Solve the System of Equations x-y-z=4 2x+y+3z=9 -3x-2y+4z=-23

$x - y - z = 4$ $2 x + y + 3 z = 9$ $- 3 x - 2 y + 4 z = - 23$

Step 1

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

Add $y$ to both sides of the equation.

$x - z = 4 + y$

$2 x + y + 3 z = 9$

$- 3 x - 2 y + 4 z = - 23$

Add $z$ to both sides of the equation.

$x = 4 + y + z$

$2 x + y + 3 z = 9$

$- 3 x - 2 y + 4 z = - 23$

$x = 4 + y + z$

$2 x + y + 3 z = 9$

$- 3 x - 2 y + 4 z = - 23$

Step 2

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

Replace all occurrences of $x$ in $2 x + y + 3 z = 9$ with $4 + y + z$ .

$2 (4 + y + z) + y + 3 z = 9$

$x = 4 + y + z$

$- 3 x - 2 y + 4 z = - 23$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$2 \cdot 4 + 2 y + 2 z + y + 3 z = 9$

$x = 4 + y + z$

$- 3 x - 2 y + 4 z = - 23$

Multiply $2$ by $4$ .

$8 + 2 y + 2 z + y + 3 z = 9$

$x = 4 + y + z$

$- 3 x - 2 y + 4 z = - 23$

$8 + 2 y + 2 z + y + 3 z = 9$

$x = 4 + y + z$

$- 3 x - 2 y + 4 z = - 23$

Simplify by adding terms.

Add $2 y$ and $y$ .

$8 + 3 y + 2 z + 3 z = 9$

$x = 4 + y + z$

$- 3 x - 2 y + 4 z = - 23$

Add $2 z$ and $3 z$ .

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

$- 3 x - 2 y + 4 z = - 23$

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

$- 3 x - 2 y + 4 z = - 23$

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

$- 3 x - 2 y + 4 z = - 23$

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

$- 3 x - 2 y + 4 z = - 23$

Replace all occurrences of $x$ in $- 3 x - 2 y + 4 z = - 23$ with $4 + y + z$ .

$- 3 (4 + y + z) - 2 y + 4 z = - 23$

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$- 3 \cdot 4 - 3 y - 3 z - 2 y + 4 z = - 23$

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

Multiply $- 3$ by $4$ .

$- 12 - 3 y - 3 z - 2 y + 4 z = - 23$

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

$- 12 - 3 y - 3 z - 2 y + 4 z = - 23$

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

Simplify by adding terms.

Subtract $2 y$ from $- 3 y$ .

$- 12 - 5 y - 3 z + 4 z = - 23$

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

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

$- 12 - 5 y + z = - 23$

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

$- 12 - 5 y + z = - 23$

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

$- 12 - 5 y + z = - 23$

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

$- 12 - 5 y + z = - 23$

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

$- 12 - 5 y + z = - 23$

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

Step 3

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

Add $12$ to both sides of the equation.

$- 5 y + z = - 23 + 12$

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

Add $5 y$ to both sides of the equation.

$z = - 23 + 12 + 5 y$

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

Add $- 23$ and $12$ .

$z = - 11 + 5 y$

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

$z = - 11 + 5 y$

$8 + 3 y + 5 z = 9$

$x = 4 + y + z$

Step 4

Replace all occurrences of $z$ with $- 11 + 5 y$ in each equation.

Replace all occurrences of $z$ in $8 + 3 y + 5 z = 9$ with $- 11 + 5 y$ .

$8 + 3 y + 5 (- 11 + 5 y) = 9$

$z = - 11 + 5 y$

$x = 4 + y + z$

Simplify the left side.

Simplify $8 + 3 y + 5 (- 11 + 5 y)$ .

Simplify each term.

Apply the distributive property.

$8 + 3 y + 5 \cdot - 11 + 5 (5 y) = 9$

$z = - 11 + 5 y$

$x = 4 + y + z$

Multiply $5$ by $- 11$ .

$8 + 3 y - 55 + 5 (5 y) = 9$

$z = - 11 + 5 y$

$x = 4 + y + z$

Multiply $5$ by $5$ .

$8 + 3 y - 55 + 25 y = 9$

$z = - 11 + 5 y$

$x = 4 + y + z$

$8 + 3 y - 55 + 25 y = 9$

$z = - 11 + 5 y$

$x = 4 + y + z$

Simplify by adding terms.

Subtract $55$ from $8$ .

$3 y - 47 + 25 y = 9$

$z = - 11 + 5 y$

$x = 4 + y + z$

Add $3 y$ and $25 y$ .

