How to find the general solution of these linear equation systems?

[popokolok]

Hi, please help me out...
There are 2 equation systems:
1)
ax + by + cz = d
fx + gy + hz = k

2)
ax + by + cz = 0
fx + gy + hz = 0

It's given that (2,-3,1) is a solution for the first and (1,0,1) and (-1,1,1) are solutions for the second.

How do I find the general solution(s) for both systems?

Thank you in advance, and very much appreciated!

 

[Deleted member 4993]

Hi, please help me out...
There are 2 equation systems:
1)
ax + by + cz = d
fx + gy + hz = k

2)
ax + by + cz = 0
fx + gy + hz = 0

It's given that (2,-3,1) is a solution for the first and (1,0,1) and (-1,1,1) are solutions for the second.

How do I find the general solution(s) for both systems?

Thank you in advance, and very much appreciated!

What are the unknowns (to be solved) and what are the constants (in the equations you have posed)?

If there are 3 unknowns and you are given two equations - you will end up with infinite solutions - related by one equation with two variables (unknowns).

For example in problem (1) you'll end up with an equation, if you eliminate "x" (assuming that was one of your intended unknowns).

Please share your work with us, indicating exactly where you are stuck - so that we may know where to begin to help you.

 

[popokolok]

Hi, thank you for replying! Actually this is exactly how the question appears in my book, so I'm not sure what to add. The unknowns are xyz, all others are constants - I'm looking for the general solution that represents the infinite solutions (I guess...). I'm stuck pretty much at the beginning, as I'm unsure of how to tackle this one - how would you solve it?

Thank you very much!

 

[HallsofIvy]

Hi, please help me out...
There are 2 equation systems:
1)
ax + by + cz = d
fx + gy + hz = k

Assuming you mean "find all values of x, y, and z that satisfy these equations for given a, b,, c, d, e, f, g, h, k, then the fact that (2, -3, 1) is a solution tells us that 2a- 3b+ c= d and 2f- 3g+ h= k. That is, the two equations can be written ax+ by+ cz= 2a- 3b+ c and fx+ gy+hz= 2f- 3g+ h. Now, for example, eliminate x by multiplying the first equation by f, the second equation by a, and subtracting: (bf- ga)y+ (cf- ah)z= -3bf+ cf+ 3ag- ah. If \(\displaystyle bf- ga\ne 0\), you can solve for y in terms of z, then get x in terms of z. If \(\displaystyle bf- ga= 0\) but \(\displaystyle cf- ah\ne 0\), you can solve for z, then solve for x in terms of y. Tedious but pretty straightforward.

2)
ax + by + cz = 0
fx + gy + hz = 0

It's given that (2,-3,1) is a solution for the first and (1,0,1) and (-1,1,1) are solutions for the second.

How do I find the general solution(s) for both systems?

Thank you in advance, and very much appreciated!

 

[popokolok]

Thank you, I see, but how would the general solution look?

Thank you!

 

[mmm4444bot]

I see

Excellent!

how would the general solution look?

How would it look? Based on what condition? :?

I'm thinking that you're probably trying to ask how it will look.

If so, then you'll know how it looks after you do the work.

Let us know, if you get stuck. Please show your efforts to that point, explain your thoughts, or ask a specific question.

Cheers :cool:

 

[mmm4444bot]

If this is being given in a beginning algebra class to 8th or 9th graders, I believe it is far enough over the top to warrant solution.

I doubt that this exercise was given in a beginner's course, if it was given at all. (Thank you for drawing my attention to the original board; I moved this thread.) :cool:

When I read "general solution", I envision statements for each of x,y,z in terms of a,b,c,d,f,g,h,k or a,b,c,f,g,h -- as the case may be. Yet, I'm not sure what the poster is trying to accomplish here.

 

[mmm4444bot]

I'm not even sure that those solutions belong to the system as posted. For all I know, the poster may have been given a number of systems and general solution method(s), and now they're fishing for some kind of shortcut. I'm going to wait for feedback.

My classical-geometry class (in the 10th grade) was taught by a stone-faced, retired naval guy. He provided us with geometry textbooks published by the US Navy (pretty much all text -- theorems, corollaries, etc -- very few diagrams). I got a D.

I still remember the day in which two classmates showed up late. The door flew open, and one of them stepped over the threshold, blew a signal whistle, and barked, "Permission to come aboard, sir!". They were both dressed in the old WWII naval undress-whites with caps and scarfs. It was the only time that I ever saw that instructor crack a hint of a smile. :cool: