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?
It's simpler than that.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
The solution in the book is
and
I didn't understand this part.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?
As I said you multiply the first equation by 3, then the RHS of theOriginally 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
Alternative method: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