2. LINEAR ALGEBRA

3. A toy manufacturer makes airplanes and boats: It costs $3 to make one airplane and $2 to make one boat. He has a total of $60. The many possibilities can be organized as solutions to 3a + 2b = 60 It takes 2 hours to make one airplane and 4 hours to make one boat. He has a total of 80 hours The many possibilities can be organized as solutions to 2a + 4b = 80 The system can be solved geometrically - the solution is the intersection of two lines (10 airplanes,15 boats)

4. A toy manufacturer makes airplanes and boats It costs $3 to make one airplane, $2 to make one boat and $1 to make one car. He has a total of $60. The many possibilities can be organized as solutions to 3a + 2b + 1c = 60 It takes 2 hours to make one airplane, 4 hours to make one boat and 6 hours to make one car. He has a total of 80 hours The many possibilities can be organized as solutions to 2a + 4b + 6c = 80 and cars

5. A toy manufacturer makes airplanes and boats It costs $3 to make one airplane, $2 to make one boat and $1 to make one car. He has a total of $60. The many possibilities can be organized as solutions to 3a + 2b + 1c = 60 It takes 2 hours to make one airplane, 4 hours to make one boat and 6 hours to make one car. He has a total of 80 hours The many possibilities can be organized as solutions to 2a + 4b + 6c = 80 and cars The system can be solved geometrically. The solutions are the points at the intersection of two planes

6. In 3 dimensional space, solutions to linear equations and systems can be organized geometrically as lines and points and planes and lines

7. A toy manufacturer makes airplanes and boats, It costs $3 to make one airplane, $2 to make one boat,$1 to make one car and $2 to make one drum. He has a total of $60. Possibilities are solutions to 3a + 2b + 1c + 2d = 60 It takes 2 hours to make one airplane, 4 hours to make one boat, 6 hours to make one car and 1 hour to make one drum. He has a total of 80 hours. Possibilities are solutions to 2a + 4b + 6c + 1d = 80 cars and drums

8. 3a + 2b + 1c + 2d = 60 2a + 4b + 6c + 1d = 80 We are now in four dimensions. It is no longer possible to organize solutions geometrically as points, lines, and planes. We can no longer visualize the solutions. However, the set of solutions has just as much structure as the solutions to the previous problems in two and three dimensions. We must rely on algebra to explore this structure.

9. The points on the line 2x - y = 0 have an obvious geometric structure and a not so obvious algebraic structure. Choose any two points on the line (2,4)and (-1,-2) The sum of these points (1,2) is also a point on the line We say that this set of points has closure An example of algebraic structure: