Solve the System of Equations x+y+z=2 6x-4y+5z=31 5x+2y+2z=13

$x + y + z = 2$ $6 x - 4 y + 5 z = 31$ $5 x + 2 y + 2 z = 13$

Step 1

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

Subtract $y$ from both sides of the equation.

$x + z = 2 - y$

$6 x - 4 y + 5 z = 31$

$5 x + 2 y + 2 z = 13$

Subtract $z$ from both sides of the equation.

$x = 2 - y - z$

$6 x - 4 y + 5 z = 31$

$5 x + 2 y + 2 z = 13$

$x = 2 - y - z$

$6 x - 4 y + 5 z = 31$

$5 x + 2 y + 2 z = 13$

Step 2

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

Replace all occurrences of $x$ in $6 x - 4 y + 5 z = 31$ with $2 - y - z$ .

$6 (2 - y - z) - 4 y + 5 z = 31$

$x = 2 - y - z$

$5 x + 2 y + 2 z = 13$

Simplify the left side.

Simplify $6 (2 - y - z) - 4 y + 5 z$ .

Simplify each term.

Apply the distributive property.

$6 \cdot 2 + 6 (- y) + 6 (- z) - 4 y + 5 z = 31$

$x = 2 - y - z$

$5 x + 2 y + 2 z = 13$

Simplify.

Multiply $6$ by $2$ .

$12 + 6 (- y) + 6 (- z) - 4 y + 5 z = 31$

$x = 2 - y - z$

$5 x + 2 y + 2 z = 13$

Multiply $- 1$ by $6$ .

$12 - 6 y + 6 (- z) - 4 y + 5 z = 31$

$x = 2 - y - z$

$5 x + 2 y + 2 z = 13$

Multiply $- 1$ by $6$ .

$12 - 6 y - 6 z - 4 y + 5 z = 31$

$x = 2 - y - z$

$5 x + 2 y + 2 z = 13$

$12 - 6 y - 6 z - 4 y + 5 z = 31$

$x = 2 - y - z$

$5 x + 2 y + 2 z = 13$

$12 - 6 y - 6 z - 4 y + 5 z = 31$

$x = 2 - y - z$

$5 x + 2 y + 2 z = 13$

Simplify by adding terms.

Subtract $4 y$ from $- 6 y$ .

$12 - 10 y - 6 z + 5 z = 31$

$x = 2 - y - z$

$5 x + 2 y + 2 z = 13$

Add $- 6 z$ and $5 z$ .

$12 - 10 y - z = 31$

$x = 2 - y - z$

$5 x + 2 y + 2 z = 13$

$12 - 10 y - z = 31$

$x = 2 - y - z$

$5 x + 2 y + 2 z = 13$

$12 - 10 y - z = 31$

$x = 2 - y - z$

$5 x + 2 y + 2 z = 13$

$12 - 10 y - z = 31$

$x = 2 - y - z$

$5 x + 2 y + 2 z = 13$

Replace all occurrences of $x$ in $5 x + 2 y + 2 z = 13$ with $2 - y - z$ .

$5 (2 - y - z) + 2 y + 2 z = 13$

$12 - 10 y - z = 31$

$x = 2 - y - z$

Simplify the left side.

Simplify $5 (2 - y - z) + 2 y + 2 z$ .

Simplify each term.

Apply the distributive property.

$5 \cdot 2 + 5 (- y) + 5 (- z) + 2 y + 2 z = 13$

$12 - 10 y - z = 31$

$x = 2 - y - z$

Simplify.

Multiply $5$ by $2$ .

$10 + 5 (- y) + 5 (- z) + 2 y + 2 z = 13$

$12 - 10 y - z = 31$

$x = 2 - y - z$

Multiply $- 1$ by $5$ .

$10 - 5 y + 5 (- z) + 2 y + 2 z = 13$

$12 - 10 y - z = 31$

$x = 2 - y - z$

Multiply $- 1$ by $5$ .

$10 - 5 y - 5 z + 2 y + 2 z = 13$

$12 - 10 y - z = 31$

$x = 2 - y - z$

$10 - 5 y - 5 z + 2 y + 2 z = 13$

$12 - 10 y - z = 31$

$x = 2 - y - z$

$10 - 5 y - 5 z + 2 y + 2 z = 13$

$12 - 10 y - z = 31$

$x = 2 - y - z$

Simplify by adding terms.

