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Thread: simultaneous equations

  1. #1
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    simultaneous equations

    I am trying to solve the simultaneous equations by algebraic method, I know the solutions for "a" and "b" are 0.46 and -5.2, I have used a few different methods and the latest is close but not correct.

    Using elimination method I came up with;

    12a + 3b = - 10
    3a - 9b = 17

    36a + 9b = - 30

    9a - 9b = 51

    36 - ( - 9)a + (9 - 9)b = -30 + 51

    41a = 21

    a = 0.512

    3a - 3b = 17

    3(0.512) - 3b = 17

    6.15 - 3b = 17

    3b = 17 - 6.15

    b = -5.38

    12(0.512) + 3(-5.38) = -10

    3(0.512) - (-16.14) = 17.69

    As can be seen by the solutions they are not a million miles out, but somewhere I am making an error?
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  2. #2
    Super Member Quacky's Avatar
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    Re: simultaneous equations

    This is very promising work! With the first situation, you start off by correctly multiplying the first equation by three, which leads to the following equations:

    $\displaystyle 3a - 9b = 17$
    $\displaystyle 36a + 9b = - 30$

    You then get confused. You multiply the first equation by $\displaystyle 3$, which isn't necessary, and forget to multiply the $\displaystyle 9b$ by $\displaystyle 3$. Instead, if at this stage you simply add the equations, you get that $\displaystyle 39a=-13$. This leads to $\displaystyle a=\frac{-13}{39}=\frac{-1}{3}$, which is not the answer you've been presented with. Are you sure you've written the question correctly?
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  3. #3
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    Re: simultaneous equations

    Yes I have made an error as you say with the multiplication, but the equations (Original) are correct, I will try again?

    Tried again a different method but can't get closer to the right solution at the moment?
    Last edited by David Green; Jan 7th 2012 at 04:12 PM. Reason: back to the drawing board on this one?
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  4. #4
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    Re: simultaneous equations

    If the original equations are:

    $\displaystyle 12a + 3b = - 10$
    $\displaystyle 3a - 9b = 17$

    ...then the given answer is incorrect.
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  5. #5
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    Re: simultaneous equations

    Quote Originally Posted by David Green View Post
    I am trying to solve the simultaneous equations by algebraic method, I know the solutions for "a" and "b" are 0.46 and -5.2, I have used a few different methods and the latest is close but not correct.

    Using elimination method I came up with;

    12a + 3b = - 10
    12(0.46)+ 3(-5.2)= 5.52- 15.6= -10.08, not -10

    3a - 9b = 17
    3(0.46)- 9(-5.2)= 1.38+ 46.8= 48.18, not 17.

    The solutions you say you "know" are incorrect.

    36a + 9b = - 30

    9a - 9b = 51

    36 - ( - 9)a + (9 - 9)b = -30 + 51

    41a = 21

    a = 0.512

    3a - 3b = 17

    3(0.512) - 3b = 17

    6.15 - 3b = 17

    3b = 17 - 6.15

    b = -5.38

    12(0.512) + 3(-5.38) = -10

    3(0.512) - (-16.14) = 17.69

    As can be seen by the solutions they are not a million miles out, but somewhere I am making an error?
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  6. #6
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    Re: simultaneous equations

    OK I have had another attemp, so far I can only get one of the two equations to work?

    The previous thread points out that the solutions I have are incorrect, these solutions were calculated using a software program for simutaneous equations, therefore if they are indeed incorrect, then the software or a mathematical technique (Idea) is different to solving these equations?

    12a + 3b = -10
    3a - 9b = 17

    I have multiplied out the first equation;

    36a + 9b = -30

    next I have subtracted the second equation;

    36a + 9b = -30
    3a - 9b = 17

    a = 1.42

    Using equation (1) I have worked out the value for b;

    12(1.42) + 3b = -10
    17 + 3b = -10
    3b = -10 - 17

    b = -9

    substitute this value into equation (1)

    12(1.42) + 3(-9) = 17 - 27 = -10

    Up to here the values for a and b work, however I am either doing something wrong, or there is another mathematical method for solving these equations that I am not aware of, the values will not work for the second equation, and I am struggling to find another method?

    any help now would be much appreciated.

    P.S.

    I have read in a Algebra book that the laws of basic arithmetic + and - do not equal - ?

    Is this true or can I take it that the book has a typo error?

    Thanks
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  7. #7
    Super Member Quacky's Avatar
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    Re: simultaneous equations

    Remember that as I showed in post#2, your given answer is wrong. You are correct up to:

    $\displaystyle 36a + 9b = -30$
    $\displaystyle 3a - 9b = 17$

    But if you subtract the equations, you get this:

    $\displaystyle 36a+9b-(3a-9b)=-30-17$
    $\displaystyle 33a+18b=-47$ ...which doesn't help us.

    Rule: If the terms you are trying to eliminate have the same sign, subtract the equations. If the terms you are trying to eliminate have opposing signs, add the equations.

    Here, the $\displaystyle 9b$ is positive in $\displaystyle 36a + 9b = -30$ but it's negative in $\displaystyle 3a - 9b = 17$. This means, using previously stated rule, we add the equations.
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  8. #8
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    Re: simultaneous equations

    following on from what you said in the previous thread, the equations;

    12a + 3b = -10
    3a - 9b = 17

    are added.

    36a + 9b = -30
    3a - 9b = 17

    This is where the confusion starts;

    -30 + 17 = -13
    + 9 - 9 = 0
    36 + 3 = 39

    a = -0.33

    b = -1.99

    = -21.87, which is of course completely wrong.

    The arithmetic I think is were I am getting this wrong.

    I was always told that the following rules to arithmetic applied;

    + and + = +
    - and - = +
    + and - = -

    here in the equations I am now unsure, no matter which way I try to solve this I get one answer correct and one incorrect as follows;

    12a + 3b = -30
    3a - 9b = 17

    36a + 9b = -30
    3a - 9b = 17

    33a = -47

    a = -1.42

    12(-1.42) + 3b = -10

    -17 + 3b = -10

    3b = -10 + 17

    b = 2.33

    12(-1.42) + 3(2.33) = -17 + 6.99 = -10

    The next equation gives a solution completely wrong at -16.71 using those same values, so at the moment I am either way off track or these equations cannot be solved using integers?
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  9. #9
    Super Member Quacky's Avatar
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    Re: simultaneous equations

    When you said that

    $\displaystyle a = -0.33$
    $\displaystyle b = -1.99$, you had the correct solution but for a rounding error. Remember where appropriate to leave your answers in an exact form. $\displaystyle a=\frac{-1}{3}$, which gives $\displaystyle b=-2$, which is the correct solution when we back substitute:

    $\displaystyle 12(\frac{-1}{3}) + 3(-2)$

    $\displaystyle =\frac{-12}{3}-6$

    $\displaystyle =-4-6$

    $\displaystyle =-10$, as expected.
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