Math 340 Homework Assignment One (Summer 24) Do the following problems showing a

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Math 340 Homework Assignment One (Summer 24)
Do the following problems showing all your work. Be sure to submit your solutions in
PDF format to the Assignment area of LEO. (Point values are given in the square
brackets.)
1. [15] Use elimination to find all solutions of the system:
� + 2� + 3� = 70
4� + 5� + 6� = 160
7� + 8� + 9� = 250
2. [15] Find all the solutions of the linear system,
� + 2� + 3� = �
� + 3� + 8� = �
� + 2� + 2� = �
where a, b, and c are arbitrary constants. (We say this system is “parameterized”
by a, b and c. Your answers for x, y and z should be in terms of a, b and c.)
3. [15] Use “pencil and paper” with Gaussian elimination to solve the system,
2�! − 3�” + 7�# + 7�$ = 0
−2�! + �% + 6�” − 6�# − 12�$ = 0
�% − 3�” + �# + 5�$ = 0
−2�% + �& + �# + �$ = 0
2�! + �% − 3�” + 8�# + 7�$ = 0
4. [15] Find a polynomial of degree three (that is, a polynomial of the form �(�) =
� + �� + ��% + ��”) whose graph goes through the points
(0,1), (1,0), (−1,0), and (2, −15). Sketch a graph of this cubic and label the
given points (or use an app).
5. [15] “A duck costs four coins, five sparrows cost one coin, and a rooster costs one
coin. Somebody buys 100 birds for 100 coins. How many of each kind of bird can
he buy?”. Find all possible solutions. [Note that you cannot buy a fractional part
of a bird]  

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