title: Getting all possible Linear Combinations of a matrix R
body: <p>suppose I have the following augmented matrix</p>
  
  <pre><code>100 100 0   100
  325 0   500 500
  </code></pre>
  
  <p>In equation terms:</p>
  
  <pre><code>c1(100) + c2(100) + c3(0) = 100
  c1(325) + c2(0) + c3(500) = 500
  </code></pre>
  
  <p>I cannot change the final column.
  How do I find all possible linear combinations?
  The obvious solution is c2 = 1 and c3 = 1.
  But how do I get it in a systemic manner?</p>
  
  <p>I tried using rref in the pracma package but I get </p>
  
  <pre><code>1 0  1.538462   1.5384615
  0 1 -1.538462  -0.5384615
  </code></pre>
  
  <p>I really can't have any fraction values.</p>
  
  <p>Any advice?</p>