
#1
Jan314, 05:41 PM

P: 82

Hi. I don't understand how a solution to a linear system is obtained (for example geometrically; don't consider the substitution method and elimination), and I am feeling very frustrated.
Say I have the following equations: y = x + 5 b = 2*a (the relation remains the same even if I change the variables) Obviously, the solution is (x=a=5 and y=b=10) or (x=b=5 and y=a=10). The first equation has the solution sets, A1={(x, x+5) : x∈R} and A2={(x+5, x) : x∈R}. The second equation has the solution sets, B1={(x, 2*x) : x∈R} and B2={(2*x, x) : x∈R} A1 ∩ B1 and A2 ∩ B2 are the solution sets to the system, if x=a and y=b. A1 ∩ B2 and A2 ∩ B1 are the solution sets to the system, if x=b and y=a. How can I prove that the variables which are meant to be equal (for example x=a), must be both the first or the second element in given pairs? For example if x=a and I consider A2, then I must consider B2, i.e. if x is the second element of the pairs x and y, then a must also be the second element of pairs a and b. Am I thinking right? I've been looking at many websites but none really cleared up my confusion. I'm really thankful if you can explain this clearly. 



#2
Jan314, 06:14 PM

Mentor
P: 21,069

y = x + 5 y = 2x then the system is trivial to solve. 


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