Question

x+y+x=53
3x-2y+z=69
-x+2y-z=-59

i got (68, -24, 9)

Answer 1

Hmm... I actually got different answers. I might be wrong though. You want to work on this together?

Answer 2

ok

Answer 3

i did y

Answer 4

Oh wait...

Answer 5

x+y+x=53 <--- are you sure it's two x's ?
3x-2y+z=69
-x+2y-z=-59

Answer 6

last one a z my bad

Answer 7

Ah ok. This will change up my answer a little. Regardless, here's what we shall do:

equation 1 =
x+y+z=53

We want to solve what Y is first, so... let's put y by itself.

y = -x - z + 53

Answer 8

x

Answer 9

Mmk.
3x-2y+z=69
3x = 2y - z + 69
x = (2y - z + 69)/3

Answer 10

To be honest, though... z would be A LOT easier to use. Should we do that instead?

Answer 11

Your choice.

Answer 12

ok

Answer 13

Awesome.

Equation 2:
3x-2y+z=69
z = -3x + 2y + 69

So far so good?

Answer 14

yea writing the steps down

Answer 15

Okie dokie. I'll go ahead and begin explaining the next step. We're going to combine this last equation into the equation we made earlier.

y = -x - z + 53 This was equation 1 in the form of y = blahblah
z = -3x + 2y + 69 This was equation 2 in the form of z = blahblah

We're going to create what I call equation 1.2 (meaning equation 1 combined with 2)
y = -x - (-3x + 2y + 69) + 53

Let me know if this step made sense. It's important that this makes sense before we do anything else.

Answer 16

ok i dont get this step

Answer 17

Okie dokie. Thanks for letting me know. ^_^

Remember how equation one was written to say
y = -x - z + 53 , right?

Answer 18

And how we made equation 2 say z = -3x + 2y + 69 ?

Answer 19

yea

Answer 20

Alright. Because equation two says, "the letter z is exactly "-3x + 2y + 69", we can replace any z that we see with "-3x + 2y + 69". Does that statement make sense?

Answer 21

yea

Answer 22

Snazzy. So what we did is go back to equation 1:
y = -x - z + 53

And replaced the z with what?

Answer 23

-3x + 2y + 69

Answer 24

Excellent.
So we're left with
y = -x - (-3x + 2y + 69) + 53 right?

Answer 25

(The stuff in parentheses replacing the original z that was there)

Answer 26

yea i see

Answer 27

Alright. Now we're going to distribute that minus that's in front of our parentheses.

Answer 28

y = -x - (-3x + 2y + 69) + 53
y = -x +3x -2y - 69 + 53
Looking good so far?

Answer 29

yea and sorry idk why it keeps putting i dont get this step every time i post

Answer 30

Yeah, I noticed lol. It's ok, this site does that kind of stuff. :P

Answer 31

Now we combine similar terms:
y = -x +3x -2y - 69 + 53
y = 2x -2y - 16 (be careful that you don't accidentally say you have 4x)
3y = 2x - 16

So far so good?

Answer 32

yea

Answer 33

Awesome. Now, we're going to want to note this equation, because it's missing a Z, which is great.

So, let's call it equation 1.2 (as in equation 1 mixed with equation 2)

***** Equation 1.2 *******
3y = 2x - 16



I'll show you in a few moments why we are going to like this equation.

Answer 34

Alright. Last thing we want to do is get x by itself in equation 3.

Equation 3:
-x+2y-z=-59

Answer 35

-x+2y-z=-59
2y -z + 59 = x
x = 2y - z + 59
Makes sense so far?

Answer 36

yea and thanks for going slow I starting to get it

Answer 37

Not a problem! I'm just happy that you're willing to learn instead of just demanding the answer. :D

Answer 38

Alright, so remember in our awesome equation 1.2, we only had x and y?

Answer 39

yep

Answer 40

3y = 2x - 16
What would happen if we plugged in x = 2y - z + 59

You should notice a problem that would occur.

Answer 41

As in,
3y = 2x - 16
3y = 2(2y - z + 59) - 16

Answer 42

we can simplify the y?

Answer 43

We technically can, but there's a bigger problem. I'll give you a hint:
"Alright, so remember in our awesome equation 1.2, we only had x and y?"

Why do you think I asked that question? :P

Answer 44

In other words, we started with
3y = 2x - 16 where there were only two variables.
But once we put in our x = 2y - z + 59, we have a problem.

Answer 45

z came into the equation?

Answer 46

That's right. How do you think we will make z go away and replace it with something nicer?

Answer 47

ok i dont get this step

Answer 48

I would say cancel it out but I don't know how that would work

Answer 49

Well, we did something earlier that let us get rid of the original z. Do you recall what we did?

