Solve the System of Equations -x+4y+3z=0 x-2y+z=-8 -2x+3y+4z=-4

$- x + 4 y + 3 z = 0$ $x - 2 y + z = - 8$ $- 2 x + 3 y + 4 z = - 4$

Step 1

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

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

$x + z = - 8 + 2 y$

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

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

Subtract $z$ from both sides of the equation.

$x = - 8 + 2 y - z$

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

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

$x = - 8 + 2 y - z$

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

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

Step 2

Replace all occurrences of $x$ with $- 8 + 2 y - z$ in each equation.

Replace all occurrences of $x$ in $- x + 4 y + 3 z = 0$ with $- 8 + 2 y - z$ .

$- (- 8 + 2 y - z) + 4 y + 3 z = 0$

$x = - 8 + 2 y - z$

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

Simplify the left side.

Simplify $- (- 8 + 2 y - z) + 4 y + 3 z$ .

Simplify each term.

Apply the distributive property.

$8 - (2 y) + z + 4 y + 3 z = 0$

$x = - 8 + 2 y - z$

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

Simplify.

Multiply $- 1$ by $- 8$ .

$8 - (2 y) + z + 4 y + 3 z = 0$

$x = - 8 + 2 y - z$

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

Multiply $2$ by $- 1$ .

$8 - 2 y + z + 4 y + 3 z = 0$

$x = - 8 + 2 y - z$

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

Multiply $- - z$ .

Multiply $- 1$ by $- 1$ .

$8 - 2 y + 1 z + 4 y + 3 z = 0$

$x = - 8 + 2 y - z$

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

Multiply $z$ by $1$ .

$8 - 2 y + z + 4 y + 3 z = 0$

$x = - 8 + 2 y - z$

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

$8 - 2 y + z + 4 y + 3 z = 0$

$x = - 8 + 2 y - z$

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

$8 - 2 y + z + 4 y + 3 z = 0$

$x = - 8 + 2 y - z$

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

$8 - 2 y + z + 4 y + 3 z = 0$

$x = - 8 + 2 y - z$

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

Simplify by adding terms.

Add $- 2 y$ and $4 y$ .

$8 + 2 y + z + 3 z = 0$

$x = - 8 + 2 y - z$

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

Add $z$ and $3 z$ .

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

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

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

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

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

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

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

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

Replace all occurrences of $x$ in $- 2 x + 3 y + 4 z = - 4$ with $- 8 + 2 y - z$ .

$- 2 (- 8 + 2 y - z) + 3 y + 4 z = - 4$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

Simplify the left side.

Simplify $- 2 (- 8 + 2 y - z) + 3 y + 4 z$ .

Simplify each term.

Apply the distributive property.

$- 2 \cdot - 8 - 2 (2 y) - 2 (- z) + 3 y + 4 z = - 4$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

Simplify.

Multiply $- 2$ by $- 8$ .

$16 - 2 (2 y) - 2 (- z) + 3 y + 4 z = - 4$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

Multiply $2$ by $- 2$ .

$16 - 4 y - 2 (- z) + 3 y + 4 z = - 4$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

Multiply $- 1$ by $- 2$ .

$16 - 4 y + 2 z + 3 y + 4 z = - 4$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

$16 - 4 y + 2 z + 3 y + 4 z = - 4$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

$16 - 4 y + 2 z + 3 y + 4 z = - 4$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

Simplify by adding terms.

Add $- 4 y$ and $3 y$ .

$16 - y + 2 z + 4 z = - 4$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

Add $2 z$ and $4 z$ .

$16 - y + 6 z = - 4$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

$16 - y + 6 z = - 4$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

$16 - y + 6 z = - 4$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

$16 - y + 6 z = - 4$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

$16 - y + 6 z = - 4$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

Step 3

Solve for $y$ in $16 - y + 6 z = - 4$ .

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

Subtract $16$ from both sides of the equation.

