## 3 variables 3 equations (2 linear 1 quadratic)

1. The problem statement, all variables and given/known data
Find three numbers x,y,z satisfying the following equations:
9x-6y-10z=1
-6x+4y+7z=0
x2+y2+z2=9

2. Relevant equations

3. The attempt at a solution
The book suggests that the best approach is to express 2 unknowns in terms of the third unknown from the first 2 linear equations and, by substituting these expressions in the last equation, obtain a quadratic equation for the third unknown.
So i try this:
4(9x-6y-10z=1)= 36x-24y-40z=4
6(-6x+4y+7z=0)= -36x+24y+42z=0
(36x-24y-40z=4) + (-36x+24y+42z=0)= 2z=4
z=2
I figure I've gone wrong somewhere because this is not the strategy outlined in the book (it's also not the correct answer for z according to the book)
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Recognitions:
Homework Help
 Quote by Appleton 1. The problem statement, all variables and given/known data Find three numbers x,y,z satisfying the following equations: 9x-6y-10z=1 -6x+4y+7z=0 x2+y2+z2=9 2. Relevant equations 3. The attempt at a solution The book suggests that the best approach is to express 2 unknowns in terms of the third unknown from the first 2 linear equations and, by substituting these expressions in the last equation, obtain a quadratic equation for the third unknown. So i try this: 4(9x-6y-10z=1)= 36x-24y-40z=4 6(-6x+4y+7z=0)= -36x+24y+42z=0 (36x-24y-40z=4) + (-36x+24y+42z=0)= 2z=4 z=2 I figure I've gone wrong somewhere because this is not the strategy outlined in the book (it's also not the correct answer for z according to the book)
If you have stated the problem correctly, you are right: z = 2, no matter what the book says. In fact, there are two solutions; they both have z = 2 but have different x,y values.
 Thanks for your reply. On further inspection I realise that the book does in fact say that z is equal to 2. However this still leaves me a bit confused about the purpose of the quadratic equation since it seems it is superfluous to the requirements for providing a solution. The problem is question 1A, page 44 (in the pdf digital copy. Or 22 in the hard copy) of this book: http://www.scribd.com/doc/39412845/M...ics-Polya-1954

Recognitions:
Homework Help

## 3 variables 3 equations (2 linear 1 quadratic)

 Quote by Appleton Thanks for your reply. On further inspection I realise that the book does in fact say that z is equal to 2. However this still leaves me a bit confused about the purpose of the quadratic equation since it seems it is superfluous to the requirements for providing a solution. The problem is question 1A, page 44 (in the pdf digital copy. Or 22 in the hard copy) of this book: http://www.scribd.com/doc/39412845/M...ics-Polya-1954
If you know z = 2 you can put this into the first and third equations (for example), to get two equations in the two unknowns x and y. One of the equations is linear, so that helps.
 Thanks for the help I'll try that.