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

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

Step 1

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

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

$y + z = - 8 + 2 x$

$4 x - y + 3 z = 0$

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

Subtract $z$ from both sides of the equation.

$y = - 8 + 2 x - z$

$4 x - y + 3 z = 0$

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

$y = - 8 + 2 x - z$

$4 x - y + 3 z = 0$

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

Step 2

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

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

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

$y = - 8 + 2 x - z$

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

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$y = - 8 + 2 x - z$

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

Simplify.

Multiply $- 1$ by $- 8$ .

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

$y = - 8 + 2 x - z$

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

Multiply $2$ by $- 1$ .

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

$y = - 8 + 2 x - z$

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

Multiply $- - z$ .

Multiply $- 1$ by $- 1$ .

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

$y = - 8 + 2 x - z$

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

Multiply $z$ by $1$ .

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

$y = - 8 + 2 x - z$

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

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

$y = - 8 + 2 x - z$

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

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

$y = - 8 + 2 x - z$

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

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

$y = - 8 + 2 x - z$

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

Simplify by adding terms.

Subtract $2 x$ from $4 x$ .

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

$y = - 8 + 2 x - z$

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

Add $z$ and $3 z$ .

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

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

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

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

Simplify.

Multiply $- 2$ by $- 8$ .

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

Multiply $2$ by $- 2$ .

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

Multiply $- 1$ by $- 2$ .

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

Simplify by adding terms.

Subtract $4 x$ from $3 x$ .

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

Add $2 z$ and $4 z$ .

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

Step 3

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

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

Subtract $16$ from both sides of the equation.

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

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

$- x = - 4 - 16 - 6 z$

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

Subtract $16$ from $- 4$ .

$- x = - 20 - 6 z$

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

$- x = - 20 - 6 z$

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

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

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

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

Simplify the left side.

Dividing two negative values results in a positive value.

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

Divide $x$ by $1$ .

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

Simplify the right side.

Simplify each term.

Divide $- 20$ by $- 1$ .

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

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

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

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

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

$x = 20 - (- 6 z)$

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

Multiply $- 6$ by $- 1$ .

$x = 20 + 6 z$

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

$x = 20 + 6 z$

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

$x = 20 + 6 z$

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

$x = 20 + 6 z$

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

$x = 20 + 6 z$

$2 x + 8 + 4 z = 0$

$y = - 8 + 2 x - z$

Step 4

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

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

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

$x = 20 + 6 z$

$y = - 8 + 2 x - z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$x = 20 + 6 z$

$y = - 8 + 2 x - z$

Multiply $2$ by $20$ .

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

$x = 20 + 6 z$

$y = - 8 + 2 x - z$

Multiply $6$ by $2$ .

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

$x = 20 + 6 z$

$y = - 8 + 2 x - z$

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

$x = 20 + 6 z$

$y = - 8 + 2 x - z$

Simplify by adding terms.

Add $40$ and $8$ .

$12 z + 48 + 4 z = 0$

$x = 20 + 6 z$

$y = - 8 + 2 x - z$

Add $12 z$ and $4 z$ .

$16 z + 48 = 0$

$x = 20 + 6 z$

$y = - 8 + 2 x - z$

$16 z + 48 = 0$

$x = 20 + 6 z$

$y = - 8 + 2 x - z$

$16 z + 48 = 0$

$x = 20 + 6 z$

$y = - 8 + 2 x - z$

$16 z + 48 = 0$

$x = 20 + 6 z$

$y = - 8 + 2 x - z$

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

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

$16 z + 48 = 0$

$x = 20 + 6 z$

Simplify the right side.

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

Simplify each term.

Apply the distributive property.

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

$16 z + 48 = 0$

$x = 20 + 6 z$

Multiply $2$ by $20$ .

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

$16 z + 48 = 0$

$x = 20 + 6 z$

Multiply $6$ by $2$ .

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

$16 z + 48 = 0$

$x = 20 + 6 z$

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

$16 z + 48 = 0$

$x = 20 + 6 z$

Simplify by adding terms.

Add $- 8$ and $40$ .

$y = 32 + 12 z - z$

$16 z + 48 = 0$

$x = 20 + 6 z$

Subtract $z$ from $12 z$ .

$y = 32 + 11 z$

$16 z + 48 = 0$

$x = 20 + 6 z$

$y = 32 + 11 z$

$16 z + 48 = 0$

$x = 20 + 6 z$

$y = 32 + 11 z$

$16 z + 48 = 0$

$x = 20 + 6 z$

$y = 32 + 11 z$

$16 z + 48 = 0$

$x = 20 + 6 z$

$y = 32 + 11 z$

$16 z + 48 = 0$

$x = 20 + 6 z$

Step 5

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

Subtract $48$ from both sides of the equation.

$16 z = - 48$

$y = 32 + 11 z$

$x = 20 + 6 z$

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

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

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

$y = 32 + 11 z$

$x = 20 + 6 z$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

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

$y = 32 + 11 z$

$x = 20 + 6 z$

Divide $z$ by $1$ .

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

$y = 32 + 11 z$

$x = 20 + 6 z$

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

$y = 32 + 11 z$

$x = 20 + 6 z$

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

$y = 32 + 11 z$

$x = 20 + 6 z$

Simplify the right side.

Divide $- 48$ by $16$ .

$z = - 3$

$y = 32 + 11 z$

$x = 20 + 6 z$

$z = - 3$

$y = 32 + 11 z$

$x = 20 + 6 z$

$z = - 3$

$y = 32 + 11 z$

$x = 20 + 6 z$

$z = - 3$

$y = 32 + 11 z$

$x = 20 + 6 z$

Step 6

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

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

$y = 32 + 11 (- 3)$

$z = - 3$

$x = 20 + 6 z$

Simplify the right side.

Simplify $32 + 11 (- 3)$ .

Multiply $11$ by $- 3$ .

$y = 32 - 33$

$z = - 3$

$x = 20 + 6 z$

Subtract $33$ from $32$ .

$y = - 1$

$z = - 3$

$x = 20 + 6 z$

$y = - 1$

$z = - 3$

$x = 20 + 6 z$

$y = - 1$

$z = - 3$

$x = 20 + 6 z$

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

$x = 20 + 6 (- 3)$

$y = - 1$

$z = - 3$

Simplify the right side.

Simplify $20 + 6 (- 3)$ .

Multiply $6$ by $- 3$ .

$x = 20 - 18$

$y = - 1$

$z = - 3$

Subtract $18$ from $20$ .

$x = 2$

$y = - 1$

$z = - 3$

$x = 2$

$y = - 1$

$z = - 3$

$x = 2$

$y = - 1$

$z = - 3$

$x = 2$

$y = - 1$

$z = - 3$

Step 7

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

$(2, - 1, - 3)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(2, - 1, - 3)$

Equation Form:

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

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

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