Add $- 5 y$ and $2 y$ .

$10 - 3 y - 5 z + 2 z = 13$

$12 - 10 y - z = 31$

$x = 2 - y - z$

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

$10 - 3 y - 3 z = 13$

$12 - 10 y - z = 31$

$x = 2 - y - z$

$10 - 3 y - 3 z = 13$

$12 - 10 y - z = 31$

$x = 2 - y - z$

$10 - 3 y - 3 z = 13$

$12 - 10 y - z = 31$

$x = 2 - y - z$

$10 - 3 y - 3 z = 13$

$12 - 10 y - z = 31$

$x = 2 - y - z$

$10 - 3 y - 3 z = 13$

$12 - 10 y - z = 31$

$x = 2 - y - z$

Step 3

Solve for $y$ in $10 - 3 y - 3 z = 13$ .

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

Subtract $10$ from both sides of the equation.

$- 3 y - 3 z = 13 - 10$

$12 - 10 y - z = 31$

$x = 2 - y - z$

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

$- 3 y = 13 - 10 + 3 z$

$12 - 10 y - z = 31$

$x = 2 - y - z$

Subtract $10$ from $13$ .

$- 3 y = 3 + 3 z$

$12 - 10 y - z = 31$

$x = 2 - y - z$

$- 3 y = 3 + 3 z$

$12 - 10 y - z = 31$

$x = 2 - y - z$

Divide each term in $- 3 y = 3 + 3 z$ by $- 3$ and simplify.

Divide each term in $- 3 y = 3 + 3 z$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{3}{- 3} + \frac{3 z}{- 3}$

$12 - 10 y - z = 31$

$x = 2 - y - z$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{3}{- 3} + \frac{3 z}{- 3}$

$12 - 10 y - z = 31$

$x = 2 - y - z$

Divide $y$ by $1$ .

$y = \frac{3}{- 3} + \frac{3 z}{- 3}$

$12 - 10 y - z = 31$

$x = 2 - y - z$

$y = \frac{3}{- 3} + \frac{3 z}{- 3}$

$12 - 10 y - z = 31$

$x = 2 - y - z$

$y = \frac{3}{- 3} + \frac{3 z}{- 3}$

$12 - 10 y - z = 31$

$x = 2 - y - z$

Simplify the right side.

Simplify each term.

Divide $3$ by $- 3$ .

$y = - 1 + \frac{3 z}{- 3}$

$12 - 10 y - z = 31$

$x = 2 - y - z$

Cancel the common factor of $3$ and $- 3$ .

Factor $3$ out of $3 z$ .

$y = - 1 + \frac{3 (z)}{- 3}$

$12 - 10 y - z = 31$

$x = 2 - y - z$

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

$y = - 1 - 1 \cdot z$

$12 - 10 y - z = 31$

$x = 2 - y - z$

$y = - 1 - 1 \cdot z$

$12 - 10 y - z = 31$

$x = 2 - y - z$

Rewrite $- 1 \cdot z$ as $- z$ .

$y = - 1 - z$

$12 - 10 y - z = 31$

$x = 2 - y - z$

$y = - 1 - z$

$12 - 10 y - z = 31$

$x = 2 - y - z$

$y = - 1 - z$

$12 - 10 y - z = 31$

$x = 2 - y - z$

$y = - 1 - z$

$12 - 10 y - z = 31$

$x = 2 - y - z$

$y = - 1 - z$

$12 - 10 y - z = 31$

$x = 2 - y - z$

Step 4

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

Replace all occurrences of $y$ in $12 - 10 y - z = 31$ with $- 1 - z$ .

$12 - 10 (- 1 - z) - z = 31$

$y = - 1 - z$

$x = 2 - y - z$

Simplify the left side.

Simplify $12 - 10 (- 1 - z) - z$ .

Simplify each term.

Apply the distributive property.

$12 - 10 \cdot - 1 - 10 (- z) - z = 31$

$y = - 1 - z$

$x = 2 - y - z$

Multiply $- 10$ by $- 1$ .

$12 + 10 - 10 (- z) - z = 31$

$y = - 1 - z$

$x = 2 - y - z$

Multiply $- 1$ by $- 10$ .

$12 + 10 + 10 z - z = 31$

$y = - 1 - z$

$x = 2 - y - z$

$12 + 10 + 10 z - z = 31$

$y = - 1 - z$

$x = 2 - y - z$

Simplify by adding terms.

