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Math Help - Simultaneous Equation with 4 Variables

  1. #1
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    Simultaneous Equation with 4 Variables

    The system of equations x + y + z + w = 4, x + 3y + 3z = 2, x + y + 2z - w = 6 has infinitely many solutions. Describe this family of solutions (in terms of the parameter t) and give the unique solutions when w = 6.

    I got as far as w = (t - 2)/2, but can't seem to go any further.

    Thanks in advance.
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  2. #2
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    Re: Simultaneous Equation with 4 Variables

    Quote Originally Posted by Fratricide View Post
    The system of equations x + y + z + w = 4, x + 3y + 3z = 2, x + y + 2z - w = 6 has infinitely many solutions. Describe this family of solutions (in terms of the parameter t) and give the unique solutions when w = 6.

    I got as far as w = (t - 2)/2, but can't seem to go any further.

    Thanks in advance.
    could use a bit more of your work here. There's nothing magic you just have to hack away at it getting say z in terms of y and x from the 2nd equation. Then you can use that to reduce the first or third, and then similar with the last.

    That will give you everything in terms of x.

    Now just replace x with t.

    For the last bit you'll have to solve for t when w=6 and then substitute that t in for the expressions for y and z. x of course is just t.

    I'm not seeing w=(t-2)/2 from my hack at this so maybe check your work.

    Post back some of your cut at this if you like and we can check for any mistakes.
    Thanks from Fratricide and topsquark
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    Re: Simultaneous Equation with 4 Variables

    I had another crack at it this morning, but still didn't get any further. I would try substituting w = (t-2)/2 and z = t into the first equation and solving for x and y, but when I substituted those x and y values into the other equations I would always end up with "x - x" or "y - y":

    x + y + t + (t-2)/2 = 4
    2x + 2y + 2t + t - 2 = 4
    2x + 2y + 3t = 6
    2x = 6 - 3t - 2y
    x = 3 - (3t/2) - y

    Substituting back into equation 1 I get:

    3 - (3t/2) - y + y + z + w = 4

    From there I could only solve for z or w, both of which I already have the values for. I also tried substituting that value for x into the second equation, but turned up nothing.

    For our reference, the answers are:

    z = t
    y = (-3(t + 2))/4
    x = (26 - 3t)/4
    w = (t - 2)/2

    And for the second part of the question:

    x = -4, y = -12, z = 14
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    Re: Simultaneous Equation with 4 Variables

    Quote Originally Posted by Fratricide View Post
    I had another crack at it this morning, but still didn't get any further. I would try substituting w = (t-2)/2 and z = t into the first equation and solving for x and y, but when I substituted those x and y values into the other equations I would always end up with "x - x" or "y - y":

    x + y + t + (t-2)/2 = 4
    2x + 2y + 2t + t - 2 = 4
    2x + 2y + 3t = 6
    2x = 6 - 3t - 2y
    x = 3 - (3t/2) - y

    Substituting back into equation 1 I get:

    3 - (3t/2) - y + y + z + w = 4

    From there I could only solve for z or w, both of which I already have the values for. I also tried substituting that value for x into the second equation, but turned up nothing.

    For our reference, the answers are:

    z = t
    y = (-3(t + 2))/4
    x = (26 - 3t)/4
    w = (t - 2)/2

    And for the second part of the question:

    x = -4, y = -12, z = 14
    Well let's see. We want to just bang away at it keeping things in terms of z since that's going to become our independent variable.

    x + y + z + w = 4
    x +3y+3z = 2
    x + y +2z - w = 6

    taking the 2nd equation

    x = 2 - 3y - 3z

    plug this into equation 1

    (2 - 3y - 3z) + y + z + w = 4

    y(-3+1) + z(-3+1) + 2 + w = 4

    -2y - 2z + w = 2

    w = 2 +2y + 2z

    plug both into equation 3

    (2-3y-3z) + y + 2z - (2+2y+2z) = 6

    (2-2) + y(-3 + 1 + -2) + z(-3 + 2 -2) = 6

    -4y -3z = 6

    y = -(6+3z)/4

    and collecting everything and letting z=t we get

    y = -(6+3t)/4

    x = 2 - 3y - 3z = 2 -3(6+3t)/4 - 3t = (26-3t)/4

    w = 2 + 2y + 2z = 2 -2(6+3t)/4 +2t = (t-2)/2

    {x, y, z, t} = {(26-3t)/4, -(6+3t)/4, t, (t-2)/2}

    I think you just need to be careful collecting terms.
    Thanks from Fratricide
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    Re: Simultaneous Equation with 4 Variables

    Quote Originally Posted by Fratricide View Post
    x + y + z + w = 4
    x + 3y + 3z = 2
    x + y + 2z - w = 6
    give the unique solutions when w = 6.
    After substituting 6 for w:
    x + y + z = -2 [1]
    x + 3y + 3z = 2 [2]
    x + y + 2z = 12 [3]

    Subtract [1] from [3]:
    x - x + y - y + 2z - z = 12 - (-2)
    z = 14

    Now you're almost done...
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