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You can put this solution on YOUR website!
\n" ); document.write( "It is rather pointless to have someone do this for you, because (1) you don't learn how to do it yourself; and (2) 100 different people will use probably at least 90 different paths to get the answer.
\n" ); document.write( "Nevertheless, I will go through one POSSIBLE sequence of steps for solving this example, with explanations of why I chose that path. If you want to learn anything about how to do this, you should then solve the problem using a different path.
\n" ); document.write( "The initial matrix is this:
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\n" ); document.write( "The matrix we want to end up with is this:
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\n" ); document.write( "The matrix in that form will tell us that the solution to the system is x=A, y=B, and z=C.
\n" ); document.write( "At each step, there are three things you can do:
\n" ); document.write( "(a) switch the order of the rows;
\n" ); document.write( "(b) multiply a row by a nonzero constant;
\n" ); document.write( "(c) add a multiple of one row to another row
\n" ); document.write( "In terms of the objectives in choosing a path to the answer, the sequence of steps you want to accomplish is this:
\n" ); document.write( "(1) get a 1 in position (1,1)
\n" ); document.write( "(2) use the 1 in position (1,1) to get 0's in the rest of column 1
\n" ); document.write( "(3) get a 1 in position (2,2)
\n" ); document.write( "(4) use the 1 in position (2,2) to get 0's in the rest of column 2
\n" ); document.write( "(5) get a 1 in position (3,3)
\n" ); document.write( "Following, then, is ONE POSSIBLE path to the solution.
\n" ); document.write( "As a preliminary step, seeing that the numbers in row 3 are all multiples of 2, I would use (b) to make the numbers in row 3 smaller:
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\n" ); document.write( "Step (1): getting a 1 in position (1,1)
\n" ); document.write( "I could of course use (b) to multiply row 1 by 1/2 -- but that would introduce fractions, which will surely complicate the process. Since rows 2 and 3 both have 1's in column 1, I can use (a) to accomplish step (1). I can rearrange the rows in any order I want; I chose this:
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\n" ); document.write( "Step (2): getting 0's in positions (2,1) and (3,1)
\n" ); document.write( "I use (c) with the 1 in (1,1) to do this: multiply row 1 by -1 and add to row 2 to get the 0 in (2,1); multiply row 1 by -2 and add to row 3 to get the 0 in (3,1).
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\n" ); document.write( "Step (3): getting a 1 in position (2,2)
\n" ); document.write( "By the path I have chosen, I already have that....
\n" ); document.write( "Step (4): getting 0's in positions (1,2) and (3,2)
\n" ); document.write( "Again use (c) with the 1 in position (2,2) to do this: multiply row 2 by -1 and add to row 1 to get the 0 in position (1,2); multiply row 2 by 1 and add to row 3 to get the 0 in position (3,2).
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\n" ); document.write( "In this problem, we can't do step (5) -- getting a 1 in position (3,3) -- because row 3 is all 0's.
\n" ); document.write( "That row of all 0's means there is no unique solution to the system of equations; there are infinitely many solutions.
\n" ); document.write( "We have done all we can do to the matrix. Translating the rows of the matrix into equations, the matrix in the final form tells us this:
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\n" ); document.write( "We use this to set up a parametric description of the infinite number of solutions to the system:
\n" ); document.write( "z=t
\n" ); document.write( "x+z=4 --> x = 4-z --> x = 4-t
\n" ); document.write( "y+z=-4 --> y = -4-z --> y = -4-t
\n" ); document.write( "The parametric solution to the system of equations is (x,y,z) = (4-t, -4,-t,t)
\n" ); document.write( "Substitute any number for t to get a solution to the system of equations:
\n" ); document.write( "t=0 --> (4,-4,0)
\n" ); document.write( "t=1 --> (3,-5,1)
\n" ); document.write( "t=-3 --> (7,-1,-3)
\n" ); document.write( "etc...
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