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Math Help - Linear system, solutions and their geometric interpretation.

  1. #1
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    Linear system, solutions and their geometric interpretation.

    Solve the following system of equations and give a geometrical interpretation of the result.
    x + y + z = 6
    2x + y − 3z = -5
    4x − 5y + z = −3
    Give a geometrical interpretation of the intersection of the planes with equations
    x + y − 3 = 0
    y + z + 5 = 0
    x + z + 2 = 0
    Determine a scalar equation for the plane that passes through the point (2, 0, −1) and is perpendicular to the line of intersection of the planes
    2x + y z + 5 = 0 and x + y + 2z + 7 = 0.
    Last edited by mr fantastic; July 29th 2010 at 04:15 PM.
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  2. #2
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    Quote Originally Posted by ilovemymath View Post
    Solve the following system of equations and give a geometrical interpretation of the result.
    x + y + z = 6
    2x + y − 3z = -5
    4x − 5y + z = −3
    What methods have you tried?
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  3. #3
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    Quote Originally Posted by pickslides View Post
    What methods have you tried?
    i havent tried anythign because i m taking online course and where my teacher doesnt teach my anything so though i would get my answer here....please help
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  4. #4
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    Has your teacher mentioned what method is preferred? I.e. using substitutions or matrices?
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  5. #5
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    substitutions
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  6. #6
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    Here's the system.

    x + y + z = 6 ...(1)
    2x + y − 3z = -5 ...(2)
    4x - 5y + z = -3 ...(3)

    Now looking at (1)

    x + y + z = 6

    z   = 6-y - x

    Now we substitute where we see z into both (2) and (3) giving

    2x + y - 3(6-y - x) = -5 ...(2)
    4x - 5y + 6-y - x = -3 ...(3)

    Neatening this up gives

    2x + y - 18+3y +3 x = -5 ...(2)
    4x - 5y + 6-y - x = -3 ...(3)

    Grouping like terms

    5x + 4y  = 13 ...(2)
    3x - 6y   = -9 ...(3)

    Do you follow? Can you use this method again on (2) and (3) to finish it?
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  7. #7
    Senior Member yeKciM's Avatar
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    isn't it easier to do determinant of matrix ? so u can see how many there are (if any) solutions of the system and so it's much easier to understand the geometrical interpretation

    how do the planes correspond between them, how do they (if they are) intersects, is it in one point or is it along the line or... and so on
    Last edited by yeKciM; July 31st 2010 at 01:22 AM.
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  8. #8
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    I agree, but the OP may not have matrices in their skill set.
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