I have a system of linear equations that has no solutions.It looks like this in augmented matrix form. 1 − 1 2 5 2 1 0 1 1 8 − 1 3 − 1 − 5 − 12 41 I need to replace right hand value in equation 4 (41) with a value that will give me a unique solution when in reduced row echelon form. How do I approach this problem in a logical way to find the possible values?

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I have a system of linear equations that has no solutions.It looks like this in augmented matrix form.                    1                    −        1                    2                    5                            2                    1                    0                    1                            1                    8                    −        1                    3                            −        1                    −        5                    −        12                    41            I need to replace right hand value in equation 4 (41) with a value that will give me a unique solution when in reduced row echelon form. How do I approach this problem in a logical way to find the possible values?

Julien Carrillo · Accepted Answer

solving the first three equation we obtain      x    1    =      4    27    ,      x    2    =      19    27    ,      x    3    =      25    9  plugging this in the left-hand side of our System in the last equation we get  −      4    27    −  5  ⋅      19    27    −  12  ⋅      25    9    =  .  .  .Can you finish?

Camille Flynn · Accepted Answer

Replacing the value 41 with an arbitrary value x the augmented matrix would be:                    1                    −        1                    2                    5                            2                    1                    0                    1                            1                    8                    −        1                    3                            −        1                    −        5                    −        12                    x            Now reduce the matrix to its echelon form:                    1                    −        1                    2                    5                            0                    3                    −        4                    −        9                            0                    0                    9                    25                            0                    0                    0                    x        +        37            For this system to be consistent, the rank of the coefficient matrix should equal to rank of augmented matrix (the rank is the maximum number of non zero rows in echelon form, obviously), and furthermore, to have unique solution, the number of variables involved should also be equal to the rank of coefficient/augmented matrix.For this,   x  +  37  =  0 which implies   x  =  −  37

I have a system of linear equations that has no solutions. It looks like this in augmented matrix

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