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Due at the beginning of class on Friday, September 5 Reading: Sections WILA, SSLE, RREF, VO, MO.

Problems: 1. Consider the system of equations 2 x + 3 y = − 7 x − 2 y = 14 (a) Find all solutions of to this system.

(b) On a single graph, draw the lines described by the two equations above, plus the lines described by any other equations that come up in the course of solving the system.

2. A booth at the farmer’s market has a sign advertising two onions and an heirloom tomato for $1 or 8 onions and 3 tomatoes for $3.50. What is the cost of a single onion? Of a single heirloom tomato?

3. Let f 1 , . . . , f m be functions of the variables you start with the system of equations x 1 , . . . , x n and let c 1 , . . . , c m be numbers. Suppose f 1 ( x 1 , . . . , x n ) = c 1 f 2 ( x 1 , . . . , x n ) = c 2 .

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f m ( x 1 , . . . , x n ) = c m and then apply Move (2) from class on 8/27 to the first equation. In other words, for some λ = 0 you create the new system of equations λf 1 ( x 1 , . . . , x n ) = λc 1 f 2 ( x 1 , . . . , x n ) = c 2 .

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f m ( x 1 , . . . , x n ) = c m (a) Show that the new system has exactly the same set of solutions as the original system.

(b) Why was it important that λ = 0. What can go wrong if possible (e.g., by giving examples).

λ = 0. Be as specific as 4. Consider the following system of linear equations: a − a a + − + 2 b 2 b b + 3 c + c + − + 4 d d d = = = 1 2 − 3 1

(a) Use equation operations to find any solutions of this system.

(b) Write down the augmented matrix that corresponds to this system of linear equations.

(c) Transform the matrix into reduced row echelon form.

(d) Use part (c) to find any solutions of the system. ( Hint: than in part (a), you did something wrong.) If you get different answers here 5. The following augmented matrices correspond to different systems of equations. For each matrix, decide whether the corresponding system has no solutions, one solution, or infinitely many solutions.

(a) 1 0 0 0 (b) 10 0 0 0 0 1 0 0 9 0 0 0 0 0 1 0 8 5 0 0 0 0 0 1 7 4 2 0 1 − 2 3 0 6 3 1 0 (c) 1 0 0 0 2 6 0 0 3 7 10 0 4 8 11 13 5 9 12 0 2