I need some help understanding some concepts (long post)

• Jul 1st 2008, 03:26 PM
jschlarb
I need some help understanding some concepts (long post)
Okay, my professor hasn't been very reliable and my midterm is on thursday so this is the only place where I can get help.

The first concept is solving variable to systems where the variable have unique, no and infinite solutions.

Here's an example:

For which a, b does the following system:

3x-y=b
7x+ay=1

have i) unique solution ii) infinite solutions iii) no solution?

The second concept (two parts) is of linear matrices.

Here are examples:

1) Solve the system:

3x+2y-z=-15
6x+2y+6z=22
-12x-8y+4z=60

Now, I know for this question we want to get the matrix to be in reduced echelon form. My question for this is: Is there any specific method I should be using to do this or should I treat it as a puzzle?

2) Write a system of linear equations that has the following matrix as its augmented matrix

( 0 1 0 -1 2 3 7)
(7 3 2 2 1 2 0)
(1 2 -1 0 0 0 0)

(sorry, thats the best I could do to make it pretty)

Any help and links would be greatly appreciated.
• Jul 1st 2008, 04:11 PM
Plato
It is often very easy to blame an instructor. As a retired chair of a department of mathematical sciences, I can tell you that only about 50% are valid.
Now what have you done on these problems?
It is only fair to help you if you can demonstrate a basic understanding of the material by showing what you have tried.
• Jul 1st 2008, 04:45 PM
jschlarb
Alright, here's what I know for the first part:

A unique solution means there is one intersection for the two lines. so there is a single x = ? and y = ? that solves the system. We have a unique solution when the two straight lines intersect

We have infinite solutions when the lines are the same. One line lies on top of the other, so there are an infinite amount of intersecting points

We have no solutions if the lines never intersect, that is, if the lines are parallel.

For the second concept, I believe our overall goal is to find the solution to the 3 equations (find x,y and/or z). So, if we were to put all 3 questions into a matrix, it would look something like this:

(3 2 -1 | -15)
(6 2 6 | 22)
(-12 -8 4 | 60)

We would perform as many row operations as required in order to reduce the matrix to the identity form as much as possible.

For the third one, I think what is being asked it that I take the set of numbers in each row and turn it into an equation

for example:

0 1 0 -1 2 3 7 = b-c+2d+3e=7

I'm not certain about that though.
• Jul 1st 2008, 05:29 PM
ticbol
Let me do the first. For the matrices, I never liked matrix so I will not touch those.

For which a, b does the following system:

3x -y = b ------(1)
7x +ay = 1 -----(2)

have i) unique solution ii) infinite solutions iii) no solution?

Your book/teacher/lesson could have ways of doing this. I don't know those ways. I will do this by, er, "common sense" only.

Since there are supposed to be two variables, x and y, (actually, as the two equations go, there 4 variables: a and b are the other two), I will express one into the other and then see what will happen.

From Eq.1, y = 3x -b
Substitute that into Eq.2,
7x +a(3x -b) = 1
7x +3ax -ab = 1
x(3a +7) = ab +1
x = (ab +1)/(3a +7) --------(3)

So, for a unique solution,
in (3), a and b must be zeros.
Then x = 1/7,
and then y = 3/7.

For infinite solutions,
in (3), a and b can be any number, including zero, but not when a and b are both zeroes.

For no solution,
in (3), x has no solution if the denominator (3a +7) = 0, or for a = -7/3.
Hence, no x, no y.