Question 1167063: so Im trying to brush up on my skills and have been presented with:
x - y - z = 5
y - x - z = 1
z - y - x = -15
x * y - z = a
so far all I can remember is to equal everything out to zero leaving:
x - y - z - 5 = 0
y - x - z - 1 = 0
z - y - x + 15 = 0
x * y - z - a = 0
what kind of problem is this exactly and how do I solve he variables?
Found 3 solutions by solver91311, ikleyn, MathTherapy:
Answer by solver91311(24713)   (Show Source):
Answer by ikleyn(52786)  (Show Source):
You can put this solution on YOUR website! .
x - y - z = 5 (1)
y - x - z = 1 (2)
z - y - x = -15 (3)
x * y - z = a (4)
The problem is to find three unknowns x, y and z from equations (1), (2) and (3)
and THEN (=after that) to find the value of "a" from equation (4).
The system of three equations (1), (2) and (3) is VERY SPECIAL, and there is VERY SPECIAL trick to solve it.
Add equations (1), (2) and (3) (=add separately their left and right sides). You will get then
(x+y+z) - 2x - 2y - 2z = 5+1-15, or
-x - y - z = -9, or
x + y + z = 9. (5)
Now add equations (1) and (5). You will get
2x = 5+9 = 14; hence, x = 14/2 = 7.
Next, add equations (2) and (5). You will get
2y = 1 + 9 = 10; hence, y = 10/2 = 5.
Finally, add equations (3) and (5). You will get
2z = -15 + 9 = -6; hence, y = -6/2 = -3.
Thus x= 7, y= 5, z= -3.
Hence, a = x*y - z = 7*5 - (-3) = 35 + 3 = 38.
ANSWER. x= 7, y= 5, z= -3, a = 38.
Solved.
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There are TWO MAJOR LESSONS for you to learn from my solution.
First is to read the problem attentively and to understand what they want from you.
Second is this SPECIAL TRICK to solve the given system of equations.
Happy learning (!)
Answer by MathTherapy(10552)  (Show Source):
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