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#1 2011-10-06 20:59:28

Marisca
Member
Registered: 2011-04-22
Posts: 53

Linear functions and Matrices

Hi, I'm having trouble with these questions and was wondering if anyone could help:

Consider the system of equations;
x+2y-z=2
2x+5y-(a+2)z=3
-x+(a-5)y+z=1

a) Solve the equations in terms of 'a', for suitable values of 'a'.
b) State the values of 'a' for which there is a unique solution.

Is this the type of question where I would need to set the three equations as matrices?

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#2 2011-10-06 21:48:17

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Linear functions and Matrices

Hi Marisca;

There is some confusion about that. You can always format the problem in terms of matrices. This is just formal though. It does not assist in the solution of the problem.

For that you must either invert a matrix ( a purely computational task ) or use iterative means to get the actual answer.

For schoolwork and small systems hand methods are used.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2011-10-06 21:57:06

Marisca
Member
Registered: 2011-04-22
Posts: 53

Re: Linear functions and Matrices

Okay that cleared some stuff up, so would I start off by rearranging the three equations?

Sorry, I'm really terrible at these questions.

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#4 2011-10-06 22:00:19

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Linear functions and Matrices

Hi;

Do you want to try it by hand?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#5 2011-10-06 22:07:45

Marisca
Member
Registered: 2011-04-22
Posts: 53

Re: Linear functions and Matrices

I'll try; do I arrange them for y?

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#6 2011-10-06 22:11:16

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Linear functions and Matrices

Hi;

First, do not be ashamed about finding this one hard. This question is the act of a sadist. If I were his/her student I would deliver a sharp blow to their probably ugly proboscis, causing them extreme discomfort.

Try to eliminate one variable, use the first equation and solve for z.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#7 2011-10-06 22:18:19

Marisca
Member
Registered: 2011-04-22
Posts: 53

Re: Linear functions and Matrices

So, would it be wrong if I removed x and solved for z in the first equation and got

z=2y+2 ?

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#8 2011-10-06 22:19:55

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Linear functions and Matrices

Hi;

Yes, that would be very wrong. You just can not make variables disappear no matter how much we hate them. Just get z by itself and show me what you get.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#9 2011-10-06 22:23:12

Marisca
Member
Registered: 2011-04-22
Posts: 53

Re: Linear functions and Matrices

Okay I got
z=x+2y-2

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#10 2011-10-06 22:26:13

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Linear functions and Matrices

That is correct!

This is a little bit tough. In the next two equations substitue for z in both of them.

Need help?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#11 2011-10-06 22:34:57

Marisca
Member
Registered: 2011-04-22
Posts: 53

Re: Linear functions and Matrices

I'm not sure but is it;

2x+5y-ax-2ay+2a=3

ay-3y=3

Edit: oh ok, I see that I wasn't supposed to expand it.

Last edited by Marisca (2011-10-06 22:37:08)

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#12 2011-10-06 22:38:23

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Linear functions and Matrices

Hi Marisca;

The second one is correct, the first one is not. You forgot to multiply by the 2 in (a+2).

Let me show you a trick. You should get this:

Notice how I did not simplify or combine. Many of the greatest moves in math were made by people who resisted the urge to simplify every time they can.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#13 2011-10-06 22:43:55

Marisca
Member
Registered: 2011-04-22
Posts: 53

Re: Linear functions and Matrices

ok that makes sense.
Not really sure what I need to do now though...

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#14 2011-10-06 22:47:27

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Linear functions and Matrices

Hi;

Remember, we do not simplify or combine unless there is a good reason to. Look at the second equation. If we combine the x and -x they will cancel and we will be left with an equation with one variable y! These we can solve easily. Do that and show me what you get.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#15 2011-10-06 22:51:55

Marisca
Member
Registered: 2011-04-22
Posts: 53

Re: Linear functions and Matrices

I got y=3/a-3

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#16 2011-10-06 22:53:59

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Linear functions and Matrices

Hi;

I think you mean y = 3 / ( a - 3 ). I see a lot of students taking the shorthand, poor notation of writing it like y=3/a-3. Resist that urge and you are ahead of them.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#17 2011-10-06 22:58:01

Marisca
Member
Registered: 2011-04-22
Posts: 53

Re: Linear functions and Matrices

oops, yes I should have done that.

From this point would I need to substitute this equation into the first equation, and then solve for a?

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#18 2011-10-06 23:04:10

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Linear functions and Matrices

Okay, let's see what we have:

We started with these:

Worked our way to these:

Now we have this:

Let's take the first equation up there and try to get another variable by itself. If you substitute for y in it what do you get?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#19 2011-10-06 23:09:03

Marisca
Member
Registered: 2011-04-22
Posts: 53

Re: Linear functions and Matrices

I got
x-z+ 6/(a-3) =2

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#20 2011-10-06 23:13:53

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Linear functions and Matrices

Hi;

Sorry, I meant the first equation in the second group. This one right here.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#21 2011-10-06 23:18:06

Marisca
Member
Registered: 2011-04-22
Posts: 53

Re: Linear functions and Matrices

Ok is it

-ax- 15/(a-3) +2a-2=3

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#22 2011-10-06 23:24:08

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Linear functions and Matrices

Hi Marisca;

Very good! That is correct! Now get x by itself.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#23 2011-10-06 23:26:09

Marisca
Member
Registered: 2011-04-22
Posts: 53

Re: Linear functions and Matrices

Thanks, I got;

x=2a-11/(a-3)

Last edited by Marisca (2011-10-06 23:28:53)

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#24 2011-10-06 23:29:39

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Linear functions and Matrices

Hi;

Mathematically,
x=2a-11/(a-3) means:

Which is not what you intended. When you write it like this:

x=(2a-11)/(a-3)

That means :

Which is what you want and is correct!


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#25 2011-10-06 23:34:28

Marisca
Member
Registered: 2011-04-22
Posts: 53

Re: Linear functions and Matrices

ah, sorry. I'll have to keep that in mind.

Ok so would I now use this to substitute back into the equation and solve for a?

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