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    djslilangel Posts: 1, Reputation: 1
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    #1

    Jan 30, 2007, 07:45 PM
    Word Problem for 3 variable system of equations
    Solve the system to find how many athletes finished in each place.

    Lawrence High prevailed in Saturdays track meet with the help of 20 individual-event placers earning a combined 68 points. A first place finish earns 5 points, a second place earns 3 points, and third place earns 1 point. Lawrence had a strong second-place showing, with as many second-place finshes as first and third place finishers combined.
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    #2

    Jan 31, 2007, 12:01 AM
    Quote Originally Posted by djslilangel
    Solve the system to find how many athletes finished in each place.

    Lawrence High prevailed in Saturdays track meet with the help of 20 individual-event placers earning a combined 68 points. A first place finish earns 5 points, a second place earns 3 points, and third place earns 1 point. Lawrence had a strong second-place showing, with as many second-place finshes as first and third place finishers combined.

    Step One - Translate word problem into formula(e):

    First one is :

    Let:
    a=First place
    b=Second
    c=Third

    Then:
    68=5a+3b+c

    Your turn for the other two, and I'll follow up with you then. I'll need to know whether we'll be using Cramer's rule or substituting and simplifying.

    Fianchetto
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    #3

    Feb 1, 2007, 04:10 AM
    Quote Originally Posted by djslilangel
    Solve the system to find how many athletes finished in each place.

    Lawrence High prevailed in Saturdays track meet with the help of 20 individual-event placers earning a combined 68 points. A first place finish earns 5 points, a second place earns 3 points, and third place earns 1 point. Lawrence had a strong second-place showing, with as many second-place finshes as first and third place finishers combined.



    Solve the system to find out how many athletes finished in each place.

    Lawrence H.S. prevailed in Saturday's track meet with the help of 20 individual-event placers earning a combined 68 points. A first place finish earns 5 points, a second place earns 3 points and a third place earns 1 point. Lawrence had a strong second-place showing, with as many second-place finishes as first and third place finishers combined.

    First write your equations:

    Let:
    a = First place finishes
    b = Second
    c = Third
    Then:

    A. 68 = 5a + 3b + c
    B. 20 = a + b + c
    C. b = a + c
    -----------------------------------------------------------

    Substitute and Simplify

    Since "C." is true, let?s plug it in for b in "A.":

    68 = 5a + 3 (a + c) + c
    68 = 5a + 3a + 3c + c
    68 = 8a + 4c
    17 = 2a + c

    Now let's do the same for b in "B."

    20 = a + (a + c) + c
    20 = 2a + 2c
    10 = a + c

    Looking back at "C.":

    b=10

    From earlier,
    Since:

    10 = a + c
    c = 10 - a

    And:

    17 = 2a + c

    Substituting again, we get:

    17 = 2a + c
    17 = 2a + (10 - a)
    17 = a + 10
    a = 7

    Last substitution yields:

    b = a + c
    10 = 7 + c
    c = 3

    So:

    a = 7
    b = 10
    c = 3
    -----------------------------------------------------------

    Cramer's Rule


    First write your equations:

    Let:
    a = First place finishes
    b = Second
    c = Third
    Then:

    A. 68 = 5a + 3b + c
    B. 20 = a + b + c
    C. b = a + c
    -----------------------------------------------------------






    | 5 3 1 |
    D = | 1 1 1 | = 5 | 1 1 | - 3 | 1 1 | + 1 | 1 1 |
    | 1 -1 1 | |-1 1 | | 1 1 | | 1 -1 |




    = 5(-1) - 3(1) + 1(-1)
    = -5 - 3 - 1
    = -9


    Next, replace each column with the "solutions" of your equations, and , one at a time, as above, lather, rinse, repeat. You will get Dx, Dy, and Dz.


    x = Dx/D; y = Dy/D; and z = Dz/D



    (for a better picture/description, see www.purplemath.com - type cramer's in the site search engine)
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    #4

    Feb 2, 2007, 03:30 PM
    Quote Originally Posted by djslilangel
    Solve the system to find how many athletes finished in each place.

    Lawrence High prevailed in Saturdays track meet with the help of 20 individual-event placers earning a combined 68 points. A first place finish earns 5 points, a second place earns 3 points, and third place earns 1 point. Lawrence had a strong second-place showing, with as many second-place finshes as first and third place finishers combined.


    Solve the system to find out how many athletes finished in each place.

    Lawrence H.S. prevailed in Saturday's track meet with the help of 20 individual-event placers earning a combined 68 points. A first place finish earns 5 points, a second place earns 3 points and a third place earns 1 point. Lawrence had a strong second-place showing, with as many second-place finishes as first and third place finishers combined.

    First write your equations:

    Let:
    a = First place finishes
    b = Second
    c = Third
    Then:

    A. 68 = 5a + 3b + c
    B. 20 = a + b + c
    C. b = a + c
    -----------------------------------------------------------

    Substitute and Simplify

    Since "C." is true, let?s plug it in for b in "A.":

    68 = 5a + 3 (a + c) + c
    68 = 5a + 3a + 3c + c
    68 = 8a + 4c
    17 = 2a + c

    Now let's do the same for b in "B."

    20 = a + (a + c) + c
    20 = 2a + 2c
    10 = a + c

    Looking back at "C.":

    b=10

    From earlier,
    Since:

    10 = a + c
    c = 10 - a

    And:

    17 = 2a + c

    Substituting again, we get:

    17 = 2a + c
    17 = 2a + (10 - a)
    17 = a + 10
    a = 7

    Last substitution yields:

    b = a + c
    10 = 7 + c
    c = 3

    So:

    a = 7
    b = 10
    c = 3
    -----------------------------------------------------------

    Cramer's Rule


    First write your equations:

    Let:
    a = First place finishes
    b = Second
    c = Third
    Then:

    A. 68 = 5a + 3b + c
    B. 20 = a + b + c
    C. b = a + c OR 0 = a ? B + c
    -----------------------------------------------------------

    5 3 1
    1 1 1
    1 ?1 1








    Next, replace each column with the "solutions" of your equations:,





    and, one at a time, as above, lather, rinse, repeat. You will get Dx, Dy, and Dz.








    x = Dx/D; y = Dy/D; and z = Dz/D



    (for a better picture/description, see www.purplemath.com - type cramer's in the site search engine)
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    #5

    Feb 26, 2007, 04:01 AM
    Quote Originally Posted by djslilangel
    Solve the system to find how many athletes finished in each place.

    Lawrence High prevailed in Saturdays track meet with the help of 20 individual-event placers earning a combined 68 points. A first place finish earns 5 points, a second place earns 3 points, and third place earns 1 point. Lawrence had a strong second-place showing, with as many second-place finshes as first and third place finishers combined.

    More cramer's rule: attached
    Attached Files
  1. File Type: zip Cramer II.zip (5.0 KB, 102 views)

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