II. In items 6-11, consider a 31-point geometry. For a model of the 31-point geometry (6...

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II. In items 6-11, consider a 31-point geometry. For a model of the 31-point geometry (6 points on any line) take as points the triples (x1, x2, x3)(0,0,0), where each of the x is from the set {0, 1, 2, 3, 4) of remainders when the natural numbers are divided by 5, and as lines the equations of the form α₁x₁ + a₂ + ax=0 where each of the a, is from the same set of remainders ex- cepting only the choice a₁ = a₂ a3 = 0. We also agree that (ja, ja, jxs), where (j = 1,2,3,4), are the same point and that j(a₁ + a₂ + axs) = 0 are the same line. 6. (5 points) Are the points (3,0,4) and (4,0,2) the same? Why or why not? 7. (5 points) What points are the same with (3,2,0)? 8. (6 points) Name 6 different lines that pass through the point (3,1,4). 9. (6 points) Name 6 different points that are on line 2x1 + x2 + 3x3 = 0. 10. (10 points) What is the common point to the lines 4x1 + x2 + 2x3 = 0 and 2x₁ + 3x2 + 4x3 = 0? 11. (10 points) What is the equation of the line, if there exists, that contains the points L(4,1,2) and M(1,3,2)? If no such line exists, explain why. II. In items 6-11, consider a 31-point geometry. For a model of the 31-point geometry (6 points on any line) take as points the triples (x1, x2, x3)(0,0,0), where each of the x is from the set {0, 1, 2, 3, 4) of remainders when the natural numbers are divided by 5, and as lines the equations of the form α₁x₁ + a₂ + ax=0 where each of the a, is from the same set of remainders ex- cepting only the choice a₁ = a₂ a3 = 0. We also agree that (ja, ja, jxs), where (j = 1,2,3,4), are the same point and that j(a₁ + a₂ + axs) = 0 are the same line. 6. (5 points) Are the points (3,0,4) and (4,0,2) the same? Why or why not? 7. (5 points) What points are the same with (3,2,0)? 8. (6 points) Name 6 different lines that pass through the point (3,1,4). 9. (6 points) Name 6 different points that are on line 2x1 + x2 + 3x3 = 0. 10. (10 points) What is the common point to the lines 4x1 + x2 + 2x3 = 0 and 2x₁ + 3x2 + 4x3 = 0? 11. (10 points) What is the equation of the line, if there exists, that contains the points L(4,1,2) and M(1,3,2)? If no such line exists, explain why.

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In the 31point geometry youve described lets address each of the questions 1Are the points 304 and 4... View the full answer