Homogeneous linear system question

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A homogeneous linear system with two equations in three unknowns always has infinitely many solutions because the equations represent planes in three-dimensional space, and their intersection forms a line containing infinitely many points. Each homogeneous equation includes the origin, guaranteeing at least one solution. In contrast, a nonhomogeneous system may have no solutions if the equations are contradictory. Therefore, the geometric interpretation of solutions differs significantly between homogeneous and nonhomogeneous systems. Understanding these concepts is crucial for solving linear systems effectively.
loli12
Hi, i have a question. Hope you guys can help~

Ques: Give a geometric explanation of why a homogeneous linear system consisting of 2 equations in 3 unknowns must have inifinitely many solutions. What are the possible numbers of solutions for a nonhomogeneous 2 x 3 linear system? Give a geometric explanation of your answer.
 
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Each equation in your set represents a plane in three dimensions. With only two equations the solutions consist of the intersection of those planes (a line)which corresponds to infinitely many points.
 
I got it, Thanks a lot!
 
A bit more care is required:
Each homogenous equation is an equation of the plane WHICH CONTAIN THE ORIGIN!
Hence, the system has always at least one solution!
This is by no means always true for an inhomogenous system.
(That is, an inhomogenous system may have no solutions at all (a self-contradictory system))
 
Thread 'Correct statement about size of wire to produce larger extension'

Thread 'Correct statement about size of wire to produce larger extension'

The answer is (B) but I don't really understand why. Based on formula of Young Modulus: $$x=\frac{FL}{AE}$$ The second wire made of the same material so it means they have same Young Modulus. Larger extension means larger value of ##x## so to get larger value of ##x## we can increase ##F## and ##L## and decrease ##A## I am not sure whether there is change in ##F## for first and second wire so I will just assume ##F## does not change. It leaves (B) and (C) as possible options so why is (C)...
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