Systems depending on a parameter

Consider the following system of two equations with two unknowns, where $k$ is a real number:

$$\begin{cases} kx + y = 3 \\ 4x + y = 2 \end{cases}$$

1. a) Determine for which value of $k$ the system does not have a unique solution.

1. b) If we choose $k = 5$, what can we say about the relative position of the two lines that represent the equations?

A small app development company charges for two types of services: maintenance ($x$) and consultancy ($y$). The system relating the income over two months is:

$$\begin{cases} x + 2y = 500 \\ 2x + ay = 1000 \end{cases}$$

1. a) Write down the matrix of coefficients $A$ and the augmented matrix $A^*$ associated with the system.
2. b) For which value of $a$ does the system have infinitely many solutions? What does this mean in the context of the services provided?

Given the system depending on the real parameter $m$:

$$\begin{cases} x + y + z = 1 \\ x - y + z = m \\ 2x + 2z = 2 \end{cases}$$

Use Gaussian elimination to show that, for any value of $m \neq 1$, the system is inconsistent. What happens if $m = 1$?

A courier company uses three types of vehicles (vans, motorbikes and drones) to make its deliveries. It is known that the daily operating cost of each vehicle depends on a variable maintenance factor $k$. The total expenditure on three different routes is given by the following system:

$$\begin{cases} x + 2y + kz = 3 \\ 3x + ky = 6 \\ x + y + z = 2 \end{cases}$$

Where $x, y, z$ represent the number of operational vans, motorbikes and drones, respectively.

1. a) Analyse the consistency of the system for the different values of $k$.

1. b) For $k = 4$, determine whether the system has a unique solution. If so, calculate it using Cramer's rule.

You are given the following three planes in three-dimensional space, whose equations depend on the parameter $m$:

$$\pi_1: mx + y + z = 0$$ $$\pi_2: x + my + z = 0$$ $$\pi_3: x + y + mz = 0$$

1. a) Determine for which values of $m$ the three planes intersect only at the origin of coordinates $(0, 0, 0)$.

1. b) Find the values of $m$ for which the system formed by the three equations is consistent and underdetermined (has infinitely many solutions). For each of those values, give the geometric interpretation of the relative position of the three planes.

An "influencer" is analyzing the evolution of his followers on three social networks (TikTok, Instagram and Twitch). Let $x, y, z$ be the net change in daily followers (gains minus losses, so it can be negative) on TikTok, Instagram and Twitch, respectively. These changes satisfy the following monthly conditions, where $m$ is a virality coefficient:

$$\begin{cases} x + y + z = m \\ 2x + 3y + mz = 5 \\ x + my + 3z = 8 \end{cases}$$

1. a) Discuss the system as a function of $m$.

1. b) If for a given month the virality coefficient is $m = 2$, what is the net change in daily followers on each social network?

Consider the following system of equations depending on the real parameter $a$:

$$ \begin{cases} x + ay + z = a \\ ax + y + z = 1 \\ x + y + az = a^2 \end{cases} $$

1. a) Discuss the system according to the different values of the parameter $a$.
2. b) Solve the system, if possible, for the case where it is consistent with infinitely many solutions and for the case $a = 0$.

Given the system of equations with two real parameters $a$ and $b$:

$$ \begin{cases} x + y + z = 1 \\ ax + 2y + 3z = 2 \\ x + ay + 3z = b \end{cases} $$

1. a) Discuss the system depending on the values of $a$ and $b$.
2. b) For the values that make the system consistent with infinitely many solutions, find the general solution, expressing it in terms of a parameter $\lambda$.