Determine whether thefollowing system has a unique solution, no solution, orinde

Question

Determine whether thefollowing system has a unique solution, no solution, orindefinitely many solutions. If a solution exists, write itdown
x1 +2x3 - x4 =5         2x1 +x2 + 3x2 =7        5x1+x2+9x3-3x4 =12 Can somebody walk me through the solution to thisproblem? Determine whether thefollowing system has a unique solution, no solution, orindefinitely many solutions. If a solution exists, write itdown
x1 +2x3 - x4 =5         2x1 +x2 + 3x2 =7        5x1+x2+9x3-3x4 =12 Can somebody walk me through the solution to thisproblem?

Explanation / Answer

It looks like you might have a typo in the second equation. Two of the variables are the same. I will assume that the3x2 in the second equation was actually a3x3, and what you meant was that it was this set: r1       x1   +   2x3 - x4 = 5 r2        2x1 +x2 + 3x3 = 7 r3        5x1 +x2 + 9x3 - 3x4 = 12 We will refer to the three rows as row1, row2, row3. Threeequations, four unknowns. Remember the rules - we can addrows together, or we can multiply a constant through one entireequation. We can also swap rows. x1 + 2x3 -x4   = 5 2x1 + x2 + 3x3       =7              Let's add (-2 times the first row) to this row. 5x1 + x2 + 9x3 - 3x4 =12           And we'll add (-5 times the first row) to this row. x1 +         2x3- x4 = 5 0x1 + x2 - 1x3+2x4 = 3 0x1 + x2 - 1x3 +2x4 =-13          Right here we can see something interesting. Let's look atrows 2 and 3. x2 - 1x3 + 2x4 = 3 x2 - 1x3 + 2x4 =-13          These are "equations". Which means that the left side shouldequal the right side. We can see that the left sides of thesetwo equations are the same. Which means that the right sidesshould be the same - but they're not. Therefore, this systemhas no solutions (because they can't be combined to makesense). We could take it a step further, by adding (-1 times row 2 to row3): x2 - 1x3 + 2x4 = 3                   0 = -16 This obviously doesn't make sense. But remember, this was based on the assumption that there was atypo in the original problem. If that wasn't the solution,this will at least give you an idea of how to manipulate the rowsto get what you want.

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