Exercises: 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 $k$.

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 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 $(0, 0, 0)$?

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).

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.

Consider the system:

$$\begin{cases} 2x - y + z = 3 \\ x + y + z = 2 \\ x - 2y = k \end{cases}$$

Find the value of the parameter $k$ such that the system is Consistent with Infinitely Many Solutions. For that value of $k$, state how many solutions the system has.

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.

Discuss the following system according to the values of the parameter $t$:

$$\begin{cases} tx + y = 1 \\ x + ty = t \\ x + y = 1 \end{cases}$$

Hint: Analyse what happens when $t=1$ and when $t \neq 1$ by observing the relationship between the three equations.

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.

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.

A logistics company organises shipments of three types of containers: A, B, and C. The shipping cost depends on a variable tax $m$ applied to management. The system relating the number of containers to the total budget is:

$\begin{cases} x + y + z = m \\ x + my + z = 1 \\ mx + y + z = 1 \end{cases}$

Discuss whether it is possible that, for some value of $m$, the company cannot uniquely determine how many containers of each type have been sent (that is, that the system is not consistent with a unique solution).

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.

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.

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.

It is known that the following system has infinitely many solutions for a certain value of $a$:

$\begin{cases} x + 2y - z = 2 \\ 2x + ay + 2z = 3 \\ x - y + 3z = a \end{cases}$

1. a) Find that value of $a$.
2. b) Without performing further calculations, explain with reasoning whether the system could be inconsistent for any other value of $a$.

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.

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.

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.

A last-mile logistics company uses three types of vehicles: electric vans ($F$), cargo tricycles ($T$), and delivery drones ($D$). Energy consumption per kilometer, maintenance costs, and load capacity are defined by the following system, where $m$ represents an energy efficiency coefficient that varies according to the outside temperature:

$\begin{cases} mF + T + D = 10 \\ F + mT + D = m + 9 \\ F + T + mD = 24 - 14m \end{cases}$

1. Discuss the feasibility of the system (consistency) depending on $m$.

2. If on a particularly cold day ($m=2$) it is known that the system is inconsistent, explain what this means from the perspective of the real-world problem.

3. Determine whether there is any value of $m$ for which it is possible to make the delivery using only drones ($F=0, T=0$).

Consider the system of linear equations depending on two real parameters $a$ and $b$:

$\begin{cases} x + ay + z = 2 \\ 2x + 4y + az = b \\ x + y + z = a \end{cases}$

1. Discuss the system in terms of the parameters $a$ and $b$.

2. For the value of $a$ that makes the system consistent with infinitely many solutions, find the required value of $b$ and solve the system.

3. Is there any value of $a$ for which the system is inconsistent regardless of the value of $b$?

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.

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.

We wish to study the intersection of four planes in space, whose equations are:

$\begin{cases} x + y + z = 2 \\ x + 2y - 3z = 8 \\ ax - y - z = 1 \\ x - y + z = -2 \end{cases}$

1. Use the Rouché–Frobenius theorem to determine whether there is any value of $a$ for which all four planes intersect at a single point.

2. If so, calculate the coordinates of that point.

3. If we replace the third equation with $ax - y - z = b$, what relation must $a$ and $b$ satisfy so that the system still has a solution?

Let the system of equations with parameters $k$ and $t$ be:

$\begin{cases} x + y + z = t \\ kx + 2y + z = 0 \\ x + ky + z = 0 \end{cases}$

1. Discuss the system according to the values of $k$ and $t$.

2. For $k=1$, what condition must $t$ satisfy for the system to have a solution? Interpret the result considering that the equations represent planes.

3. For $k=2$, solve the system in terms of $t$ and determine the value of $t$ for which the solution satisfies $x + y = z$.