Why a set of linear equations has one answer, none, or infinitely many
Two straight lines on a page can cross once, run parallel forever, or lie exactly on top of each other. That picture explains every system of linear equations: any such system has a single solution, no solution, or infinitely many, and never, say, exactly two.
A linear system is simply several linear equations sharing the same unknowns, and a solution is a set of values satisfying all of them at once. A small example is 2x plus 3y equals 6 alongside 4x plus 9y equals 15. One way in is substitution: rearrange the first equation for x, plug that into the second, and you are left with a single equation in y. The same idea extends to larger systems.
Geometry makes the three outcomes intuitive. With two unknowns, each equation is a line and the solutions are where the lines meet: a point, a whole line, or nothing. With three unknowns, each equation is a plane; three parallel planes share no point, while three planes through a common line share infinitely many. In higher dimensions the equations become hyperplanes and the solutions form a flat region of lower dimension.
A rough rule of thumb comes from counting. Usually fewer equations than unknowns leaves infinitely many solutions, an underdetermined system; equal numbers usually pin down one; and more equations than unknowns usually leaves none, an overdetermined system. Particular coefficients can break these patterns. Algebraists also rewrite the whole system as a single matrix equation, Ax equals b, and read each unknown as a weight on a column vector.
These systems sit at the heart of linear algebra, and algorithms for solving them matter in engineering, physics, chemistry, computer science and economics. Even messy nonlinear problems are often approximated by linear ones when building models or computer simulations.
Source: System of linear equations
