Solve the System of Equations x+2y+3z=2 2x+4y+6z=4 3x+6y+12z=6

$x + 2 y + 3 z = 2$ $2 x + 4 y + 6 z = 4$ $3 x + 6 y + 12 z = 6$

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

Subtract $2 y$ from both sides of the equation.

$x + 3 z = 2 - 2 y$

$2 x + 4 y + 6 z = 4$

$3 x + 6 y + 12 z = 6$

Subtract $3 z$ from both sides of the equation.

$x = 2 - 2 y - 3 z$

$2 x + 4 y + 6 z = 4$

$3 x + 6 y + 12 z = 6$

$x = 2 - 2 y - 3 z$

$2 x + 4 y + 6 z = 4$

$3 x + 6 y + 12 z = 6$

Replace all occurrences of $x$ with $2 - 2 y - 3 z$ in each equation.

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

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

$x = 2 - 2 y - 3 z$

$3 x + 6 y + 12 z = 6$

Simplify the left side.

Simplify $2 (2 - 2 y - 3 z) + 4 y + 6 z$ .

Simplify each term.

Apply the distributive property.

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

$x = 2 - 2 y - 3 z$

$3 x + 6 y + 12 z = 6$

Simplify.

Multiply $2$ by $2$ .

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

$x = 2 - 2 y - 3 z$

$3 x + 6 y + 12 z = 6$

Multiply $- 2$ by $2$ .

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

$x = 2 - 2 y - 3 z$

$3 x + 6 y + 12 z = 6$

Multiply $- 3$ by $2$ .

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

$x = 2 - 2 y - 3 z$

$3 x + 6 y + 12 z = 6$

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

$x = 2 - 2 y - 3 z$

$3 x + 6 y + 12 z = 6$

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

$x = 2 - 2 y - 3 z$

$3 x + 6 y + 12 z = 6$

Combine the opposite terms in $4 - 4 y - 6 z + 4 y + 6 z$ .

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

$4 - 6 z + 0 + 6 z = 4$

$x = 2 - 2 y - 3 z$

$3 x + 6 y + 12 z = 6$

Add $4 - 6 z$ and $0$ .

$4 - 6 z + 6 z = 4$

$x = 2 - 2 y - 3 z$

$3 x + 6 y + 12 z = 6$

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

$4 + 0 = 4$

$x = 2 - 2 y - 3 z$

$3 x + 6 y + 12 z = 6$

Add $4$ and $0$ .

$4 = 4$

$x = 2 - 2 y - 3 z$

$3 x + 6 y + 12 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

$3 x + 6 y + 12 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

$3 x + 6 y + 12 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

$3 x + 6 y + 12 z = 6$

Replace all occurrences of $x$ in $3 x + 6 y + 12 z = 6$ with $2 - 2 y - 3 z$ .

$3 (2 - 2 y - 3 z) + 6 y + 12 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

Simplify the left side.

Simplify $3 (2 - 2 y - 3 z) + 6 y + 12 z$ .

Simplify each term.

Apply the distributive property.

$3 \cdot 2 + 3 (- 2 y) + 3 (- 3 z) + 6 y + 12 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

Simplify.

Multiply $3$ by $2$ .

$6 + 3 (- 2 y) + 3 (- 3 z) + 6 y + 12 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

Multiply $- 2$ by $3$ .

$6 - 6 y + 3 (- 3 z) + 6 y + 12 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

Multiply $- 3$ by $3$ .

$6 - 6 y - 9 z + 6 y + 12 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

$6 - 6 y - 9 z + 6 y + 12 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

$6 - 6 y - 9 z + 6 y + 12 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

Simplify by adding terms.

Combine the opposite terms in $6 - 6 y - 9 z + 6 y + 12 z$ .

Add $- 6 y$ and $6 y$ .

$6 - 9 z + 0 + 12 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

Add $6 - 9 z$ and $0$ .

$6 - 9 z + 12 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

$6 - 9 z + 12 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

Add $- 9 z$ and $12 z$ .

$6 + 3 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

$6 + 3 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

$6 + 3 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

$6 + 3 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

$6 + 3 z = 6$

$4 = 4$

$x = 2 - 2 y - 3 z$

Remove any equations from the system that are always true.

$6 + 3 z = 6$

$x = 2 - 2 y - 3 z$

Solve the System of Equations x+2y+3z=2 2x+4y+6z=4 3x+6y+12z=6

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