College algebra

system of linear equation

The sum of 3 numbers is 14
the largest is 4 times the smallest
the sum of the smallest and twice the largest is 18
what are the 3 numbers? show work?

A. how many unknowns you have?
-how many equations need to be solve for that many unknown?

B. what are the equations?

C. used the system of equations using matrices.

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  1. x+y+z=14
    z=4x
    x+2z=18
    =================
    x+y+4x = 14 or 5x+y = 14
    x+8x = 18
    or x = 2
    so
    y = 14 - 10 = 4
    z = 4*2=8
    3 unknowns, three equations
    I did it with substitution, however can form augmented matrix and use Gauss Jordan:
    +1 1 1 14
    -4 0 1 0
    +1 0 2 18

    then plug and chug

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  2. post it.

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  3. matix solution
    |1 1 1 | [x y z]'= [14 0 18]'
    |4 0 1 |
    |1 0 2 |

    H*[x y z]' = [14 0 18]' (' stands for transpose; in this case makes row vector a column vector)

    Answer

    inv(H)*H*[x y z]'= inv(H)*[14 0 18]'

    [x y z] = inv(H)*[14 0 18]'

    inverting a 3x3 matrix

    inv(H)=1/|H|*adj(H) |H|=det(H)

    adj(H)=

    |+(0-0) -(-8-1) +(0-0)|'
    |-(2-0) -(2-1) -(0-1)|
    |+(1-0) -(1+4) +(0+4)|

    |0 9 0|'
    |-2 -1 1|
    |1 -5 4|

    =|0 -2 1|
    |9 -1 -5|
    |0 1 4|

    det(H)= use the second column
    since it has 2 zeros it makes
    it easy.

    det(H)=-1(-8-1)+0*(2-1)-0*(1+4)=9

    inv(H)=1/9*|0 -2 1|
    |9 -1 -5|
    |0 1 4|

    [x y z]'= inv(H)*[14 0 18]'

    =[18/9 (9*14-5*18)/9 (18*4)/9]'
    =[ 2 4 8 ]'

    hope this helps

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