MHB Solve System of Equations: 4x^2+25y^2+9z^2-10xy-15yz-6xz=0

AI Thread Summary
The discussion focuses on solving the system of equations defined by the quadratic equation 4x^2 + 25y^2 + 9z^2 - 10xy - 15yz - 6xz = 0 and the linear equation x + y + z = 5. Participants emphasize the complexity of finding solutions to this system, noting that it may have limited or no solutions. The quadratic equation suggests a conic section, which complicates the solution process. Ultimately, the consensus is that no other solutions exist beyond the ones identified. The thread concludes with the assertion that the system is solvable under specific conditions.
kaliprasad
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Solve the system of equations
$4x^2+25y^2 +9z^2 - 10xy -15yz - 6xz = 0\cdots(1)$
$x+y+z=5\cdots(2)$
 
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kaliprasad said:
Solve the system of equations
$4x^2+25y^2 +9z^2 - 10xy -15yz - 6xz = 0\cdots(1)$
$x+y+z=5\cdots(2)$
my solution:
rearrange (1) we have :
$2x(2x-5y)+5y(5y-3z)+3z(3z-2x)=0--(3)$
for any $2x=5y=3z, (3)$ will be zero
so we have :$x:y:z=15:6:10---(4)$
from $(2)$ we have :$x=\dfrac{75}{31},y=\dfrac{30}{31},z=\dfrac{50}{31}$
 
Albert said:
my solution:
rearrange (1) we have :
$2x(2x-5y)+5y(5y-3z)+3z(3z-2x)=0--(3)$
for any $2x=5y=3z, (3)$ will be zero
so we have :$x:y:z=15:6:10---(4)$
from $(2)$ we have :$x=\dfrac{75}{31},y=\dfrac{30}{31},z=\dfrac{50}{31}$

Above is a right solution. kindly find other solutions or prove that no other solution exists
 
kaliprasad said:
Solve the system of equations
$4x^2+25y^2 +9z^2 - 10xy -15yz - 6xz = 0\cdots(1)$
$x+y+z=5\cdots(2)$
no other solution exists
$(1)\times 2$ we have:
$8x^2+50y^2+18z^2-20xy-30yz-12zx=0$
or $(2x-5y)^2+(5y-3z)^2+(3z-2x)^2=0$
$\therefore 2x=5y=3z$ must hold
 
kaliprasad said:
Solve the system of equations
$4x^2+25y^2 +9z^2 - 10xy -15yz - 6xz = 0\cdots(1)$
$x+y+z=5\cdots(2)$

The first equation is a quadric.
The first step is to analyze it and categorize it.
Let:
$$A = \begin{bmatrix}4& -5 & -3 \\ -5 & 25 & -15/2 \\ -3 & -15/2 & 9 \end{bmatrix}$$
Then (1) can be written as:
$$\mathbf x^T A \mathbf x = 0$$

$A$ is a real symmetric matrix, meaning it is diagonalizable with real eigenvalues and an orthonormal set of eigenvectors.
Its characteristic equation is:
$$\lambda^3 - 38\lambda^2 + \frac{1083}{4}\lambda = 0$$
This implies we have eigenvalues $\lambda_1 = 0$, and $\lambda_2, \lambda_3 > 0$.
So:
$$\mathbf x^T A \mathbf x = \mathbf x^T B^T \begin{bmatrix}0& 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & \lambda_3 \end{bmatrix} B \mathbf x = 0$$
where $B$ is an orthogonal matrix formed by the corresponding eigenvectors.

Consequently, we're looking at a degenerate quadric that is similar to:
$$\lambda_2 (y')^2 + \lambda_3 (z')^2 = 0$$
This is a line through the origin, since $\lambda_2, \lambda_3 > 0$.

The direction of the line is given by the eigenvector for $0$, and the line is also the kernel of $A$.
Solving $A\mathbf x=0$, we find:
$$\mathbf x = t\begin{pmatrix}15 \\ 6\\ 10\end{pmatrix}$$

Combining with (2) tells us that:
$$x=\frac{75}{31},y=\frac{30}{31},z=\frac{50}{31}$$

Just like Albert found. ;)
 
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