Directions: You may work to solve these problems in groups, but all written work must be your own. Show all work; no credit for solutions without work..

1. [1.1.38] Suppose \(a\), \(b\), \(c\), and \(d\) are constants such that \(a\) is nonzero and the system below is consistent for all possible values of \(f\) and \(g\). What can you say about the numbers \(a\), \(b\), \(c\), and \(d\)? Justify your answer. \[\begin {array}{*{5}{r}} ax_1 &+& bx_2 &=& f\\ cx_1 &+& dx_2 &=& g \end {array}\]

2. [1.2.{7-11}] Find the general solutions of the systems whose augmented matrices are given below.

   1. \(\left [ \begin {array}{*{4}{r}} 1 & 3 & 4 & 7\\ 3 & 9 & 7 & 6 \end {array}\right ]\)
   2. \(\left [ \begin {array}{*{4}{r}} 1 & 4 & 0 & 7\\ 2 & 7 & 0 & 11 \end {array}\right ]\)
   3. \(\left [ \begin {array}{*{4}{r}} 0 & 1 & -6 & 5\\ 1 & -2 & 7 & -4\\ \end {array}\right ]\)
   4. \(\left [ \begin {array}{*{4}{r}} 1 & -2 & -1 & 3\\ 3 & -6 & -2 & 2\\ \end {array}\right ]\)
   5. \(\left [ \begin {array}{*{4}{r}} 3 & -4 & 2 & 0\\ -9 & 12 & -6 & 0\\ -6 & 8 & -4 & 0 \end {array}\right ]\)
   6. \(\left [ \begin {array}{*{5}{r}} 1 & -7 & 0 & 6 & 5\\ 0 & 0 & 1 & -2 & -3\\ -1 & 7 & -4 & 2 & 7 \end {array}\right ]\)
3. Determine a cubic polynomial \(f(t) = a + bt + ct^2 + dt^3\) such that \(f(-1)=-1\), \(f(0)=0\), \(f(1)=1\), and \(f'(1)=0\). Hint: use the four given pieces of information about \(f\) to write four linear equations in the variables \(a\), \(b\), \(c\), and \(d\).