SOLUTION: -2x+3y+z=3 3x-y+2z=2 x+2y-3z=1 the answer they provided us is 13/21,25/21,2/3 i do not know how to get there..

Algebra ->  Expressions-with-variables -> SOLUTION: -2x+3y+z=3 3x-y+2z=2 x+2y-3z=1 the answer they provided us is 13/21,25/21,2/3 i do not know how to get there..      Log On


   

Question 425094: -2x+3y+z=3
3x-y+2z=2
x+2y-3z=1
the answer they provided us is 13/21,25/21,2/3 i do not know how to get there..
Found 2 solutions by Edwin McCravy, emargo19:
Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!

(A)     -2x + 3y +  z = 3
(B)      3x -  y + 2z = 2
(C)       x + 2y - 3z = 1

Multiply equation (C) by 2

(D)      2x + 4y - 6z = 2

Add eq. (D) term by term to eq. (A)

(A)     -2x + 3y +  z = 3
(D)      2x + 4y - 6z = 2
-------------------------
(E)           7y - 5z = 5

Multiply equation (C) by -3

(F)     -3x - 6y + 9z = -3

Add eq. (F) term by term to eq. (B)

(B)      3x -  y +  2z =  2
(F)     -3x - 6y +  9z = -3
---------------------------
(G)          -7y + 11z = -1 

Add eq. (G) term by term to eq. (E)

(E)           7y -  5z =  5
(G)          -7y + 11z = -1  
---------------------------
                    6z =  4
                     z = 6/4
                     z = 3/2

Multiply eq. (E) by 11

(H)          77y - 55z = 55

Multiply eq. (G) by 5

(I)         -35y + 55z = -5

Add eq. (I) term by term to eq. (H)

(H)          77y - 55z = 55
(I)         -35y + 55z = -5
---------------------------
             42y       = 50
                     y = 50/42
                     y = 25/21

 -------------------------------------
(A)     -2x + 3y +  z = 3
(B)      3x -  y + 2z = 2
(C)       x + 2y - 3z = 1

Multiply equation (B) by 3

(J)      9x - 3y + 6z = 6

Add eq. (J) term by term to eq. (A)

(A)     -2x + 3y +  z = 3
(J)      9x - 3y + 6z = 6
-------------------------
(K)      7x      + 7z = 9

Multiply equation (B) by 2

(L)      6x - 2y + 4z =  4

Add eq. (L) term by term to eq. (C)

(C)       x + 2y - 3z =  1
(J)      6x - 2y + 4z =  4
--------------------------
(M)      7x      +  z =  5

Multiply eq. (M) by -7

(N)    -49x      - 7z = -35

Add eq. (N) term by term to eq. (K) 

(K)      7x      + 7z =   9
(N)    -49x      - 7z = -35
---------------------------
       -42x           = -26
                    x = -26/(-42)
                    x = 13/21 
                             
Edwin

Answer by emargo19(101) About Me  (Show Source):
You can put this solution on YOUR website!
The below is for your understanding.
-2x%2B3y%2Bz=3---eq1
3x-y%2B2z=2----eq2
x%2B2y-3z=1----eq3
Lets eliminate z So take eq 1 & 2
3x-y%2B2z=2---eq2
-2x%2B3y%2Bz=3+---eq 1 (multiply 2 )

GIVES
3x-y%2B2z=2---eq2
-4x%2B6y%2B2z=6---eq1
To eliminate z we need to change the signs in the equation
So when solved
4x-6y-2z=-6
3x-y%2B2z=2
ans is
+7x-7y=-4 ----eq 4
Next lets again eliminate z using eq 1 & 3
-2x%2B3y%2Bz=3---eq1
x%2B2y-3z=1----eq3
Multiply eq 1 with -3 [i.e., coeff of z of (eq 3)]
6x-9y-3z=-9 ---eq1
x%2B2y-3z=1 ---eq3
----------
Where it is + in eq 3---> x%2B2y-3z=1 make it - and vice versa Therefore it becomes -x-2y%2B3z=-1
GIVES
5x-11y=-10---(eq 5)

NOW Lets find the value of y by eliminating x.
+7x-7y=-4 ----(eq 4)
5x-11y=-10---(eq 5)
To do so lets mutiply (eq 4) with 5 and (eq 5) with 7
+7x-7y=-4+ ----(eq 4) multiply by 5
5x-11y=-10---(eq 5)multiply by 7

35x-35y=-20
35x-77y=-70
Where it is + in (35x-77y=-70) make it - and vice versa Therefore it becomes
-35x%2B77y=70
So
42y=50
y=25%2F21
NEXT lets find the value of x by using eq 5
5x-11y=-10
5x-%2811%2A25%29%2F21=-10 (We substituted y in this eq)
5x-275%2F21=-10
5x=%28-210%2B275%29%2F21
x=65%2F21%2A1%2F5
x=13%2F21
Let us use eq 1 to find value of z
-2x%2B3y%2Bz=3---eq1 (substitute x and y in eq 1)
-26%2F21%2B75%2F21%2Bz=3
49%2F21%2Bz=3
z=3-%2849%2F21%29
z=%2863-49%29%2F21
z=14%2F21
z=2%2F3

So x=13%2F21
y=25%2F21
z=2%2F3
***************************************************************
You can write this
-2x%2B3y%2Bz=3---eq1
3x-y%2B2z=2----eq2
x%2B2y-3z=1----eq3
****
3x-y%2B2z=2---eq2
-2x%2B3y%2Bz=3+---eq 1 (multiply 2 )
So on changing signs in the 2nd eq it is
3x-y%2B2z=2
4x-6y-2z=-6
gives
+7x-7y=-4 ----eq 4
****
Next eliminate z using eq 1 & 3
-2x%2B3y%2Bz=3---eq1
x%2B2y-3z=1----eq3
Multiply eq 1 by 3
6x-9y-3z=-9 ---eq1
x%2B2y-3z=1 ---eq3
Next on changing signs in the 2nd eq it is
6x-9y-3z=-9 ---eq1
-x-2y%2B3z=-1 ---eq3
Gives
5x-11y=-10---(eq 5)
*****
+7x-7y=-4+ (multiply by 5)
5x-11y=-10 (multiply by 7)
35x-35y=-20
35x-77y=-70
Next after change of signs :
35x-35y=-20
-35x%2B77y=70
+42y=50%7D%7D%0D%0A%7B%7B%7By=25%2F21
*** To find x use eq :
5x-11y=-10
5x-%2811%2A25%29%2F21=-10 (We substituted y= 25/21 in this eq)
5x-275%2F21=-10
5x=%28-210%2B275%29%2F21
x=65%2F21%2A1%2F5
x=13%2F21
*****use eq 1 to find value of z
-2x%2B3y%2Bz=3---eq1 (substitute x and y in eq 1)
-26%2F21%2B75%2F21%2Bz=3
49%2F21%2Bz=3
z=3-%2849%2F21%29
z=%2863-49%29%2F21
z=14%2F21
z=2%2F3
I hope this helps.