I have to determine for which values of parameters p and q the system has indefinite solutions.

System is:

(p-q)x + (3p-5)y = 2pq

(p+q)x + (q-7)y = 6pq

I have solved equation for x:

What next?

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- Dec 24th 2005, 03:45 PMBobanSystem of linear equattions
I have to determine for which values of parameters p and q the system has indefinite solutions.

System is:

(p-q)x + (3p-5)y = 2pq

(p+q)x + (q-7)y = 6pq

I have solved equation for x:

What next? - Dec 25th 2005, 03:15 AMCaptainBlackQuote:

Originally Posted by**Boban**

These equations represent a pair if lines in the plane.

They have no solutions when the lines are parallel, but not the same line;

they have a single unique solution if they are not parallel, and an infinite

number of solutions if they represent the same line.

These equations represent the same line when they are a multiple of

one another.

If

Then the RHS of the second equation is times the RHS of the

first equation, so the same must be true of the LHS. So multiplying

the first equation by 3 and then equating coefficients of and

, which gives:

,

.

Which is a pair of linear equations which can be solved for and

to give and which give an infinite number of solutions

to the original equations.

If

Both equations reduce to , so there are an infinite

number of solutions (that is any point on the x-axis is a solution).

If

The equations become:

,

which implies that .

If

Left as an exercise to the reader.

RonL - Dec 25th 2005, 06:08 AMBoban
The solution in the book is

and

Quote:

If

Then the RHS of the second equation is times the RHS of the

first equation, so the same must be true of the LHS. So multiplying

the first equation by 3 and then equating coefficients of and

, which gives:

,

.

You have multiple first equation by 3 and how did you get this? What happened to x and y? - Dec 25th 2005, 08:20 AMCaptainBlackQuote:

Originally Posted by**Boban**

solutions , . so

the answer in the book is incomplete.

RonL - Dec 25th 2005, 08:37 AMCaptainBlackQuote:

Originally Posted by**Boban**

two equations are equal. Then as these are to represent the same

line the the coefficients of and on the LHS must be the same

in each equation.

For example if

with represent the same line then and

.

Note the book's solution is in fact the solution of the equations:

,

.

RonL - Dec 25th 2005, 08:59 AMCaptainBlackQuote:

Originally Posted by**Boban**

Assume what you have done so far is correct. Then for any values of

and you have a well defined value of , except when:

you solve this equation for and to give candidate values for

the original equations to have an infinite number of solutions. These need to

be substituted back into the original equations to check that they give the

required type of solutions.

Then repeat the procedure solving for in terms of and .

You can try this method but I know I'd rather use the other.

RonL - Dec 25th 2005, 11:19 AMBoban
Thanks for help!