Budget planners in two towns, Alphaville and Betaville, developed models to determine the budget surplus (in dollars) for a year based on the tax revenue (in thousands of dollars) for the year. Using historical data, Alphaville’s planner produced the model -2x2+ 500x, while Betaville’s planner produced the model x2- 100x + 10,000. What expression gives how much greater Alphaville’s annual budget surplus is than Betaville’s for a particular amount of tax revenue? If the tax revenue that year in each town is $75,000, how much greater is Alphaville’s budget surplus than Betaville’s that year?

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answer: a video game has three difficulty levels: easy, medium, and expert. the game's scoring works in such a way that a player has to score a fixed number of points to advance to the next level, and then the scoring starts over. the total score needed to clear all three difficulty levels is 7,500. two times the number of points needed to clear the easy level is 500 more than the number of points needed to clear the medium level. twice the number of points needed to clear the medium difficulty level is 1,500 more than the number of points needed to clear the expert level. if x, y, and z represent the easy, medium, and expert levels, respectively, and a is the coefficient matrix of the system of equations modeling this situation, identify the inverse matrix, a-1, and the solution matrix, x, of the system of equations.

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