1. 1.1 The row picture of a linear system with 3 variables

2. Problem 1 Solve the system of equations What picture comes to mind as you solve this system of equations? x +2y =1 2x+3y =1

3. Solution to problem 1 x +2y =1 2x+3y =1

4. Row picture and Column picture Problem 1 Row picture Column picture x +2y =1 2x+3y =1

5. Problem 3 Consider the following system of equations 2x + 4y = 3 3x + 6y = 2

6. Problem 3 Solution This system has no solution The graphing interpretation that we used in previous classes was the row picture. Two lines that do not intersect What does the column picture look like? From the point of view of the columns, why does this system have no solution? In general what would the column picture look like for systems of equations with no solution?

7. Row Picture and column picture Row picture Column Picture For what values of b could the equation Ax= b be solved? 2x + 4y = 3 3x + 6y = 2

8. Problem 4 Solve the system of equations. What picture comes to mind? 2x+4y = 2 3x+6y = 3

9. Problem 4

10. Row picture of problem 4 Row picture Column picture 2x+4y = 2 3x+6y = 3

11. Problem 6 Solve the following system of equations What does the row picture look like? What does the column picture look like?

12. Solution to problem 6 What does this solution mean from the column perspective?

13. Row picture Column Picture Row and Column Picture Problem 6

14. What if the system had no solutions or infinitely many solutions? What would the row picture look like? What would the equations look like when simplified? What would the Column picture look like?

15. Row picture of infinitely many solutions and no solutions in 3-D

16. Column picture of infinitely many solutions or no solutions in 3-D For what values of b could the equation Ax = b be solved? Case 1 The 3 column vectors are parallel Case 2 The 3 column vectors lie on the same plane

17. Homework Page 5 1-17,20,23,25