Which quick algebraic test allows you determine whether a two-line system has zero, one, or infinitely many solutions without needing to graph anything?

Graph both equations on paper and use a ruler to physically measure if the lines meet at any point; if they seem to coincide because of the ruler alignment then declare that there are infinite solutions, otherwise count the intersections to decide.

Convert both equations to slope-intercept form: y = mx + b and compare slopes (m) and intercepts (b): different slopes ⇒ one solution; same slopes different intercepts ⇒ zero solutions; identical slopes and intercepts ⇒ infinitely many solutions.

Solve numerically for x in both equations and compare the magnitude of the resulting roots: if one root is larger than the system is independent, if they are equal then they will be inconsistent, and if they match under scaling then they will be dependent.

Compute the product of the coefficients on x and y for each equation; if the products are equal then the system will have one solution, if they are different then the system will be inconsistent, and if they are both zero then the system will be dependent.