Question 202709
2X+2Y+4Z=10
3X+3Y=15
4X+12Y+4Z=44


Let's take two equations and eliminate one variable.  I'd suggest the variable "Z" because the 2nd equation does not have a "z".  SO, let's work with equation 1 and 3....


To get rid of the "Z" variable, let's multiply this equation 4X+12Y+4Z=44 by a negative 1.   


2X+2Y+4Z=10
4X+12Y+4Z=44  (multiply by -1)


Two new equations....
2X  + 2Y+ 4Z =  10
-4x -12y -4z = -44
_________________  now add these two equations
-2x - 10y    = -34


Now let's take this new equation and look at it along with the other equation that only had "X" and "Y" variables.........


-2x - 10y = -34  (What if we multiply this by 3?)
3X + 3Y = 15  (What if we multiply this by 2?)


If we do the multiplications I suggest above, our new equations will be:


-6x -30y =   -102
6x + 6y = 30
_________________ add these equations
    -24y = -72  (divide both sides by -24)

y = 3


Now that we know what Y is, let's plug it into the equation with only 2 variables...........


3X+3Y=15
3x+3(3) = 15 (plug in the 3 for the y variable)
3x + 9 = 15  (multiply 3 times 3)
3x = 6  (subtract 9 from both sides)
x =2 (divide both sides by 3)


Ok, now we know x = 2 and y = 3.  Let's plug those numbers into one of the equations that has three variables.  I will choose the first equation:


2X+2Y+4Z=10
2(2) + 2(3) + 4z = 10 (plug in 2 for the x variable and 3 for the y variable)
4 + 6+ 4z = 10 (multiply 2 times 2.  Multiply 2 times 3)
10 + 4z = 10 (added 4 and 6)
4z = 0 (subtracted 10 from both sides)
z = 0 (divided both sides by 4)


Now we know x = 2, y = 3 and z = 0.  If we plug these into our 3 equations, will they all check out?  Let's try it:



First equation:
2X+2Y+4Z=10
2(2) + 2(3) + 4(0) = 10
4 + 6 + 0 = 10
10= 10  
Yay... the 1st equation works.



Second equation:
3X+3Y=15
3(2) + 3(3) = 15
6 + 9 = 15
15 = 15
Yay... the 2nd equation works. :-) Let's go for the 3rd.......


Third equation:
4X+12Y+4Z=44
4(2) + 12(3) + 4(0)=44
8 + 36 + 0 = 44
44 = 44
Yay... the 3rd equation works.  SO cool.  


Hope this helps...    :-)