Systems with parameters

On a streaming platform, the price of three types of subscriptions (Basic, Standard and Premium) varies according to a monthly promotion represented by the parameter $k$. The system of equations relating the number of users of each type to total revenue is:

$$\begin{cases} x + y + z = 10 \\ x + ky + z = 12 \\ x + y + kz = 14 \end{cases}$$

Discuss the consistency of the system according to the values of the real parameter $k$.

Given the system of equations:

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

1. a) Determine for which values of $a$ the system is consistent with a unique solution.
2. b) Is there any value of $a$ for which the system represents three planes intersecting along the same line? Justify your answer using the concept of rank.

Let $A$ be the coefficient matrix and $B$ be the matrix of constant terms:

$A = \begin{pmatrix} 1 & 1 & 1 \\ 0 & a & 1 \\ 1 & 1 & a \end{pmatrix}, \quad B = \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}$

1. a) For which values of $a$ does the inverse matrix $A^{-1}$ not exist?
2. b) For the values found in the previous part, discuss the system $A \times X = B$.
3. c) Solve the system for $a = 2$ using Cramer's rule.

Let the matrix $A$ and the vector $B$ be defined by:

$A = \begin{pmatrix} 1 & 0 & -1 \\ 0 & m & 3 \\ 4 & 1 & -m \end{pmatrix}, \quad B = \begin{pmatrix} 1 \\ -1 \\ m \end{pmatrix}$

1. Determine the values of $m$ for which the matrix $A$ is non-invertible.

2. For the values of $m$ for which $A$ is invertible, solve the matrix equation $AX = B$ using Cramer's rule.

3. For $m=3$, study whether the system $AX=B$ is consistent or inconsistent without calculating the inverse.

Determine for which value of the parameter $a$ the following system of linear equations is inconsistent:

$$\begin{cases} x + 2y = 3 \\ ax + 4y = 1 \end{cases}$$

Justify your answer by analysing the proportionality of the coefficients and the constant terms.

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

$\begin{cases} kx + y + z = 1 \\ x + ky + z = k \\ x + y + kz = k^2 \end{cases}$

1. a) Discuss the system according to the values of the parameter $k$.
2. b) Solve the system in the case where it is consistent with infinitely many solutions.

Consider the following homogeneous system of equations, where $m$ is a real parameter:

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

1. a) Calculate the determinant of the coefficient matrix $A$.
2. b) For which values of $m$ does the system have solutions other than the trivial solution $(0, 0, 0)$?

Study the consistency of the following homogeneous system depending on the parameter $a$:

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

Calculate all its solutions for the value of $a$ that makes it consistent with infinitely many solutions.

Given the homogeneous system:

$\begin{cases} ax + y + 3z = 0 \\ x + ay + 2z = 0 \\ x + y + az = 0 \end{cases}$

1. Determine the values of $a$ for which the system admits non-trivial solutions.

2. For the largest value of $a$ found in the previous part, obtain the general solution and find a particular solution such that the sum of its components is $10$.

3. If $a=0$, describe the solution set geometrically.

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

$\begin{cases} kx + y + z = k^2 \\ x + ky + z = k \\ x + y + kz = 1 \end{cases}$

You are required to:

1. Discuss the system according to the different values of the parameter $k$.

2. Solve the system in the cases where it is consistent with infinitely many solutions, expressing the solution in parametric form.

3. If the system is consistent with a unique solution, is there any value of $k$ for which the solution is $(x, y, z) = (1, 1, 1)$? Justify your answer.

Given the system of equations with parameter $a \in \mathbb{R}$:

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

Explain with reasons why this system can never be Consistent with a Unique Solution, regardless of the value that $a$ takes.

It is known that the triplet $(x, y, z) = (1, 2, -1)$ is a solution to the following system of equations depending on the parameter $b$:

$$\begin{cases} bx + y + z = 3 \\ x + by - z = 6 \\ x - y + bz = -3 \end{cases}$$

1. a) Substitute the solution into the equations to find the value of $b$.
2. b) With the obtained value of $b$, discuss whether the given solution is unique.

Study the following system of linear equations depending on the parameters $a$ and $b$:

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

1. a) Explain why the system can never be consistent with a unique solution.
2. b) If $a = 1$, what condition must $b$ satisfy for the system to be consistent with infinitely many solutions?
3. c) Geometrically interpret what these two equations represent in space.

A video game designer is programming the collisions of three projectiles whose trajectories in space are defined by the following planes, where $\lambda$ is a power adjustment variable:

$$\pi_1: x + y + z = 2$$ $$\pi_2: 2x + 3y + z = 3$$ $$\pi_3: \lambda x + 10y + 4z = 11$$

Find the value of $\lambda$ such that the three projectiles meet at a single point in space (Consistent System with a Unique Solution).

Consider the three planes defined by the following equations:

$\pi_1: x - y + z = 2$

$\pi_2: 2x + y - z = 1$

$\pi_3: x + 2y + kz = -1$

1. a) Find the value of $k$ so that the three planes have infinitely many points in common.
2. b) For $k = -2$, determine the relative position of the three planes.

Three planes in space $\mathbb{R}^3$ are given by the following equations, where $a$ is a real parameter:

$\pi_1: ax + y + z = 1$

$\pi_2: x + ay + z = a$

$\pi_3: x + y + az = a^2$

Analyse the relative position of the three planes depending on the values of $a$. In particular, determine for which values of $a$ the planes intersect along a single line and for which values the three planes have no point in common simultaneously.