x = 4 - 3 z$

$y = 4 - 5 z$

Simplify the left side.

Cancel the common factor of $- 26$ .

Cancel the common factor.

$\frac{- 26 z}{- 26} = \frac{- 26}{- 26}$

$x = 4 - 3 z$

$y = 4 - 5 z$

Divide $z$ by $1$ .

$z = \frac{- 26}{- 26}$

$x = 4 - 3 z$

$y = 4 - 5 z$

$z = \frac{- 26}{- 26}$

$x = 4 - 3 z$

$y = 4 - 5 z$

$z = \frac{- 26}{- 26}$

$x = 4 - 3 z$

$y = 4 - 5 z$

Simplify the right side.

Divide $- 26$ by $- 26$ .

$z = 1$

$x = 4 - 3 z$

$y = 4 - 5 z$

$z = 1$

$x = 4 - 3 z$

$y = 4 - 5 z$

$z = 1$

$x = 4 - 3 z$

$y = 4 - 5 z$

$z = 1$

$x = 4 - 3 z$

$y = 4 - 5 z$

Step 6

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

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

$x = 4 - 3 \cdot 1$

$z = 1$

$y = 4 - 5 z$

Simplify the right side.

Simplify $4 - 3 \cdot 1$ .

Multiply $- 3$ by $1$ .

$x = 4 - 3$

$z = 1$

$y = 4 - 5 z$

Subtract $3$ from $4$ .

$x = 1$

$z = 1$

$y = 4 - 5 z$

$x = 1$

$z = 1$

$y = 4 - 5 z$

$x = 1$

$z = 1$

$y = 4 - 5 z$

Replace all occurrences of $z$ in $y = 4 - 5 z$ with $1$ .

$y = 4 - 5 \cdot 1$

$x = 1$

$z = 1$

Simplify the right side.

Simplify $4 - 5 \cdot 1$ .

Multiply $- 5$ by $1$ .

$y = 4 - 5$

$x = 1$

$z = 1$

Subtract $5$ from $4$ .

$y = - 1$

$x = 1$

$z = 1$

$y = - 1$

$x = 1$

$z = 1$

$y = - 1$

$x = 1$

$z = 1$

$y = - 1$

$x = 1$

$z = 1$

Step 7

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

$(1, - 1, 1)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(1, - 1, 1)$

Equation Form:

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

Solve the System of Equations 3x-2y-z=4 , x-y-2z=0 , 4x+3y+z=2

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