Solutions vs Answers

for David L Ragozin's sections of Math 308, Spring 2004

• A solution to a problem should:
• Supply reasons or explanations or other notes to clarify the work presented.
• Demonstrate understanding of  the methods involved.
• Communicate what has been done and why:
• Any written solution, when read by you, other classmates, the instructor or grader,  will lead the reader to say `I understand how that was done', based on what is written.
• Numerical, or yes/no answers,  like the answers at the back of the book are not solutions.

Here is an example of a  typical linear algebra problem with an answer, a "professionals" solution and  then a solution in the style of this class.

Find numbers x,y which satisfy the equations:

E₁ :   x + 2y =  1
E₂ :  2x + 7y = -1

An answer :   x = 3 , y = -1.
A professionals solution : Gaussian elimination yields    x = 3 , y = -1.
A solution:

Subtract 2 * E₁  from E₂ to eliminate x, yielding

E₁ :    x + 2y =  1
E'₂ :                3y = -3

Divide E'₂  by 3 to solve for y:

E₁ :    x + 2y =  1
E''₂ :            y = -1

Subtract 2 * E''₂  from E₁ to eliminate y, yielding the solution

E₁   :    x    =  3
E'''₂ :                y = -1

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Note that the answer provides no reasons, explanations.  Although a reader who knows something about equations may be able to check the answer.