Mixing
Show that all solutions are of the form $(t,t^2,t^3)$
$begingroup$
I want to show that there exists a global map of a submanifold described as the solution of the equations below. So I found one map, but I need to know if it contains all possible solutions.
How does one show that all real solutions of the system of equations:
$x^2+xy-y-z = 0$
$2x^2+3xy-2y-3z = 0$
are of the form: $(t,t^2,t^3)$ ?
real-analysis systems-of-equations submanifold
asked Jan 31 at 17:38
SlyderSlyder
316
$endgroup$
add a comment |
$begingroup$
I want to show that there exists a global map of a submanifold described as the solution of the equations below. So I found one map, but I need to know if it contains all possible solutions.
How does one show that all real solutions of the system of equations:
$x^2+xy-y-z = 0$
$2x^2+3xy-2y-3z = 0$
are of the form: $(t,t^2,t^3)$ ?
real-analysis systems-of-equations submanifold
asked Jan 31 at 17:38
SlyderSlyder
316
$endgroup$
1
$begingroup$
Solve one equation or the other for $z$ and substitute, then the result becomes a linear equation in $y$ with the free parameter $x$. The given solution parametrizes everything by $x$ itself.
$endgroup$
– Ian
Jan 31 at 17:42
$begingroup$
Thank you!.....
$endgroup$
– Slyder
Jan 31 at 17:46
add a comment |
$begingroup$
I want to show that there exists a global map of a submanifold described as the solution of the equations below. So I found one map, but I need to know if it contains all possible solutions.
How does one show that all real solutions of the system of equations:
$x^2+xy-y-z = 0$
$2x^2+3xy-2y-3z = 0$
are of the form: $(t,t^2,t^3)$ ?
real-analysis systems-of-equations submanifold
asked Jan 31 at 17:38
SlyderSlyder
316
$endgroup$
I want to show that there exists a global map of a submanifold described as the solution of the equations below. So I found one map, but I need to know if it contains all possible solutions.
How does one show that all real solutions of the system of equations:
$x^2+xy-y-z = 0$
$2x^2+3xy-2y-3z = 0$
are of the form: $(t,t^2,t^3)$ ?
real-analysis systems-of-equations submanifold
real-analysis systems-of-equations submanifold
asked Jan 31 at 17:38
SlyderSlyder
316
asked Jan 31 at 17:38
SlyderSlyder
316
asked Jan 31 at 17:38
SlyderSlyder
316
asked Jan 31 at 17:38
SlyderSlyder
316
asked Jan 31 at 17:38
SlyderSlyder
316
316
1
$begingroup$
Solve one equation or the other for $z$ and substitute, then the result becomes a linear equation in $y$ with the free parameter $x$. The given solution parametrizes everything by $x$ itself.
$endgroup$
– Ian
Jan 31 at 17:42
$begingroup$
Thank you!.....
$endgroup$
– Slyder
Jan 31 at 17:46
add a comment |
1
$begingroup$
Solve one equation or the other for $z$ and substitute, then the result becomes a linear equation in $y$ with the free parameter $x$. The given solution parametrizes everything by $x$ itself.
$endgroup$
– Ian
Jan 31 at 17:42
$begingroup$
Thank you!.....
$endgroup$
– Slyder
Jan 31 at 17:46
1
1
$begingroup$
Solve one equation or the other for $z$ and substitute, then the result becomes a linear equation in $y$ with the free parameter $x$. The given solution parametrizes everything by $x$ itself.
$endgroup$
– Ian
Jan 31 at 17:42
$begingroup$
Solve one equation or the other for $z$ and substitute, then the result becomes a linear equation in $y$ with the free parameter $x$. The given solution parametrizes everything by $x$ itself.
$endgroup$
– Ian
Jan 31 at 17:42
$begingroup$
Thank you!.....
$endgroup$
– Slyder
Jan 31 at 17:46
$begingroup$
Thank you!.....
$endgroup$
– Slyder
Jan 31 at 17:46
add a comment |
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$begingroup$
Solve one equation or the other for $z$ and substitute, then the result becomes a linear equation in $y$ with the free parameter $x$. The given solution parametrizes everything by $x$ itself.
$endgroup$
– Ian
Jan 31 at 17:42
$begingroup$
Thank you!.....
$endgroup$
– Slyder
Jan 31 at 17:46