3 years ago

Why do axioms exist in geometry

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Our three axioms A1,A2 and A3 for affine plane geometry are consistent since there is at least one model for these axioms. However this system is incomplete because in this system there exist relevant statements which cannot be either proven or disproven. Consider for example the statement A7.

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What is parallel to 2x-y=1 and contains (4, -6)? Please show your work!

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$k:\ y=m_1x+b_1\\\\l:\ y=m_2x+b_2\\\\l\ ||\ k\iff m_1=m_2\\\\\text{We have}\\\\k:\ 2x-y=1\ \ \ |-2x\\-y=-2x+1\ \ \ \ |\cdot(-1)\\y=2x-1\to m_1=2\\\\l:\ y=m_2x+b\\\\l\ ||\ k\iff m_2=2\\\\l:\ y=2x+b\\\\\text{The line l contains (4, -6)}\\\\\text{Put the values of coordinates to the equation of line l}\\\\-6=2(4)+b\\-6=8+b\ \ \ \ \ |-8\\b=-14\\\\Answer:\ y=2x-14\to 2x-y=14$