Homogeneous systems

A group of Year 13 students is analysing different types of mathematical models. They encounter the following system of equations:

$\begin{cases} 3x - 2y + z = 0 \\ x + 4y - 5z = 0 \\ 2x - y + 3z = 0 \end{cases}$

Give reasons to explain why this system is classified as a homogeneous system and check whether the triple $(x, y, z) = (0, 0, 0)$ is a solution to it. What is the technical name for this solution?

Determine whether the following homogeneous system of two equations in two unknowns has only the trivial solution or if, on the contrary, it has infinitely many solutions:

$\begin{cases} 2x + 5y = 0 \\ 4x + 10y = 0 \end{cases}$

To do so, use the determinant of the coefficient matrix $A$.

Solve the following homogeneous system expressing its solutions in terms of two parameters (parametric form):

$\begin{cases} x - 2y + z = 0 \\ 2x - 4y + 2z = 0 \end{cases}$

Geometrically, each equation of a system of three equations in three unknowns represents a plane in space. In the case of homogeneous systems, all the planes share a common characteristic regarding their position relative to the coordinate axes.

Observe the following graph representing three intersecting planes:

If the highlighted point is the origin of coordinates, is it possible for a homogeneous system to be inconsistent (have no solution)? Justify your answer based on the geometric interpretation.

An architectural firm designs a structure based on three support beams whose internal stresses, represented by the variables $x$, $y$ and $z$, must be in equilibrium. The resulting mathematical model is the following homogeneous system:

$\begin{cases} x + y + z = 0 \\ 2x - y + 3z = 0 \\ 4x + y + 5z = 0 \end{cases}$

1. a) Justify, without solving the system, why it always admits at least one solution.
2. b) Calculate the rank of the coefficient matrix and determine whether the structure only has the "resting solution" (trivial) or if it allows other equilibrium configurations.
3. c) If non-trivial solutions exist, find their general expression.

Three technology companies, Alpha, Beta and Gamma, exchange consultancy services monthly. The cost of the services is offset in such a way that the net cash flow at the end of the quarter is zero for the global system. The balance of outstanding payments is modelled using the following homogeneous system, where $x, y, z$ represent the monthly payments of each company:

$$ \begin{cases} 2x - ay + z = 0 \\ ax - 4y + (a-2)z = 0 \\ x + 2y - z = 0 \end{cases} $$

Determine the value of the parameter $a$ for which the companies can make non-zero payments without breaking the system's equilibrium. For this value, calculate the ratio of proportionality between the payments of the three companies.

In a chemistry laboratory, three components are mixed to stabilize a reagent. The relative proportions $x, y, z$ must satisfy the system:

$\begin{cases} x + y - z = 0 \\ x + 3y + z = 0 \\ 3x + 10y + 4z = 0 \end{cases}$

Show that the system is underdetermined (consistent with infinitely many solutions) and find its solutions in terms of a parameter $\lambda$. If the laboratory requires that $z = 5$, what would be the value of the other two components?

A traffic control system in a low-emission zone of a Spanish city analyses the flow of vehicles in three main streets. The number of cars entering and leaving a central junction must satisfy the condition of zero net balance (entries = exits). The system is described by:

$$ \begin{cases} ax + y + z = 0 \\ x + ay + z = 0 \\ x + y + az = 0 \\ 2x + 2y + 2z = 0 \end{cases} $$

Analyse the consistency of this system of 4 equations in 3 unknowns depending on the parameter $a$. Is it possible for the system to be consistent dependent despite having more equations than unknowns? Justify your answer using the rank of the augmented matrix.