$- y + 6 z = - 4 - 16$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

Subtract $6 z$ from both sides of the equation.

$- y = - 4 - 16 - 6 z$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

Subtract $16$ from $- 4$ .

$- y = - 20 - 6 z$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

$- y = - 20 - 6 z$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

Divide each term in $- y = - 20 - 6 z$ by $- 1$ and simplify.

Divide each term in $- y = - 20 - 6 z$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 20}{- 1} + \frac{- 6 z}{- 1}$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 20}{- 1} + \frac{- 6 z}{- 1}$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

Divide $y$ by $1$ .

$y = \frac{- 20}{- 1} + \frac{- 6 z}{- 1}$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

$y = \frac{- 20}{- 1} + \frac{- 6 z}{- 1}$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

Simplify the right side.

Simplify each term.

Divide $- 20$ by $- 1$ .

$y = 20 + \frac{- 6 z}{- 1}$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

Move the negative one from the denominator of $\frac{- 6 z}{- 1}$ .

$y = 20 - 1 \cdot (- 6 z)$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

Rewrite $- 1 \cdot (- 6 z)$ as $- (- 6 z)$ .

$y = 20 - (- 6 z)$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

Multiply $- 6$ by $- 1$ .

$y = 20 + 6 z$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

$y = 20 + 6 z$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

$y = 20 + 6 z$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

$y = 20 + 6 z$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

$y = 20 + 6 z$

$8 + 2 y + 4 z = 0$

$x = - 8 + 2 y - z$

Step 4

Replace all occurrences of $y$ with $20 + 6 z$ in each equation.

Replace all occurrences of $y$ in $8 + 2 y + 4 z = 0$ with $20 + 6 z$ .

$8 + 2 (20 + 6 z) + 4 z = 0$

$y = 20 + 6 z$

$x = - 8 + 2 y - z$

Simplify the left side.

Simplify $8 + 2 (20 + 6 z) + 4 z$ .

Simplify each term.

Apply the distributive property.

$8 + 2 \cdot 20 + 2 (6 z) + 4 z = 0$

$y = 20 + 6 z$

$x = - 8 + 2 y - z$

Multiply $2$ by $20$ .

$8 + 40 + 2 (6 z) + 4 z = 0$

$y = 20 + 6 z$

$x = - 8 + 2 y - z$

Multiply $6$ by $2$ .

$8 + 40 + 12 z + 4 z = 0$

$y = 20 + 6 z$

$x = - 8 + 2 y - z$

$8 + 40 + 12 z + 4 z = 0$

$y = 20 + 6 z$

$x = - 8 + 2 y - z$

Simplify by adding terms.

Add $8$ and $40$ .

$48 + 12 z + 4 z = 0$

$y = 20 + 6 z$

$x = - 8 + 2 y - z$

Add $12 z$ and $4 z$ .

$48 + 16 z = 0$

$y = 20 + 6 z$

$x = - 8 + 2 y - z$

$48 + 16 z = 0$

$y = 20 + 6 z$

$x = - 8 + 2 y - z$

$48 + 16 z = 0$

$y = 20 + 6 z$

$x = - 8 + 2 y - z$

$48 + 16 z = 0$

$y = 20 + 6 z$

$x = - 8 + 2 y - z$

Replace all occurrences of $y$ in $x = - 8 + 2 y - z$ with $20 + 6 z$ .

$x = - 8 + 2 (20 + 6 z) - z$

$48 + 16 z = 0$

$y = 20 + 6 z$

Simplify the right side.

Simplify $- 8 + 2 (20 + 6 z) - z$ .

Simplify each term.

Apply the distributive property.

$x = - 8 + 2 \cdot 20 + 2 (6 z) - z$

$48 + 16 z = 0$

$y = 20 + 6 z$

Multiply $2$ by $20$ .

$x = - 8 + 40 + 2 (6 z) - z$

$48 + 16 z = 0$

$y = 20 + 6 z$

Multiply $6$ by $2$ .

