Systems of Three Equations in Three Variables

Solving a 3×3 system

With three equations and three unknowns, the strategy is to eliminate one variable at a time, reducing the system step by step: 3 equations → 2 equations → 1 equation → solution.

Step-by-step solution

Given:

• Eq 1: $x + y + z = 6$
• Eq 2: $2x - y + z = 3$
• Eq 3: $x + 2y - z = 2$

Step 1 — Eliminate $z$ by adding equations.

Add Eq 1 and Eq 3 (the $z$ and $-z$ cancel):

$(x + y + z) + (x + 2y - z) = 6 + 2$ $2x + 3y = 8 \quad \text{... call this Eq 4}$

Add Eq 2 and Eq 3 ($z$ and $-z$ cancel):

$(2x - y + z) + (x + 2y - z) = 3 + 2$ $3x + y = 5 \quad \text{... call this Eq 5}$

Step 2 — Solve the 2×2 system (Eq 4 and Eq 5).

From Eq 5: $y = 5 - 3x$. Substitute into Eq 4:

$2x + 3(5 - 3x) = 8$ $2x + 15 - 9x = 8$ $-7x = -7$ $x = 1$

Step 3 — Back-substitute for $y$.

$y = 5 - 3(1) = 2$

Step 4 — Back-substitute for $z$. Using Eq 1:

$1 + 2 + z = 6 \implies z = 3$

Solution: $(x, y, z) = (1, 2, 3)$.

Step 5 — Check in ALL three equations.

• Eq 1: $1 + 2 + 3 = 6$ ✓
• Eq 2: $2(1) - 2 + 3 = 3$ ✓
• Eq 3: $1 + 2(2) - 3 = 2$ ✓

Geometric interpretation

Each equation defines a plane in 3D space. Two planes intersect in a line; three planes (in general) intersect at a single point. That point is the solution.

Try it in the visualization

The 3D view shows three planes. Rotate to see them converge at the solution point. Adjust the constants to move the planes — the intersection point updates automatically.