When does a system of equations have a unique solution?

[uranium_235]

I know how to determine the determinant, and the inverse, and how to sovle a system with the inverse of a matrice, but I have no idea what these two questions are talking about:
1. Find the values of $$k$$ for which the system
$$kx+2y=1$$
$$3x+(k-1)y=1$$
does not have a unique solution. If $$k$$ does not have these values, find the unique solution. For each value of $$k$$ for which no unique solution exists, determine whether or not any solution of the system exists.

what is this question asking me for? What is a unique solution?

2. Find the values of $$k$$ for which the system
$$x+2y+2z=1$$
$$2x+y+2z=4$$
$$3x+3y+kz=5$$
has a unique solution. Find this unique solution and solve the system for any values of $$k$$ for which the unique solution does not exist.

again, wtf?

 

[dextercioby]

What's the condition for a nonhomogenous system to have unique solution?

Daniel.

 

[uranium_235]

I do not know what a unique solution is, nor have I ever heard the term nonhomogenous.

 

[dextercioby]

Then my advice is to study theory first and then get busy (trying to) solve(ing) problems...

Daniel.

 

[uranium_235]

this is not some independent study topic. This is from a practice booklet the teacher gave us, a question which the tacher recomnded we do. So, my current knowledge should suffice in answering this question.
What I know: -determinants
-inverses
-simple matrix operations
-solving systems with matrix inverses
-there being no solution if a system forms a singular matrix

 

[dextercioby]

The IV-th item of your "list" is the key.

Yes,setting the determinant of coefficients to zero will ensure nonuniqueness of solutions.

Daniel.

 

[uranium_235]

For the 2nd question, if nonuniqueness equates to there being no solution, the that is easy, k = 4, but for a unique solution, could not that be anything but for in the field of k?

 

[dextercioby]

I'm sorry,i really didn't understand your question.Could you please rephrase it using other words...?

Daniel.

 

[uranium_235]

I really don't know what a unique solution is. One of the answers it gives in the answers section is k cannot equal 4, which is the restriction should there be a solution to the system, if k = 4 detA = 0. So if k=4 means the system has no unique solutions, what does it mean if it does have unique solutions? How could you figure that out? Would not any value of k other than 4 bring about a system with unique solutions?

 

[dextercioby]

Yes,any value different than 4 would make unique solution.

Daniel.

 

[HallsofIvy]

I am sorely tempted to ask "Do you know what a "dictionary" is"?
(but I won't).

"Solution" a value (in this case a vector or set of numbers) that satisfies the equation.

"unique" only one.

You don't actually need to know anything about matrices to answer these questions.

kx+ 2y= 1
3x+ (k-1)y= 1

Try to solve the pair of equations. For what values of k can you NOT get a single (unique) solution? If you can get a single solution what is it?

Yes, you could write this as a matrix equation. You can solve for a single (unique) result as long as you can take the inverse of the matrix- as long as the determinant is not 0. What value of k makes the determinant 0?

 

[xanthym]

Also keep in mind that for a system of Linear Equations, there are only 3 possibilities:
1) No Solutions
2) 1 Solution (The "Unique" Solution)
3) Infinite Number of Solutions

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