$x = - 8 + 40 + 12 z - z$

$48 + 16 z = 0$

$y = 20 + 6 z$

$x = - 8 + 40 + 12 z - z$

$48 + 16 z = 0$

$y = 20 + 6 z$

Simplify by adding terms.

Add $- 8$ and $40$ .

$x = 32 + 12 z - z$

$48 + 16 z = 0$

$y = 20 + 6 z$

Subtract $z$ from $12 z$ .

$x = 32 + 11 z$

$48 + 16 z = 0$

$y = 20 + 6 z$

$x = 32 + 11 z$

$48 + 16 z = 0$

$y = 20 + 6 z$

$x = 32 + 11 z$

$48 + 16 z = 0$

$y = 20 + 6 z$

$x = 32 + 11 z$

$48 + 16 z = 0$

$y = 20 + 6 z$

$x = 32 + 11 z$

$48 + 16 z = 0$

$y = 20 + 6 z$

Step 5

Solve for $z$ in $48 + 16 z = 0$ .

Subtract $48$ from both sides of the equation.

$16 z = - 48$

$x = 32 + 11 z$

$y = 20 + 6 z$

Divide each term in $16 z = - 48$ by $16$ and simplify.

Divide each term in $16 z = - 48$ by $16$ .

$\frac{16 z}{16} = \frac{- 48}{16}$

$x = 32 + 11 z$

$y = 20 + 6 z$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 z}{16} = \frac{- 48}{16}$

$x = 32 + 11 z$

$y = 20 + 6 z$

Divide $z$ by $1$ .

$z = \frac{- 48}{16}$

$x = 32 + 11 z$

$y = 20 + 6 z$

$z = \frac{- 48}{16}$

$x = 32 + 11 z$

$y = 20 + 6 z$

$z = \frac{- 48}{16}$

$x = 32 + 11 z$

$y = 20 + 6 z$

Simplify the right side.

Divide $- 48$ by $16$ .

$z = - 3$

$x = 32 + 11 z$

$y = 20 + 6 z$

$z = - 3$

$x = 32 + 11 z$

$y = 20 + 6 z$

$z = - 3$

$x = 32 + 11 z$

$y = 20 + 6 z$

$z = - 3$

$x = 32 + 11 z$

$y = 20 + 6 z$

Step 6

Replace all occurrences of $z$ with $- 3$ in each equation.

Replace all occurrences of $z$ in $x = 32 + 11 z$ with $- 3$ .

$x = 32 + 11 (- 3)$

$z = - 3$

$y = 20 + 6 z$

Simplify the right side.

Simplify $32 + 11 (- 3)$ .

Multiply $11$ by $- 3$ .

$x = 32 - 33$

$z = - 3$

$y = 20 + 6 z$

Subtract $33$ from $32$ .

$x = - 1$

$z = - 3$

$y = 20 + 6 z$

$x = - 1$

$z = - 3$

$y = 20 + 6 z$

$x = - 1$

$z = - 3$

$y = 20 + 6 z$

Replace all occurrences of $z$ in $y = 20 + 6 z$ with $- 3$ .

$y = 20 + 6 (- 3)$

$x = - 1$

$z = - 3$

Simplify the right side.

Simplify $20 + 6 (- 3)$ .

Multiply $6$ by $- 3$ .

$y = 20 - 18$

$x = - 1$

$z = - 3$

Subtract $18$ from $20$ .

$y = 2$

$x = - 1$

$z = - 3$

$y = 2$

$x = - 1$

$z = - 3$

$y = 2$

$x = - 1$

$z = - 3$

$y = 2$

$x = - 1$

$z = - 3$

Step 7

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

$(- 1, 2, - 3)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 1, 2, - 3)$

Equation Form:

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

Solve the System of Equations -x+4y+3z=0 x-2y+z=-8 -2x+3y+4z=-4

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