Linear equations are named for the lines they draw
Put an equation in the form of a linear polynomial set to zero, and its solutions in two variables sketch a straight line in the plane. That geometric picture—not the algebra alone—is why we call the equation linear.
Coefficients are parameters that may be arbitrary expressions so long as they avoid the unknowns; typically they are real numbers, though any field works. Solutions are the values that make the equality true. With one variable and a nonzero leading coefficient there is exactly one solution—the classic "unknown." With two variables, solutions are points of a Euclidean plane forming a line (if not both coefficients of the variables vanish); conversely every line arises this way. In n variables the solution set is a hyperplane of dimension n − 1. Linear equations flood physics and engineering partly because nonlinear systems are often well approximated by them.
Writing ax + by + c = 0 with real a and b typically yields infinitely many solutions. Solving for y when b ≠ 0 produces a function whose graph is a line with a definite slope. Calculus often calls any straight-graph function "linear," but linear algebra reserves linear maps for functions that send sums to sums and pass through the origin—so the general line is affine, and only the through-origin case is a linear map. If b = 0 and a ≠ 0 the line is vertical; if a = 0 and b ≠ 0 it is horizontal.
Many textbook forms—slope-intercept, point-slope, two-point, intercept—are just different ways to write the same line as a linear equation. The article's focus is a single equation over the reals; systems of several simultaneous linear equations are the next chapter, solved by elimination and matrices. The naming lesson sticks: linearity here means the geometry of a flat cut, not merely the absence of x².
Source: Linear equation
