SOLUTION: Solve for X, Y, and Z in the following systems of three equations using either substition or elimination methods: 10X + Y + Z = 12, 8X + 2Y +Z = 11, 20X - 10Y - 2Z = 8

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: Solve for X, Y, and Z in the following systems of three equations using either substition or elimination methods: 10X + Y + Z = 12, 8X + 2Y +Z = 11, 20X - 10Y - 2Z = 8       Log On


   

Question 256489: Solve for X, Y, and Z in the following systems of three equations using either substition or elimination methods: 10X + Y + Z = 12, 8X + 2Y +Z = 11, 20X - 10Y - 2Z = 8
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
system%2810X+%2B+Y+%2B+Z+=+12%2C+8X+%2B+2Y+%2BZ++=+11%2C+20X+-+10Y+-+2Z+=+8%29

Solve one equation for one letter,
Substitute that in both the other equations, and simplify them.

Pick the second equation to solve for Z

8X+%2B+2Y+%2BZ++=+11
Z=11-8X-2Y

Substitute %2811-8X-2Y%29 for Z in the other two

10X+%2B+Y+%2B+Z+=+12
10X+%2B+Y+%2B+%2811-8X-2Y%29+=+12
10X+%2B+Y+%2B+11-8X-2Y+=+12
2X+-+Y+%2B+11+=+12
2X+-+Y+=+1


20X+-+10Y+-+2Z+=+8
20X+-+10Y+-+2%2811-8X-2Y%29+=+8
20X+-+10Y+-+22%2B16X%2B4Y+=+8
36X+-6Y+-+22+=+8
36X+-6Y+=+30
Notice that you can divide that equation through by 6
6X-Y=5

So now you have this system with only 2 equations
and 2 unknowns:

system%282X+-+Y+=+1%2C6X-Y=5%29

Solve the first equation for Y:

2X+-+Y+=+1
-Y=1-2X
Y=-1%2B2X

Substitute %28-1%2B2X%29 for Y in
6X-Y=5
6X-%28-1%2B2X%29=5
6X%2B1-2X=5
4X%2B1=5
4X=4
X=1

Substitute 1 for X in

Y=-1%2B2X
Y=-1%2B2%281%29
Y=-1%2B2
Y=1

Substitute 1 for  X and 1 for Y in 

Z=11-8X-2Y
Z=11-8%281%29-2%281%29
Z=11-8-2
Z=1

So (X,Y,Z) = (1,1,1)

Edwin