The discussion revolves around solving a system of equations represented by .5= (x-40)/(x-y) and .5=(40-y)/(x-y). Participants are exploring methods to manipulate these equations to find values for x and y.
The conversation includes attempts to clarify the problem and explore different solving techniques. Some participants have provided partial manipulations of the equations, while others have noted potential issues with the independence of the equations, leading to the possibility of infinite solutions. There is no explicit consensus on the best approach or final outcome.
Participants are working under the constraints of a homework forum, where complete solutions are discouraged. There is a noted concern about providing answers rather than guiding through the reasoning process.
HallsofIvy said:Personally, I would get rid of all fractions by multiplying each equation by 2(x- y).
That gives x- y= 2(x- 40) and x- y= 2(40- y). Multiplying those out, x- y= 2x- 80 or x+ y= 80, and x- y= 80- 2y or x+ y= 80. Those two equations are the same so the original equations were not "independent". There are an infinite number of solutions. Given any x, (x, y)= (x, 80- x) is a solution (except (40, 40) for which x- y= 0 and the original equation makes no sense).