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How to Solve It (1945) is a small volume by mathematician George PÃ³lya describing methods of problem solving.
Four principles
How to Solve It suggests the following steps when solving a mathematical problem:
 First, you have to understand the problem.
 After understanding, then make a plan.
 Carry out the plan.
 Look backon your work.How could it be better?
If this technique fails, PÃ³lya advises: "If you can't solve a problem, then there is an easier problem you can solve: find it." Or: "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"
First principle: Understand the problem
"Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. In order to remedy this oversight, PÃ³lya taught teachers how to prompt each student with appropriate questions, depending on the situation, such as:
 What are you asked to find or show?
 Can you restate the problem in your own words?
 Can you think of a picture or a diagram that might help you understand the problem?
 Is there enough information to enable you to find a solution?
 Do you understand all the words used in stating the problem?
 Do you need to ask a question to get the answer?
The teacher is to select the question with the appropriate level of difficulty for each student to ascertain if each student understands at their own level, moving up or down the list to prompt each student, until each one can respond with something constructive.
Second principle: Devise a plan
PÃ³lya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:
 Guess and check
 Make an orderly list
 Eliminate possibilities
 Use symmetry
 Consider special cases
 Use direct reasoning
 Solve an equation
Also suggested:
 Look for a pattern
 Draw a picture
 Solve a simpler problem
 Use a model
 Work backward
 Use a formula
 Be creative
 Use your head/noggin
Third principle: Carry out the plan
This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don't be misled; this is how mathematics is done, even by professionals.
Fourth principle: Review/extend
PÃ³lya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn't. Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.
The book contains a dictionarystyle set of heuristics, many of which have to do with generating a more accessible problem. For example:
The technique "have I used everything" is perhaps most applicable to formal educational examinations (e.g., n men digging m ditches) problems.
The book has achieved "classic" status because of its considerable influence (see the next section).
Other books on problem solving are often related to more creative and less concrete techniques. See lateral thinking, mind mapping, brainstorming, and creative problem solving.
Influence
 It has been translated into several languages and has sold over a million copies, and has been continuously in print since its first publication.
 Marvin Minsky said in his influential paper Steps Toward Artificial Intelligence that "everyone should know the work of George PÃ³lya on how to solve problems."
 PÃ³lya's book has had a large influence on mathematics textbooks as evidenced by the bibliographies for mathematics education.
 Russian physicistZhores I. Alfyorov, (Nobel laureate in 2000) praised it, saying he was very pleased with PÃ³lya's famous book.
 Russian inventor Genrich Altshuller developed an elaborate set of methods for problem solving known as TRIZ, which in many aspects reproduces or parallels PÃ³lya's work.
