Systems of Linear Equations in Three Variables

1. Use two equations to eliminate one variable → 2-variable system.
2. Solve the 2-variable system.
3. Back-substitute.

$x + y + z = 6$, $2x - y + z = 3$, $x + 2y - z = 4$.

Add (1) and (3): $2x + 3y = 10$. Add (1) and (2): $3x + 2z = 9$ — but we need to eliminate $z$ properly.

(1) + (3): $2x + 3y = 10$. (2) + (3): $3x + y = 7$.

From $3x + y = 7$: $y = 7 - 3x$. Sub: $2x + 3(7-3x) = 10$ → $-7x = -11$ → $x = \frac{11}{7}$.

Continue to find $y$ and $z$.

Enter coefficients into a system solver or use row reduction (rref).

• No solution: inconsistent system (parallel planes).
• Infinite solutions: dependent system (planes intersect in a line).