Solve the System of Equations x+y+z=2 5x+5y+5z=3 4x+y-3z=-6

$x + y + z = 2$ $5 x + 5 y + 5 z = 3$ $4 x + y - 3 z = - 6$

Step 1

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

Subtract $y$ from both sides of the equation.

$x + z = 2 - y$

$5 x + 5 y + 5 z = 3$

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

Subtract $z$ from both sides of the equation.

$x = 2 - y - z$

$5 x + 5 y + 5 z = 3$

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

$x = 2 - y - z$

$5 x + 5 y + 5 z = 3$

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

Step 2

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

Replace all occurrences of $x$ in $5 x + 5 y + 5 z = 3$ with $2 - y - z$ .

$5 (2 - y - z) + 5 y + 5 z = 3$

$x = 2 - y - z$

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

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$5 \cdot 2 + 5 (- y) + 5 (- z) + 5 y + 5 z = 3$

$x = 2 - y - z$

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

Simplify.

Multiply $5$ by $2$ .

$10 + 5 (- y) + 5 (- z) + 5 y + 5 z = 3$

$x = 2 - y - z$

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

Multiply $- 1$ by $5$ .

$10 - 5 y + 5 (- z) + 5 y + 5 z = 3$

$x = 2 - y - z$

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

Multiply $- 1$ by $5$ .

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

$x = 2 - y - z$

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

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

$x = 2 - y - z$

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

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

$x = 2 - y - z$

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

Combine the opposite terms in $10 - 5 y - 5 z + 5 y + 5 z$ .

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

$10 - 5 z + 0 + 5 z = 3$

$x = 2 - y - z$

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

Add $10 - 5 z$ and $0$ .

$10 - 5 z + 5 z = 3$

$x = 2 - y - z$

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

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

$10 + 0 = 3$

$x = 2 - y - z$

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

Add $10$ and $0$ .

$10 = 3$

$x = 2 - y - z$

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

$10 = 3$

$x = 2 - y - z$

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

$10 = 3$

$x = 2 - y - z$

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

$10 = 3$

$x = 2 - y - z$

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

Replace all occurrences of $x$ in $4 x + y - 3 z = - 6$ with $2 - y - z$ .

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

$10 = 3$

$x = 2 - y - z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$4 \cdot 2 + 4 (- y) + 4 (- z) + y - 3 z = - 6$

$10 = 3$

$x = 2 - y - z$

Simplify.

Multiply $4$ by $2$ .

$8 + 4 (- y) + 4 (- z) + y - 3 z = - 6$

$10 = 3$

$x = 2 - y - z$

Multiply $- 1$ by $4$ .

$8 - 4 y + 4 (- z) + y - 3 z = - 6$

$10 = 3$

$x = 2 - y - z$

Multiply $- 1$ by $4$ .

$8 - 4 y - 4 z + y - 3 z = - 6$

$10 = 3$

$x = 2 - y - z$

$8 - 4 y - 4 z + y - 3 z = - 6$

$10 = 3$

$x = 2 - y - z$

$8 - 4 y - 4 z + y - 3 z = - 6$

$10 = 3$

$x = 2 - y - z$

Simplify by adding terms.

Add $- 4 y$ and $y$ .

$8 - 3 y - 4 z - 3 z = - 6$

$10 = 3$

$x = 2 - y - z$

Subtract $3 z$ from $- 4 z$ .

$8 - 3 y - 7 z = - 6$

$10 = 3$

$x = 2 - y - z$

$8 - 3 y - 7 z = - 6$

$10 = 3$

$x = 2 - y - z$

$8 - 3 y - 7 z = - 6$

$10 = 3$

$x = 2 - y - z$

$8 - 3 y - 7 z = - 6$

$10 = 3$

$x = 2 - y - z$

$8 - 3 y - 7 z = - 6$

$10 = 3$

$x = 2 - y - z$

Step 3

Since $10 = 3$ is not true, there is no solution.

No solution

Solve the System of Equations x+y+z=2 5x+5y+5z=3 4x+y-3z=-6

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