2D Systems Two variables
2D Systems of Linear Equations
A system of two linear equations in two variables has the form:
a₁x + b₁y = c₁
a₂x + b₂y = c₂
a₂x + b₂y = c₂
Each equation represents a line in the plane. The solution corresponds to the intersection point.
Three Possible Outcomes
- Unique Solution — Lines intersect at one point. Consistent & independent.
- No Solution — Lines are parallel. Inconsistent system.
- Infinite Solutions — Lines are coincident. Consistent & dependent.
Dragging
Click and drag the numbered circles on each line to shift it. The constant term c updates as you drag.
↔ Click and drag the numbered circles on each line to move it
Equation 1 — Blue Line
x
+
y
=
Equation 2 — Pink Line
x
+
y
=
Solution
—
Drag lines to see solution
[
2.0
1.0
]
[
x
]
=
[
5.0
]
[
1.0
-1.0
]
[
y
]
=
[
0.0
]
3D Systems Three variables
3D Systems of Linear Equations
A system of three equations in three variables:
a₁x + b₁y + c₁z = d₁
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃
Each equation represents a plane in 3D space. The solution is the intersection of all three planes.
Possible Outcomes
- Unique Solution — Planes intersect at exactly one point.
- No Solution — Planes have no common intersection.
- Line of Solutions — Planes intersect along a line.
- Plane of Solutions — All three planes coincide.
Navigation
Click and drag to orbit. Scroll to zoom in/out. The green sphere marks the unique solution point.
⊙ Drag to orbit the 3D view. Scroll to zoom. Green sphere = solution point
Equation 1 — Cyan Plane
x
+
y
+
z
=
Equation 2 — Magenta Plane
x
+
y
+
z
=
Equation 3 — Yellow Plane
x
+
y
+
z
=
Solution
✓ Unique
Calculating…
[
1.0
1.0
1.0
]
[
x
]
=
[
6.0
]
[
2.0
1.0
-1.0
]
[
y
]
=
[
1.0
]
[
1.0
-1.0
2.0
]
[
z
]
=
[
3.0
]