The Lineweaver-Burk equation comes from taking reciprocals and rearranging. K m , the Michaelis constant, accounts for the previous rate constants. This set of simultaneous differential equations was used in a JAVA applet for you to experiment with concentrations and coefficients. The rates of substrate decline and product formation are nearly linear throughout much of the simulation. This alone has some teaching value because a student can verify that the textbook results make sense.
Additional insight comes from tinkering with the rate coefficients to see their effect.
It must be appreciated that one reaction with one enzyme in vitro is not typical of living biochemistry where there are complicated pathways and intermediates tend not to accumulate. Actual metabolic pathways have branches and many steps, and the over accumulation of intermediates is prevented by feedback inhibition where a concentration of a biochemical further along in the pathway inhibits or represses the activity of the enzyme for an early step.