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Unformatted text preview: Chapter 11 Linear Equations The most important task in technical computing. I am thinking of two numbers. Their average is 3. What are the numbers? Please remember the first thing that pops into your head. I will get back to this problem in a few pages. Solving systems of simultaneous linear equations is the most important task in technical computing. It is not only important in its own right, it is also a fun- damental, and often hidden, component of other more complicated computational tasks. The very simplest linear equations involve only one unknown. Solve 7 x = 21 The answer, of course, is x = 21 7 = 3 Now solve ax = b The answer, of course, is x = b a But what if a = 0? Then we have to look at b . If b 6 = 0 then there is no value of x that satisfies x = b Copyright c 2009 Cleve Moler Matlab R is a registered trademark of The MathWorks, Inc. TM August 8, 2009 1 2 Chapter 11. Linear Equations The solution does not exist. On the other hand, if b = 0 then any value of x satisfies x = 0 The solution is not unique. Mathematicians have been thinking about existence and uniqueness for centuries. We will see that these concepts are also important in modern technical computing. Here is a toy story problem. Alice buys three apples, a dozen bananas, and one cantaloupe for $2.36. Bob buys a dozen apples and two cantaloupes for $5.26. Carol buys two bananas and three cantaloupes for $2.77. How much do single pieces of each fruit cost? Let x 1 ,x 2 , and x 3 denote the unknown price of each fruit. We have three equations in three unknowns....
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