Expert: Sherman D. - 11/2/2006

Question
Hi! We are learning how to solve system of equations using three variables. I understand it for the most part but get stuck towards the end. Help would be greatly appreciated! Here are the two problems:
6a-2b=18            x+y+z=-1
3b+5c=34            3x-2y-4z=19
a+6c=-28    and     2x-y+z=19
Thanks a lot!

Answer
6a - 2b = 18
3b + 5c = 34
a + 6c = -28

6a - 2b = 18
3a - b = 9
-b = -3a + 9
b = 3a - 9

a + 6c = -28
6c = -a - 28
c = (-1/6)a - (14/3)

so now you have b and c

3(3a - 9) + 5((-1/6)a - (14/3)) = 34
9a - 27 + (-5/6)a - (70/3) = 34
Multiply everything by 6
54a - 162 - 5a - 140 = 204
49a - 302 = 204
49a = 506
a = (506/49)

b = 3a - 9
b = 3(506/49) - 9
b = (1518/49) - 9
b = (1518/49) - (441/49)
b = (1077/49)

c = (-1/6)a - (14/3)
c = (-1/6)(506/49) - (14/3)
c = (-506/294) - (14/3)
c = (-506/294) - (1372/294)
c = (-313/49)

a = (506/49)
b = (1077/49)
c = (-313/49)

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

x + y + z = -1
3x - 2y - 4z = 19
2x - y + z = 19

x + y + z = -1
2x - y + z = 19

3x + 2z = 18
3x = -2z + 18
x = (-2/3)z + 6

3((-2/3)z + 6) - 2y - 4z = 19
-2z + 18 - 2y - 4z = 19
-2y - 6z + 18 = 19
-2y - 6z = 1
-2y = 6z + 1
y = -3z - (1/2)

x + y + z = -1
((-2/3)z + 6) + (-3z - (1/2)) + z = -1
(-2/3)z + 6 - 3z - (1/2) + z = -1
(-8/3)z + (11/2) = -1

Multiply everything by 6

-16z + 33 = -6
-16z = -39
z = (39/16)

x = (-2/3)z + 6
x = (-2/3)(39/16) + 6
x = (-78/48) + 6
x = (-13/8) + 6
x = (-13/8) + (48/6)
x = (35/8)

y = -3z - (1/2)
y = -3(39/16) - (1/2)
y = (-117/16) - (1/2)
y = (-117/16) - (8/16)
y = (-125/16)

x = (35/8)
y = (-125/16)
z = (39/16)

www.quickmath.com, click Solve under Equations, then click on Advanced, type in your problem and make sure that you have the right variables set on the right, click solve and it will give you the same values.