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Math Help - [SOLVED] Is this system consistant?

  1. #1
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    [SOLVED] Is this system consistant?

     <br />
x_{1}-x_{2}+3x_{3}+2x_{4}=4<br />
    <br />
2x_{1}-x_{2}+3x_{3}+x_{4}=6<br />
     <br />
3x_{1}-2x_{2}+6x_{3}+3x_{4}=15<br />

    When put into rref from i get
    <br />
\left(\begin{array}{cccc|c}1&0&0&-1&2\\0&1&-3&-3&-2\\0&0&0&0&5\\\hline\end{array}\right)

    Now this says 0 = 5, so presumably this system is inconsistant as 0 in my experiance doesn't equal 5.

    Is there any way to find out if the system is inconsistant before i do 5 or more elementary row operations, which could be vital time wasted in an exam?

    Thanks
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  2. #2
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    Quote Originally Posted by Rapid_W View Post
     <br />
x_{1}-x_{2}+3x_{3}+2x_{4}=4<br />
    <br />
2x_{1}-x_{2}+3x_{3}+x_{4}=6<br />
     <br />
3x_{1}-2x_{2}+6x_{3}+3x_{4}=15<br />

    When put into rref from i get
    <br />
\left(\begin{array}{cccc|c}1&0&0&-1&2\\0&1&-3&-3&-2\\0&0&0&0&5\\\hline\end{array}\right)

    Now this says 0 = 5, so presumably this system is inconsistant as 0 in my experiance doesn't equal 5.

    Is there any way to find out if the system is inconsistant before i do 5 or more elementary row operations, which could be vital time wasted in an exam?

    Thanks
    Assuming your calculations are correct, your statement in red is correct. No further work is required.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    Assuming your calculations are correct, your statement in red is correct. No further work is required.

    I used maple and only elementary row operations, so it is correct.

    But I was asking if it is possible to find out if the system is inconsistent before finding the rref form?
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  4. #4
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    Quote Originally Posted by Rapid_W View Post
     <br />
x_{1}-x_{2}+3x_{3}+2x_{4}=4<br />
    <br />
2x_{1}-x_{2}+3x_{3}+x_{4}=6<br />
     <br />
3x_{1}-2x_{2}+6x_{3}+3x_{4}=15<br />

    When put into rref from i get
    <br />
\left(\begin{array}{cccc|c}1&0&0&-1&2\\0&1&-3&-3&-2\\0&0&0&0&5\\\hline\end{array}\right)

    Now this says 0 = 5, so presumably this system is inconsistant as 0 in my experiance doesn't equal 5.

    Is there any way to find out if the system is inconsistant before i do 5 or more elementary row operations, which could be vital time wasted in an exam?

    Thanks
    Why did you have to do 5 elementary operations? After the very first one, when you bring the first column (below the entry 1-1) to zero, the second and third rows are identical EXCEPT for the right most entry, which in the 2nd row is -2 and in the 3rd row is 3.
    This already tells you the system is inconsistent since in the next step you will substract row 3 from row 2 and all the entries BUT the last one at the right in the 3rd row will vanish.

    This says that 0*x1 + 0*x2 + 0*x3 + 0*x4 = 5, which of course is nonsense (unless you're working with a field of characteristic 5. If you don't understand this remark erase it right now from your brain: it isn't important for you right now)
    Thus at most two steps, thought the result is visible already after the first one.

    Tonio
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