We're covering first order systems, so decoupled systems, partially decoupled and special systems.

III. In class we considered the lines ax+by=e, cx+dy=f. We assumed that bit) and (#0 and showed under this assumption that the lines have the same slope if and only if ad-bc=0. Now let's look at the cases where b or d is 0. First, suppose b=0. We will then also assume agaEO because otherwise ax+by=e is not the equation of a line. So, assuming b=0 and aqO, show that ad-bc=0 if and only if the lines have the same slope. You may argue geometrically using what you know about slopes and lines. What if instead we assume d=0, c.7450? IV. Solve the following systems of equations give all solutions or show that no solution exists. For this problem, try to solve algebraically without using the general theory from class: a. 3x+4y=10, x-2y=8 b. 3x+4y=10, 6x+8y=20 c. 3x+4y=10, 6x+8y=5 V. For each of the following algebraic systems, use the general theory from class to determine (without attempting to solve) whether or not it is guaranteed to have exactly one solution. If it is, state this and stop. If not, determine all pairs (e,f) for which there will be innite solutions. a. 4x+2y=e 3x-5y=f b. 2x+4y=e -x 2y =f c. 5x+2y=e 1 5x+6y=f (1. 3x y=e y=f VI. Rewrite each of the algebra systems of part V as equations about linear combinations of vectors