# Gaussian Elimination: Translating text into equations

1. Feb 27, 2010

### Matty R

Hello

I have a homework question regarding Gaussian Elimination, where I am supposed to use information in a block of text to get equations, and form an augmented matrix.

There are 4 unknowns, but I can only seem to get 3 equations and 3 "checks", to see if the values I get are correct.

I'm fine with Gaussian Elimination itself. I'm just having trouble translating the information into equations.

1. The problem statement, all variables and given/known data

Ann, Bea, Claire and Dawn have joint birthdays.
The sum of their ages is exactly 100 years.
The sum of Ann's and Dawn's ages is the same as the sum of Bea's and Claire's.
The difference between the ages of Claire and Bea is twice Ann's age.
When Claire is as old as Dawn is now, Bea will be twice as old as Ann currently is.
Claire is older than Bea.
How old is each person?

2. Relevant equations

3. The attempt at a solution

a + b + c + d = 100

a + d = b + c

a - b - c + d = 0

2a = c - b

2a + b - c = 0

c = d $$\Rightarrow$$ b = 2a

c > b

I end up with 4 unknowns in three equations, and apparently that means there are infinite solutions. I've been told a set of answers, and they fit all of the equations I've worked out, along with the "checks". I just can't seem to work out the fourth equation.

I'd be grateful for any and all help.

Thanks

2. Feb 27, 2010

### tiny-tim

Hello Matty R!
No, that doesn't mean anything, does it?

Hint: what will Bea's age be when Claire is as old as Dawn is now?

3. Feb 27, 2010

### HallsofIvy

Staff Emeritus
"When Claire is as old as Dawn is now, Bea will be twice as old as Ann currently is.
Claire is older than Bea."
Claire will be as old as Dawn is now in d- c years. Bea's age then will be b+ (d- c) and that will be twice Ann's current age: b+ d- c= 2a or 2a- b+ c- d= 0.

You have four equations:
The sum of their ages is exactly 100 years.
a+ b+ c+ d= 100

The sum of Ann's and Dawn's ages is the same as the sum of Bea's and Claire's.
a- b- c+ d= 0

The difference between the ages of Claire and Bea is twice Ann's age.
2a+ b- c= 0
("Claire is older than Bea" tells you that the difference between the ages of Claire and Bea is c- b, not b- c).

When Claire is as old as Dawn is now, Bea will be twice as old as Ann currently is.
2a- b+ c- d= 0

4. Feb 28, 2010

### Matty R

Thanks for the replies.

I'd never have got that. I completely see how to get it now, but I just couldn't understand it before.