SOLUTION: 7. Solve for X,Y and Z 2X+2Y+4Z=10 3X+3Y=15 4X+12Y+4Z=44

Question 202709: 7. Solve for X,Y and Z
2X+2Y+4Z=10
3X+3Y=15
4X+12Y+4Z=44

Answer by PRMath(133) (Show Source): You can put this solution on YOUR website!
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... :-)
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