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# system

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Md Nur Uddin is a student of Mathematics Discipline Of Khulna University. Now, you have to find his student ID by solving the following system of linear equations:

\begin{align*} -3x+2y-6z &=6 \\ 5x+7y-5z &=6 \\ x+4y-2z &=8. \end{align*}

If the solution satisfies $x+y=z$, what is the value of $x+y,$ or equivalently $z$?

Feb 4, 2021

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-3x  +  2y  - 6z   =  6

5x  +  7y  - 5z   =   6

x +  4y   -   2z    =  8

Lookig at the bottom equation    ....mutiply  it through  by   3

3x  + 12y  - 6z  =  24             add this to the  first equation

14y  - 12z  = 30     divide through  by 2   ⇒  7y - 6z  = 15      (a)

Then, mutiply it through by    -5

-5x  - 20y  + 10z  =  -40         add this to the second equation

-13y  + 5z   =  -34       (b)

Mutiply (a)  through  by   5  and  (b)  through by  6

35y  - 30z  = 75

-78y  + 30z  = -204          add these

- 43 y  = -129

y=  -129 / - 43  =   3

Using (a)    7(3)  - 6z = 15

21 -  6z  =  15

21 -  15  = 6z

6  = 6z

z    = 1

And

x   + 4y  -2z  = 8

x + 4(3)   -2(1)  =  8

x  + 12 -2  = 8

x  + 10  = 8

x =  8 -10   =  -2

So   { x, y , z }  = { -2, 3 ,  1 }   Feb 4, 2021