1 · IB Maths AA
Systems of linear equations
Each equation ax + by + cz = d is a plane. Change the coefficients, rotate the 3D view, and see when the three planes meet at one point, along a line, or not at all.
Drag or use the arrow keys to rotate. The box shows −5 ≤ x, y, z ≤ 5; dashed lines are where pairs of planes meet.
Equation i: aᵢx + bᵢy + cᵢz = dᵢ
Matrix form: A x = b
Unique solution ⇔ det(A) ≠ 0
Consistent ⇔ rank(A) = rank(A|b)
Try this
- Choose “Unique point”, then set a₃ to −2.5: det(A) becomes 0 and a prism appears. Which value of d₃ turns it into a line of solutions?
- Choose “Prism” and rotate until you look straight along the three dashed lines: the planes now look like the sides of a triangle.
- Choose “Two parallel planes” and change b₂ by 0.5. What happens to rank(A), and what kind of solution set appears?
Exam tip: when det(A) = 0 your calculator cannot give a unique answer, so row-reduce: a row 0 0 0 | k with k ≠ 0 means no solution, while a full row of zeros means infinitely many, written with a parameter λ.