Question 207711: Solve for X, Y, and Z in the following systems of three equations using either substition or elimination methods:
a. X + 2Y + Z = 6
X + Y = 4
3X + Y + Z = 8
b. 10X + Y + Z = 12
8X + 2Y +Z = 11
20X - 10Y - 2Z = 8
c. 22X + 5Y + 7Z = 12
10X + 3Y + 2Z = 5
9X + 2Y + 12Z = 14
Answer by pjalgeb(9)   (Show Source):
You can put this solution on YOUR website!
PROBLEM A
GIVEN:
X + 2Y + Z = 6 --> 
X + Y = 4 --> 
3X + Y + Z = 8 --> 
REQUIRED: FIND X, Y AND Z
SOLUTION:
FROM :
X + Y = 4
Y = 4-X --> 
MULTIPLY WITH -1 AND ADD TO TO ELIMINATE  :
-X -2Y - Z = -6 --> 
ADD AND :
-X -2Y - Z = -6 -->
3X + Y + Z = 8 -->
_______________
2X -Y = 2
2X -Y = 2 --> 
NOW WE CAN SOLVE FOR  USING AND BY SUBSTITUTION:
2X -Y = 2 -->
2X -(4-X) = 2
2X -4 + X = 2
3X = 2 + 4
3X = 6
X = 6/3
NOW THAT WE HAVE , WE CAN SOLVE FOR  USING AND  BY SUBSTITUTION:
X + Y = 4 -->
2 + Y = 4
Y = 4-2
NOW THAT WE HAVE  AND , WE CAN SOLVE FOR USING BY SUBSTITUTION:
X + 2Y + Z = 6 -->
2 + 2(2) + Z = 6
2 + 4 + Z = 6
6 + Z = 6
Z = 6-6
PROBLEM B.
GIVEN:
10X + Y + Z = 12 -->
8X + 2Y + Z = 11 -->
20X - 10Y - 2Z = 8 -->
REQUIRED: FIND X, Y AND Z
SOLUTION:
MULTIPLY WITH -1 AND ADD TO TO ELIMINATE :
-10X - Y -Z = -12 --> *(-1)
8X +2Y +Z = 11 -->
_________________
-2X + Y = -1
-2X + Y = -1 -->
FROM :
-2X + Y = -1 -->
Y = 2X-1 -->
NOW MULTIPLY WITH  AND ADD TO TO ELIMINATE :
16X + 4Y + 2Z = 22 --> *(2)
20X - 10Y - 2Z = 8 -->
___________________
36X - 6Y = 30
36X - 6Y = 30 -->
NOW WE CAN SOLVE FOR USING IN :
36X - 6Y = 30 -->
36X - 6(2X-1) = 30
36X -12X + 6 = 30
24X = 30-6
24X = 24
X = 24/24
NOW THAT HAVE HAVE , WE CAN SOLVE FOR USING :
Y = 2X -1 -->
Y = 2(1)-1
Y = 2 - 1
NOW THAT HAVE HAVE AND , WE CAN SOLVE FOR USING , OR  , LETS TRY :
10X + Y + Z = 12 -->
10(1) + 1 + Z = 12
10 + 1 + Z = 12
11 + Z = 12
Z = 12-11
PROBLEM C
GIVEN:
22X + 5Y + 7Z = 12 -->
10X + 3Y + 2Z = 5 -->
9X + 2Y + 12Z = 14 -->
REQUIRED: FIND X, Y AND Z
SOLUTION:
MULTIPLY WITH  AND ADD TO AND TO ELIMINATE :
-22X - 5Y - 7Z = -12 --> *(-1)
10X + 3Y + 2Z = 5 -->
9X + 2Y + 12Z = 14 -->
____________________
-3X + 7Z = 7
-3X + 7Z = 7 -->
FROM :
-3X + 7Z = 7 -->
3X = 7Z - 7
X = (7Z -7)/3 -->
MULTIPLY WITH  AND ADD TO TO ELIMINATE :
-(27/2)x - 3Y - 18z = -21 --> * (-3/2)
10X + 3Y + 2Z = 5 -->
_________________________
(-7/2)x - 16z = -16
(-7/2)x - 16z = -16 --> 
FROM :
[(-7/2)x - 16z = -16] * 2 --> *2
-7X - 32Z = -32 --> 
NOW WE CAN SOLVE FOR USING IN :
7X - 32Z = -32 -->
-7[(7Z-7)/3] - 32Z = -32
[(-49Z + 49)/3 - 32Z = -32] *3
-49Z + 49 - 96Z = -96
-145Z = -145
Z = -145/-145
NOW THAT WE HAVE WE CAN SOLVE FOR USING :
X = (7Z -7)/3 -->
X = [7(1) - 7]/3
X = (7-7)/3
X = 0/3
NOW THAT HAVE HAVE AND , WE CAN SOLVE FOR USING , OR , LETS TRY :
22X + 5Y + 7Z = 12 -->
22(0) + 5Y + 7(1) = 12
0 + 5Y + 7 = 12
5Y = 12-7
5Y = 5
Y = 5/5
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