How Many Solutions? Lines and planes

Each challenge checks itself. Work on this keeps it in view above the controls.

Change the coefficients of two or three equations and watch lines in the plane, or planes in space, meet in one point, along a line, or not at all; the augmented matrix and its row-reduced form update live.

How to use

• Each equation a x + b y = c is a line in the plane. Change the three numbers of an equation and its line moves.
• Choose 2 or 3 equations. A solution of the system is a point that lies on every line.
• The status box under the picture says whether the lines meet in one point (exactly one solution), never meet (no solution, the system is inconsistent), or coincide (infinitely many solutions). Screen readers announce it.
• The augmented matrix and its reduced row echelon form update live. Pivots are shown in bold with a box. A row that reads 0 0 | nonzero means "0 = nonzero": no solution.
• Keyboard: Tab moves between boxes; Up/Down change a number by 1. Nothing needs a mouse.
• Lines are told apart by pattern and label, not only colour: L1 is solid, L2 is dashed, L3 is dotted.
• Planes in space. Each equation a x + b y + c z = d is a plane. Two or three planes can meet in one point, along a line, in a whole plane, or not at all (parallel planes, or a "prism": every pair meets, but the three lines of intersection are parallel). Turn and tilt the box, or drag it, to see how they meet. The solution is drawn in black: a dot, a thick line, or an outlined plane.

Accessibility: every control is a standard form control you can reach with Tab and change with the arrow keys; the status box is announced by screen readers; the picture has a text description (press Describe) that updates as you work; animations are skipped when your system asks for reduced motion.