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Math Help - Simultaneous Equations with way too many unknowns..

  1. #1
    Junior Member Fnus's Avatar
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    Simultaneous Equations with way too many unknowns..

    Yea, so equations aren't my bestfriends, and I'm now stuck with two simultaneous equations, and I just don't know where to start, so if someone could help me and show me what to do, I'd appreciate it.

    'Find the values of m such that these equations have no solutions':

    3x - my = 4
    x + y = 12

    'Find the values of m and a such that these equations have infinite solution sets':

    4x - my = a
    2x + y = 4

    So if someone could patiently guide me through step by step, that'd be awesome.. I hate not knowing what I'm supposed to do.
    Thanks a lot
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Fnus View Post
    Yea, so equations aren't my bestfriends, and I'm now stuck with two simultaneous equations, and I just don't know where to start, so if someone could help me and show me what to do, I'd appreciate it.

    'Find the values of m such that these equations have no solutions':

    3x - my = 4
    x + y = 12

    'Find the values of m and a such that these equations have infinite solution sets':

    4x - my = a
    2x + y = 4

    So if someone could patiently guide me through step by step, that'd be awesome.. I hate not knowing what I'm supposed to do.
    Thanks a lot
    If the equations have an infinite solution set then basically the two equations say the same thing: that is to say, one equation is a multiple of the other.

    So since the coefficient of x in the first equation is twice the coefficient of x in the second equation, then "-m" must be twice the coefficient of y in the second equation, so
    -m = 2 \cdot 1
    or m = -2.

    Similarly a = 8.

    -Dan
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  3. #3
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    there are no solutions if both equations are lines with the same slope (parallel) and different y-intercepts. thus they will never cross.
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  4. #4
    Junior Member Fnus's Avatar
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    I'm really clueless about this, but it's two different questions:

    Question 1:
    'Find the values of m such that these equations have no solutions':

    3x - my = 4
    x + y = 12

    Question 2:
    'Find the values of m and a such that these equations have infinite solution sets':

    4x - my = a
    2x + y = 4

    Does that change anything? <.<
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  5. #5
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    x+y=12\Rightarrow y=-x+12 so the slope is -1.

    3x-my=4\Rightarrow y=\frac{3}{m}x-\frac{4}{m}

    now solve \frac{3}{m}=-1 you should get m=-3
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  6. #6
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    for the second one we have

    4x-my-a=0
    and
    2(2x+y)=2(4)\Rightarrow 4x+2y-8=0

    from here it should be obvious what values to make a and m.
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  7. #7
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    Hello, Fnus!

    Find the values of m such that these equations have no solutions:
    . . \begin{array}{cc}(1)\\(2)\end{array}<br />
\begin{array}{cc}3x - my \\ x + y \end{array}<br />
\begin{array}{cc} = \\ = \end{array}<br />
\begin{array}{cc}4 \\ 12\end{array}

    If you are familiar with determinants, there is a simple solution.

    A system has no solution if its determinant equal zero.

    The determinant is: . \begin{vmatrix} 3 & \text{-}m \\ 1 & 1\end{vmatrix} \:=\:(3)(1) - (\text{-}m)(1) \:=\:3 + m

    Therefore: . 3 + m \:=\;0\quad\Rightarrow\quad\boxed{m = -3}



    Otherwise, we can try to solve the system.

    We have equation (1): . 3x - my \:=\:4
    . . .Multiply (2) by m:\;\;mx + my \:=\:12m

    . . . . . . . . . . . . Add: . mx + 3x \:=\:12m + 4

    . . . . . . . . . . .Factor: . (m + 3)x \:=\:12m + 4

    . . . . . . . . . . . Then: . . . . . .  x\:=\:\frac{12m + 4}{m + 3}


    We see that x is undefined if m = -3.

    Therefore, the system has no solutions for m = -3.

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  8. #8
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by topsquark View Post
    If the equations have an infinite solution set then basically the two equations say the same thing: that is to say, one equation is a multiple of the other.

    So since the coefficient of x in the first equation is twice the coefficient of x in the second equation, then "-m" must be twice the coefficient of y in the second equation, so
    -m = 2 \cdot 1
    or m = -2.

    Similarly a = 8.

    -Dan
    Quote Originally Posted by putnam120 View Post
    there are no solutions if both equations are lines with the same slope (parallel) and different y-intercepts. thus they will never cross.
    But if one equation is merely a multiple of the other, they represent the same line, so any point (x,y) on the line is a solution. So we have an infinite number of solutions.

    -Dan
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