6
votes
Accepted
Cartesian equation for $((\sin\theta)^3+\cos\theta,(\cos\theta)^3+\sin\theta)$
To find a polynomial $P(x,y)$ such that:
$$
P\left(\sin^3\theta+\cos\theta,\,\cos^3\theta+\sin\theta\right) = 0
\quad \quad \forall\,\theta \in [0,2\pi)
$$
there are at least two automated approaches.
...
- 4,938
4
votes
System of equations on $\mathbb R$
If you write your system of equations in a reverse order :
$$\begin{cases}tx+12y-3z&=&4\\-12x+ty+4z&=&3\\3x-4y+tz&=&12\end{cases} \iff $$
$$\begin{cases}12y-3z&=&4-tx\\-...
- 74.9k
3
votes
Accepted
System of equations on $\mathbb R$
Solve the set of equations you have $$3x-4y+tz=12$$ $$-12x+ty+4z=3$$ $$tx+12y-3z=4$$for $x$, $y$ and $z$ in terms of $t$. It looks as if it will be horrible, but in fact you can eliminate $y$, for ...
- 2,733
2
votes
On the system of functional equations $f(x^y)=\sqrt[y]{f(x)}$ and $ f(x^y)f(x^{-y})=1 $
I know Gonçalo's answer has been accepted already, but here is another approach on how to solve this problem, perhaps in a more systematic way.
First things first, it is to be noted that the second ...
- 3,781
2
votes
Accepted
On the system of functional equations $f(x^y)=\sqrt[y]{f(x)}$ and $ f(x^y)f(x^{-y})=1 $
Here is a partial solution: assuming differentiability of $f$, if we take the derivative of both sides of the first equation with respect to $y$ we obtain
$$
f'(x^y)\,x^y\ln x = -\frac{1}{y^2}\ln f(x)\...
- 1,656
2
votes
Accepted
Solve the equation $x^4-2x^3-21x^2+22x+40=0$ whose roots are in A.P.
Generally when considering AP's with 3,4,5 terms respectively, it makes it easier if we write the terms as: (which makes calculating terms of the AP much faster(especially sum and product) )
$3: a-...
- 737
1
vote
System of equations on $\mathbb R$
I don’t know why you equate $\Delta$ to zero. Instead, at that juncture, you could have easily apply Cramer's rule as shown below.
$$x=\dfrac{
\begin{bmatrix}
12 & -4& t \\
3 & t & 4 \\...
- 3,493
1
vote
Cartesian equation for $((\sin\theta)^3+\cos\theta,(\cos\theta)^3+\sin\theta)$
The Maple function eliminate() eliminates the parameter immediately:
...
- 89.4k
1
vote
Cartesian equation for $((\sin\theta)^3+\cos\theta,(\cos\theta)^3+\sin\theta)$
An experimental answer, in need of explanations :
I used Geogebra for sketching the initial curve (green curve above). It is made of 2 symmetric arcs, one obtained for $\theta \in [0,\pi)$, the other ...
- 74.9k
1
vote
Accepted
Given x, y are real number that are not equal to 1. Find all x, y that satisfy:
The resultant of the two polynomials $f=x^8+y^7-x(1+y^7)$ and $g=y^8+x^7-y(1+x^7)$ is given by a polynomial in $x$, which has four real roots, namely $x=0,1,-1$ and $x=0.502017055178$, which is the ...
- 125k
1
vote
Accepted
Solving a system of linear equations involving tensors
This system is linear for $D_{i,j}$ variables. there are $n^2$ variables and $n^2$ equations so you can solve it with changing this to linear system. you should just rearrange matrix notations in $M_{...
- 1,075
1
vote
Elements sum rule for special matrix
Once you have defined $S_1$, $S_2$ and $M$, you can forget the fact that you have a matrix. It will not give you any more structure.
Now, Rodrigo is right in that your system of equations are ...
- 1,729
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