Question

Help please!!: Show all your work please so i can understand, this is a linear solution:
3x-2y+2z=30
-x+3y-4z=-33
2x-4y+3z=42

Answer 1

I like to number my equations, it makes them easier to remember which one is which!
(1) 3x-2y+2z=30
(2) -x+3y-4z=-33
(3) 2x-4y+3z=42

First we look to see if any equations, when added, have a variable that cancels. In this case, we don't xD So, let's just decide to use equations 1 and 2 first.
(1) 3x-2y+2z=30
(2) -x+3y-4z=-33

How can we modify one to allow one of the variables to cancel. Well, if we multiply equation 2 by 3, then the x's will cancel.
(1) 3x-2y+2z=30
(2) -3x+9y-12z=-99

Now we add 1 and 2 to get.
0x+7y-10z=-69
(4) 7y-10z=-69 ; Let's call this equation 4.

Now we need to go back to our 3 original equations and choose another pair, but not the same pair, and then solve for x as well. Let's just use 2 and 3.
(2) -x+3y-4z=-33
(3) 2x-4y+3z=42

How can we modify one to allow one of the variables to cancel. Well, if we multiply equation 2 by 2, then the x's will cancel.
(2) -2x+6y-8z=-66
(3) 2x-4y+3z=42

Now we add 2 and 3 to get.
0x+2y-5z=-24
(5) 2y-5z=-24 ; Let's call this equation 5.

Now that we have equations 4 and 5 which only have two variables, we now need to add them together and cancel yet another variable to isolate just one.
(4) 7y-10z=-69
(5) 2y-5z=-24

How can we modify one to allow one of the variables to cancel. Well, if we multiply equation 5 by -2, then the z's will cancel.
(4) 7y-10z=-69
(5) -4y+10z=48

Now we add 4 and 5 to get.
3y+0z=-21
(6) 3y=-21 ; Let's call this equation 6.

Now that we've isolated one variable (y), we can solve for it in equation 6.
(6) 3y=-21
y=-7

We now know that y=-7. In order to find the others, we first go back to either equation 4 or 5, since they had only two variables. There are two unknowns, we know one now, so we can then solve for the other. I'm just going to choose equation 5. No reason.
(5) 2y-5z=-24
y=-7
2(-7)-5z=-24
-14-5z=-24
-5z=-10
z=2

Now that we've solved for y=-7 and z=2, last thing we do is just choose one of the 3 original to plug in and find x! I'm just going to use equation 1.
(1) 3x-2y+2z=30
y=-7, z=2
3x-2(-7)+2(2)=30
3x+14+4=30
3x+18=30
3x=12
x=4

So.... After the long problem (which all of these are usually) we get:

x=4
y=-7
z=2

Hope this explains it all :D