Answer 50

Hint: Scroll up about halfway to where I said, "Snazzy. So what we did is go back to equation 1: y = -x - z + 53 And replaced the z with what?"
And you answered it correctly.

Answer 51

oh we replaced z with -3x + 2y + 69

Answer 52

Exactly. :D

So here's what we have now, after switching out x:

3y = 2(2y - z + 59) - 16

What will our new problem look like after we replace z?

Answer 53

Hint: Make sure that you put the stuff that's replacing z in parentheses, or your negative won't carry through correctly.

Answer 54

um
3y=2(2y-(-3x=2y+69)+59)-16?

Answer 55

3y = 2(2y - (-3x + 2y + 69) + 59) - 16

Answer 56

Though, I'm guessing your keyboard messed up on that = instead of +, right?

Answer 57

yea i think

Answer 58

Awesome. We're almost done!
Now we just begin simplifying.
3y = 2(2y - (-3x + 2y + 69) + 59) - 16
3y = 2(2y +3x - 2y - 69 + 59) - 16

So far so good? Please double check this to be safe. It'd suck if we get messed up on this step.

Answer 59

it looks good when i went over it i didnt see anything that looked wrong

Answer 60

Awesome.

3y = 2(2y +3x - 2y - 69 + 59) - 16
3y = 2(0y + 3x - 10) - 16
3y = 0y + 6x -20 - 16
3y = 6x -36
So far so good?

Answer 61

Note that the 0y can just be removed since "no y's" is the same as "nothing".

Answer 62

ok

Answer 63

Awesome.
Now, notice there's something we can do on this step to make it a lot prettier. What do you think that is?

Answer 64

(Think greatest common factors)

Answer 65

3y = 6x -36

Answer 66

something with 6 and 36?

Answer 67

Something even easier. Hint: It has to do with a 3.

Answer 68

3 and 6?

Answer 69

We can just divide everything by 3, right?

Answer 70

yea

Answer 71

3y = 6x -36
y = 2x - 12

Looks a lot nicer, ya?

Answer 72

yea it does

Answer 73

Alright, here's where it gets a little challenging. I'm going to let you figure this out (I'll give hints if you need them). We want to replace x with something that has just constants and a y, so that we can have only one variable. You know that we have to have an equation in the form of "x = yblahblah". Which equation will we use to get such a thing?

I'll give you a hint if you need it.

Answer 74

In other words, besides the one we're working with right now, which equation has only x's and y's? We do NOT want any z's messing with us.

Answer 75

3y = 6x -36?

Answer 76

But that's the same equation we're working with. If we put that in there, we'd get something that will lead us to 0=0 because everything would cancel out. We need a different equation.

Answer 77

Hint: I said something about a "super awesome equation" earlier, right? ;) Now's her time to show us her magic!

Answer 78

I didnt see it. Gimmy min let me look through my notebook.

Answer 79

I'll go ahead and give it to you.
3y = 2x - 16

This was that special 'equation 1.2' that had only two variables. Remember that one?

Answer 80

yea

Answer 81

Ok. So what are we going to do to that equation (3y = 2x - 16) to help us solve
(y = 2x - 12) ?

Answer 82

plug y in?

Answer 83

Here's a useful hint:
NORMALLY, you want to get it in the form of x = blah blah. But in this case, you can find a similar variable in both equations. Do you see anything repeated exactly the same?

Answer 84

Hmm.... actually, we could do that too. I'll show you both ways.

Answer 85

Method 1: (what I was planning on doing)
Since 2x appears in both equations, you could do this:
given: (3y = 2x - 16) and (y = 2x - 12)
3y = 2x - 16
3y + 16 = 2x
2x = 3y + 16
Since 2x = 3y + 16, we can change equation 1.2 from
(y = 2x - 12) into (y = 3y + 16 - 12)
Then simplify into
(y = 3y + 4)
(-2y = 4)
y = -2

Does that make sense?

Answer 86

yea

Answer 87

Method 2: (your easier way :P)
given: 3y = 2x - 16 and y = 2x - 12
Plug in y into the first equation:
3(2x - 12 ) = 2x - 16
6x - 36 = 2x - 16
4x = 20
x = 5

Answer 88

Oooo nice! Do you see what happened?

Answer 89

you found what x = so no you can plug it in or something else?

Answer 90

Yup!
Let's use the easiest equation for that.

x+y+x=53

Answer 91

Oops. I copied it from the original question. Here's the correct one:
x+y+z=53

Answer 92

so be 5+y+z=53?

Answer 93

But we also found y earlier, remember?

Answer 94

-2

Answer 95

Exactly.

Answer 96

5+-2+z=53

Answer 97

Perfect. So 3 + z = 53.

Now put in the last jigsaw on this beautiful puzzle. :P

Answer 98

z=50 but thats not to big?