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 $z$?
+128460 -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 }