$28 y - 47 = 9$

$z = - 11 + 5 y$

$x = 4 + y + z$

$28 y - 47 = 9$

$z = - 11 + 5 y$

$x = 4 + y + z$

$28 y - 47 = 9$

$z = - 11 + 5 y$

$x = 4 + y + z$

$28 y - 47 = 9$

$z = - 11 + 5 y$

$x = 4 + y + z$

Replace all occurrences of $z$ in $x = 4 + y + z$ with $- 11 + 5 y$ .

$x = 4 + y - 11 + 5 y$

$28 y - 47 = 9$

$z = - 11 + 5 y$

Simplify $x = 4 + y - 11 + 5 y$ .

Simplify the left side.

Remove parentheses.

$x = 4 + y - 11 + 5 y$

$28 y - 47 = 9$

$z = - 11 + 5 y$

$x = 4 + y - 11 + 5 y$

$28 y - 47 = 9$

$z = - 11 + 5 y$

Simplify the right side.

Simplify $4 + y - 11 + 5 y$ .

Subtract $11$ from $4$ .

$x = y - 7 + 5 y$

$28 y - 47 = 9$

$z = - 11 + 5 y$

Add $y$ and $5 y$ .

$x = 6 y - 7$

$28 y - 47 = 9$

$z = - 11 + 5 y$

$x = 6 y - 7$

$28 y - 47 = 9$

$z = - 11 + 5 y$

$x = 6 y - 7$

$28 y - 47 = 9$

$z = - 11 + 5 y$

$x = 6 y - 7$

$28 y - 47 = 9$

$z = - 11 + 5 y$

$x = 6 y - 7$

$28 y - 47 = 9$

$z = - 11 + 5 y$

Step 5

Solve for $y$ in $28 y - 47 = 9$ .

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

Add $47$ to both sides of the equation.

$28 y = 9 + 47$

$x = 6 y - 7$

$z = - 11 + 5 y$

Add $9$ and $47$ .

$28 y = 56$

$x = 6 y - 7$

$z = - 11 + 5 y$

$28 y = 56$

$x = 6 y - 7$

$z = - 11 + 5 y$

Divide each term in $28 y = 56$ by $28$ and simplify.

Divide each term in $28 y = 56$ by $28$ .

$\frac{28 y}{28} = \frac{56}{28}$

$x = 6 y - 7$

$z = - 11 + 5 y$

Simplify the left side.

Cancel the common factor of $28$ .

Cancel the common factor.

$\frac{28 y}{28} = \frac{56}{28}$

$x = 6 y - 7$

$z = - 11 + 5 y$

Divide $y$ by $1$ .

$y = \frac{56}{28}$

$x = 6 y - 7$

$z = - 11 + 5 y$

$y = \frac{56}{28}$

$x = 6 y - 7$

$z = - 11 + 5 y$

$y = \frac{56}{28}$

$x = 6 y - 7$

$z = - 11 + 5 y$

Simplify the right side.

Divide $56$ by $28$ .

$y = 2$

$x = 6 y - 7$

$z = - 11 + 5 y$

$y = 2$

$x = 6 y - 7$

$z = - 11 + 5 y$

$y = 2$

$x = 6 y - 7$

$z = - 11 + 5 y$

$y = 2$

$x = 6 y - 7$

$z = - 11 + 5 y$

Step 6

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

Replace all occurrences of $y$ in $x = 6 y - 7$ with $2$ .

$x = 6 (2) - 7$

$y = 2$

$z = - 11 + 5 y$

Simplify the right side.

Simplify $6 (2) - 7$ .

Multiply $6$ by $2$ .

$x = 12 - 7$

$y = 2$

$z = - 11 + 5 y$

Subtract $7$ from $12$ .

$x = 5$

$y = 2$

$z = - 11 + 5 y$

$x = 5$

$y = 2$

$z = - 11 + 5 y$

$x = 5$

$y = 2$

$z = - 11 + 5 y$

Replace all occurrences of $y$ in $z = - 11 + 5 y$ with $2$ .

$z = - 11 + 5 (2)$

$x = 5$

$y = 2$

Simplify the right side.

Simplify $- 11 + 5 (2)$ .

Multiply $5$ by $2$ .

$z = - 11 + 10$

$x = 5$

$y = 2$

Add $- 11$ and $10$ .

$z = - 1$

$x = 5$

$y = 2$

$z = - 1$

$x = 5$

$y = 2$

$z = - 1$

$x = 5$

$y = 2$

$z = - 1$

$x = 5$

$y = 2$

Step 7

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

$(5, 2, - 1)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(5, 2, - 1)$

Equation Form:

$x = 5, y = 2, z = - 1$

Solve the System of Equations x-y-z=4 2x+y+3z=9 -3x-2y+4z=-23

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