SOLUTION: 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 every

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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): You can put this solution on YOUR website!


You need to solve the first three equations simultaneously to find the values of x, y, and z. You then use those values to calculate the value of a.

To solve the 3X3 system, look up Gauss-Jordan elimination. If you get stuck, write back showing the work you have done to solve it, successful or not, and I'll help you from there. Hint: Make sure the variables in all three equations are in the same order.


John

My calculator said it, I believe it, that settles it


I > Ø
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.


------------

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): You can put this solution on YOUR website!
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?
This is a system of equations in 3 VARIABLES. You need to find the value of each of the 3 variables....in this case, x, y, and z. Then determine the value of a, which is: x * y - z.
 
There're may ways that this system can be solved. The easiest, in my opinion, is the following:
x - y - z = 5 -------- eq (i)
y - x - z = 1 -------- eq (ii)
z - y - x = -15 ------ eq (iii)

Rearrange eqs (ii) & (iii) to get: 
- 2z = 6 ----- Adding eqs (i) & (ii)



- 2x = - 14 ----- Adding eqs (ii) & (iii)



y - 7 - (- 3) = 1 ------ Substituting 7 of x, and - 3 for z in eq (ii)
y - 7 + 3 = 1
y - 4 = 1


Shouldn't pose a problem now, to find "a."
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