I want a solution that explains why three lines that intersect at only one point is not independent. And is there a way of proving that only one of them is dependent? How do we differentiate between dependent and independent linear systems?
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Mathematics is human defined. Hence, for each concept there is some definition. For dependent and independent concepts, there is also a definition.
Let me first explain to you the meaning of independent and dependent:
If n variables x1, x2, x3, ..., xn are dependent, then
a1*x1 + a2*x2 + a3*x3 + ... + an*xn = 0
where a1, a2, a3, ..., an are non-zero constants.
Here, x1 or x2 or x3 ... can be written in terms of other variables:
x1 = - (1/a1) (a2*x2 + ...
This solution provides proof that three lines that intersect at only one point are dependent. Included is a definition of independent and dependent as it applies to the question and an in depth explanation of what distinguishes dependent and independent linear systems.