Final answer:
A linear equation's solution is unique when the equation represents a line that does not coincide with any other line. This unique solution occurs at the point where the line intersects an axis or another non-parallel line, resulting in a single intersection, which is considered the 'trivial solution' for the case of the line itself.
Step-by-step explanation:
When we talk about linear equations and their solutions, we're often dealing with an equation in the format y = a + bx, which represents a straight line. Each combination of a and b values describes a unique line with a different slope and intercept on a graph. Now, when we find a solution to a linear equation, we're finding the point(s) at which the line represented crosses the x-axis (when y=0), or the y-axis (when x=0).
The question of a unique solution being associated with a 'trivial solution' relates to the intersection of two lines. If we have two non-parallel lines, they will intersect at a single point, giving us a unique solution. However, if two lines are completely overlapping, they are essentially the same line and have an infinite number of intersecting points - hence, we don't have a unique solution; instead, we have infinitely many solutions, which is the 'trivial' case.
Consequently, a unique solution implies that the equation must describe a line that does not coincide with any other line, hence the solution is unique if and only if there is just the 'trivial solution' (that is, no other line is coinciding with it, delivering a unique intersection point).