Simple systems of linear equations question

[eugenius]

Hey, I am very new at using matrices to solve systems of equations. My question isn't actually how to invert a matrix or anything, its much simpler.

Homework Statement

If you add the age of 6 people, Anna, Barbara, Cathy, Dan, Eric, Fred, the total is 92. Anna is twice the age of Cathy. The total age of Barbara and Eric equals that of Fred. The total age of Anna and Cathy equals the total age of Barbara and Fred. Cathy is 8 years younger than Fred. If you subtract the age of Eric and Cathy from Dan the result is 10.

What is the age of each person?

Homework Equations

I need to write a linear equation for each of the above conditions and then convert the equations into matrices of form Ax=b.

Then I need to write a program that will solve the matrices for the unknowns. You see I don't have to solve the matrices myself, my program will do that for me. And besides once I can write out the matrices, I know what to do.

What I can't seem to do is construct appropriate linear equations from the above conditions. Here is what I tried. This is the best I can do so far.

The Attempt at a Solution

Each letter stands for the name of the person, A = Anna and etc.

A+B+C+D+E+F = 92

A= 2C

B+E=F

A+C= B+F

F-8= C

E+C-D = 10

Here I have 6 linear equations (I think), and 6 unknowns. Meaning 6 by 6 matrix. But what are the coefficients that I put in the matrix? They look like mostly 1's. This can't be right. Help me out please.

 

[Defennder]

Yes they're mostly 1's and 0s though you have some other numbers as well. Those are the co-efficients you write in the matrix, then solve it by row-reduction.

 

[bdforbes]

Do you understand what the vectors x and b refer to in this case? The statement Ax=b means that the 6x6 matrix A, when multiplied by the column vector x, gives the column vector b.

 

[eugenius]

So are you telling me that my equations are correct? Please read over the Alice is twice the age of Jim and etc again, and see if my equations make sense.

I'm surprised if they are correct because its rather odd to have so many ones don't you think?

So you are saying my matrix would be

1 1 1 1 1 1 92

1 2 This is (A= 2C) ? The coefficients below coorespond to the rest

1 1 1


1 1 1 1

1 -8 1

1 1 1 10

To me this doesn't look right. Is it?

 

[bdforbes]

You should expect a lot of ones, since the statements refer to for example "Barbara + Eric = Fred". In this statement, there is only a factor of one for each age.

Your matrix is close but not perfect. The whole idea here, is you write x=(A B C D E F), a column vector. You put all of the coefficients of the indeterminates (A,B,C,D,E,F) in the matrix A. Then you put the result of each linear equation into the column vector b. You should rewrite the equations with all the indeterminates on one side:

A + B + C + D + E + F = 92
A     - 2C            = 0
    B         + E - F = 0
A - B + C         - F = 0
      - C         + F = 8
        C - D + E     = 10

 

[eugenius]

You should rewrite the equations with all the indeterminates on one side:

A + B + C + D + E + F = 92
A     - 2C            = 0
    B         + E - F = 0
A - B + C         - F = 0
      - C         + F = 8
        C - D + E     = 10

Yes exactly. I asked my professor also. That was my problem. I didn't know that you should put all the unknowns on the same side. Or rather I did know it subconciously, but forgot.

Thanks this helps.