Add $12$ and $10$ .

$22 + 10 z - z = 31$

$y = - 1 - z$

$x = 2 - y - z$

Subtract $z$ from $10 z$ .

$22 + 9 z = 31$

$y = - 1 - z$

$x = 2 - y - z$

$22 + 9 z = 31$

$y = - 1 - z$

$x = 2 - y - z$

$22 + 9 z = 31$

$y = - 1 - z$

$x = 2 - y - z$

$22 + 9 z = 31$

$y = - 1 - z$

$x = 2 - y - z$

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

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

$22 + 9 z = 31$

$y = - 1 - z$

Simplify the right side.

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

Simplify each term.

Apply the distributive property.

$x = 2 + 1 + z - z$

$22 + 9 z = 31$

$y = - 1 - z$

Multiply $- 1$ by $- 1$ .

$x = 2 + 1 + z - z$

$22 + 9 z = 31$

$y = - 1 - z$

Multiply $- - z$ .

Multiply $- 1$ by $- 1$ .

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

$22 + 9 z = 31$

$y = - 1 - z$

Multiply $z$ by $1$ .

$x = 2 + 1 + z - z$

$22 + 9 z = 31$

$y = - 1 - z$

$x = 2 + 1 + z - z$

$22 + 9 z = 31$

$y = - 1 - z$

$x = 2 + 1 + z - z$

$22 + 9 z = 31$

$y = - 1 - z$

Simplify by adding terms.

Combine the opposite terms in $2 + 1 + z - z$ .

Subtract $z$ from $z$ .

$x = 2 + 1 + 0$

$22 + 9 z = 31$

$y = - 1 - z$

Add $2 + 1$ and $0$ .

$x = 2 + 1$

$22 + 9 z = 31$

$y = - 1 - z$

$x = 2 + 1$

$22 + 9 z = 31$

$y = - 1 - z$

Add $2$ and $1$ .

$x = 3$

$22 + 9 z = 31$

$y = - 1 - z$

$x = 3$

$22 + 9 z = 31$

$y = - 1 - z$

$x = 3$

$22 + 9 z = 31$

$y = - 1 - z$

$x = 3$

$22 + 9 z = 31$

$y = - 1 - z$

$x = 3$

$22 + 9 z = 31$

$y = - 1 - z$

Step 5

Solve for $z$ in $22 + 9 z = 31$ .

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

Subtract $22$ from both sides of the equation.

$9 z = 31 - 22$

$x = 3$

$y = - 1 - z$

Subtract $22$ from $31$ .

$9 z = 9$

$x = 3$

$y = - 1 - z$

$9 z = 9$

$x = 3$

$y = - 1 - z$

Divide each term in $9 z = 9$ by $9$ and simplify.

Divide each term in $9 z = 9$ by $9$ .

$\frac{9 z}{9} = \frac{9}{9}$

$x = 3$

$y = - 1 - z$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 z}{9} = \frac{9}{9}$

$x = 3$

$y = - 1 - z$

Divide $z$ by $1$ .

$z = \frac{9}{9}$

$x = 3$

$y = - 1 - z$

$z = \frac{9}{9}$

$x = 3$

$y = - 1 - z$

$z = \frac{9}{9}$

$x = 3$

$y = - 1 - z$

Simplify the right side.

Divide $9$ by $9$ .

$z = 1$

$x = 3$

$y = - 1 - z$

$z = 1$

$x = 3$

$y = - 1 - z$

$z = 1$

$x = 3$

$y = - 1 - z$

$z = 1$

$x = 3$

$y = - 1 - z$

Step 6

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

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

$y = - 1 - (1)$

$z = 1$

$x = 3$

Simplify the right side.

Simplify $- 1 - (1)$ .

Multiply $- 1$ by $1$ .

$y = - 1 - 1$

$z = 1$

$x = 3$

Subtract $1$ from $- 1$ .

$y = - 2$

$z = 1$

$x = 3$

$y = - 2$

$z = 1$

$x = 3$

$y = - 2$

$z = 1$

$x = 3$

$y = - 2$

$z = 1$

$x = 3$

Step 7

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

$(3, - 2, 1)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(3, - 2, 1)$

Equation Form:

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

Solve the System of Equations x+y+z=2 6x-4y+5z=31 5x+2y+2z=13

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