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Answers:S+O=29 S*ws+O*wo = 252 S*ps +O*po=30,000 Ican only write 3 equations for 6 unknowns so I'd better hope the historian's information is inconsistent S+O=29 8S+12O = 252 1000S+2000O=30,000 Quickly solving the first two S=24 O =5 On solving the second 2 S=36 O=3 On solving 1 and 3 and ignoring 2 S=28 O=1 Obviously inconsistent (2) Same idea 30*dE+20*dF+20*dS =340 10*(dE+dF+dS)=140 20*dE+30*dF+20*dS=320 3 equations 3 unknowns, there is a unique solution Write as matrix 30 20 20 deE 340 10 10 10 dF  = 140 20 30 20 dS  320 Solve dE =6 dF =4 dS =4 Hope that helps
Answers:I worked out your problem using a math program. The result is a PDF file stored here... https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B93wLOqLZwn_NmI1NGY4OWUtYjNlNi00NzJiLTg2OWMtNTQxYmU5NmJkOGQx&hl=en&authkey=CIXQybgL Hope this helps
Answers:HINT: Write what you know. Let h = Hilary's money Let p = Paul's money Let j = Jeff's money ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ METHOD: Elimination/Substitution ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Given: Hilary has three times as much money as Paul. Means: h = 3p Given: Jeff has $4 less than Hilary Means: j = h  4 Given: Jeff has $5 more than Paul. Means: j = p + 5 You have 3 equations. h = 3p j = h  4 j = p + 5 Substitute h with 3p in the 2nd equation. j = h  4 j = 3p  4 Look at this equation and the 3rd equation. j = 3p  4 j = p + 5 Multiply the 2nd of these equations by 1. 1(j) = 1(p + 5) j = 1(p)  1(5) j = p  5 Add this to the other equation. j = 3p  4 j = p  5  0 = 2p  9 9 = 2p 9 / 2 = p 4.5 = p Plug this into h = 3p to find h. h = 3p h = 3(4.5) h = 13.5 Plug this into j = h  4 to find j. j = h  4 j = 13.5  4 j = 9.5 ANSWER: Hilary has $13.50, Paul has $4.50, and Jeff has $9.50. CHECK: h = 3p 13.5 = 3(4.50)? 13.5 = 13.5? TRUE j = h  4 9.5 = 13.5  4? 9.5 = 9.5? TRUE j = p + 5 9.5 = 4.5 + 5? 9.5 = 9.5? TRUE ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ METHOD: Cramer's rule & matrices ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ h = 3p ==> h  3p + 0j = 0 j = h  4 ==> h + 0p  j = 4 j = p + 5 ==> 0h  p + j = 5 The matrix for D is: ......[1 3 0] D = [1 0 1] ......[0 1 1] Repeat the first two columns and evaluate for D. ......[1 3 0 1 3] D = [1 0 1 1 0] ......[0 1 1 0 1] D = (1)(0)(1) + (3)(1)(0) + (0)(1)(1)  (0)(0)(0)  (1)(1)(1)  (1)(1)(3) = 0 + 0 + 0  0  1  3 = 2 The matrix for Dh is: ........[0 3 0] Dh = [4 0 1] ........[5 1 1] Repeat the first two columns and evaluate for Dx. ........[0 3 0 0 3] Dh = [4 0 1 4 0] ........[5 1 1 5 1] Dh = (0)(0)(1) + (3)(1)(5) + (0)(4)(1)  (5)(0)(0)  (1)(1)(0)  (1)(4)(3) = 0 + 15 + 0  0  0  12 = 27 The matrix for Dp is: ........[1 0 0] Dp = [1 4 1] ........[0 5 1] Repeat the first two columns anDy evaluate for Dy. ........[1 0 0 1 0] Dp = [1 4 1 1 4] ........[0 5 1 0 5] Dp = (1)(4)(1) + (0)(1)(0) + (0)(1)(5)  (0)(4)(0)  (5)(1)(1)  (1)(1)(0) = 4 + 0 + 0  0  5  0 = 9 The matrix for Dj is: ........[1 3 0] Dj = [1 0 4] ........[0 1 5] Repeat the first two columns anDz evaluate for Dz. ........[1 3 0 1 3] Dj = [1 0 4 1 0] ........[0 1 5 0 1] Dj = (1)(0)(5) + (3)(4)(0) + (0)(1)(1)  (0)(0)(0)  (1)(4)(1)  (5)(1)(3) = 0 + 0 + 0  0  4  15 = 19 Evalute for h, p, and j. h = Dh / D = 27 / 2 = 13.5 p = Dp / D = 9 / 2 = 4.5 j = Dj / D = 19 / 2 = 9.5 (13.5, 4.5, 9.5) ANSWER: Hilary has $13.50, Paul has $4.50, and Jeff has $9.50.
Answers:No cause what you have given is an impossible situation. if x is a 1x2 matrix, its transpose is a 2x1 matrix... you cant multiply that by a 2x2 matrix.... ima assume you meant its a 2x1 matrix. This takes a long time so im just gunna tell you how to do it. (x y) {a1 a2 ; a3 a4} {x ;y} = (3x^22xy+y^2) a ; indicates a new line. Multiply it out and solve for all the a's
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