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Math Help - Equations with no solutions

  1. #1
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    Equations with no solutions

    Hey,
    Can someone please tell me how to know if an equation has no solution.....

    cuz the question goes like:
    3x- 4y = 10
    6x+wy = 16
    For which of the following values of w will the system of equations above have no solutions?
    A)-8
    B)-4
    C)0
    D)4
    E)8

    What should I do? Any attempt to help me would be very much appreciated (-:
    Thanks
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  2. #2
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    Re: Equations with no solutions

    The last one, if you plot the two equations on a graph you will see that the lines are parrellel, ie they cannot cross.

    solve 3x+4y=10, 6x +8y = 16 - Wolfram|Alpha
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  3. #3
    Senior Member BAdhi's Avatar
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    Re: Equations with no solutions

    the system won't have a solution when two different values are equal to a same statement. For example

    5x+6y=10
    10x+12y=8

    now, take the second equation. when you divide it by 2, the left hand side will be same as in the first equation, but the right hand side is different from the first. eventually when we divide the second equation by 2 and subtract it with the first equation you'll get 0=6 which is not true. That says the system does not have a solution
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  4. #4
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    Re: Equations with no solutions

    Quote Originally Posted by IBstudent View Post
    Hey,
    Can someone please tell me how to know if an equation has no solution.....

    cuz the question goes like:
    3x- 4y = 10
    6x+wy = 16
    For which of the following values of w will the system of equations above have no solutions?
    A)-8
    B)-4
    C)0
    D)4
    E)8

    What should I do? Any attempt to help me would be very much appreciated (-:
    Thanks
    You should think through what you get in each case.

    For (A)
    3x-4y=10
    6x-8y=16

    The second equation is 2(3x-4y)=2(8)
    so 3x-4y=8

    and if this is the case, then the first equation becomes 8=10.

    For (B) we get
    3x-4y=10
    6x-4y=16 so 3x+3x-4y=16
    The first equation gives us 3x+10=16
    which is ok.

    For (C)
    3x-4y=10
    6x=16 so 3x=8
    leaving us with 8-4y=10
    which is also ok.

    For (D)
    3x-4y=10
    6x+4y=16
    Adding these gives 9x=26
    which is fine.

    For (E)
    3x-4y=10
    6x+8y=16 so 2(3x+4y)=2(8)
    giving
    3x-4y=10
    3x+4y=8
    Adding gives 6x=18, fine again.
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  5. #5
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    Re: Equations with no solutions

    Thanks!
    I got it now :-)
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  6. #6
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    Re: Equations with no solutions

    From there, you may answer the question much faster
    by thinking in terms of the graphs.

    3x-4y=10
    6x+wy=16

    are "linear" equations (straight line graphs).
    If the slopes of these lines differ,
    then they intersect at a single (x,y) point.
    If the lines intersect then the equations have a "solution",
    a specific x and specific y that obey both equations.

    Hence, if w=-8, the lines have the same slope.
    The equations would both represent the same line if we had 6x-8y=20
    for the 2nd equation (instead of 6x-8y=16).
    This is because 3x-4y=10 is 2(3x-4y)=2(10).

    Rearranging the equations into y=mx+c form gives

    3x-10=4y\Rightarrow\ y=\frac{3}{4}x-2.5

    6x+wy=16\Rightarrow\ y=-\frac{6}{w}x+\frac{16}{w}

    When x=0, the constant gives the y-axis crossing point.
    The slopes are the same only if w=-8
    and this value of w gives different constants,
    so the lines are parallel.

    If this is clear, then you don't even need to rearrange the equations into y=mx+c form
    to know whether or not they intersect.

    ax+by=c

    ax+by=d

    are parallel and have no point of intersection for c\ne\ d

    ax+by=c

    kax+kby=kc

    are the same line

    Equations with one of "a" or "b" different,
    when written in simplified form,
    are not parallel and intersect and so have a solution.

    Then by inspection, only (A) has no solution.
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