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.

Lucía solves on the board the discussion of the following system depending on the real parameter $m$:

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

After calculating the determinant of the coefficient matrix, she obtains $|A| = m - 1$ and concludes verbatim: "Since for $m = 1$ the determinant is zero, Cramer's rule cannot be applied and, therefore, the system is inconsistent. On the other hand, if $m \ne 1$, since $|A| \ne 0$, the system is consistent independent".

1. Identify the conceptual error in Lucía's statement and explain in detail why not being able to apply Cramer's rule does not imply that a system has no solution.

2. Discuss the system correctly and rigorously for all real values of $m$ by applying the Rouché-Frobenius theorem.

3. Solve the system for $m = 1$, clearly indicating the degrees of freedom of the solution.

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

$$\begin{cases} x + 2y - z = 3 \\ 2x + 4y - 2z = a \\ 3x + y + z = 1 \end{cases}$$

1. Estimate: Before calculating any determinant or applying Gaussian elimination, look carefully at the coefficients of the first two equations. Is it possible for this system to be consistent independent for any value of $a$? For what specific value of $a$ do you estimate that the first two equations will not be contradictory to each other?

2. Check: Calculate with justification the rank of the coefficient matrix $A$ and the rank of the augmented matrix $A^*$ as a function of $a$. Check formally whether your prediction was correct by applying the Rouché-Frobenius theorem.

3. Geometrically interpret the relative position of the three planes defined by the equations of the system in the cases $a = 6$ and $a = 0$.

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 $A \cdot X = B$.
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.

A recycling plant in the Valencian Community processes three types of municipal waste daily: Plastic ($x$), Glass ($y$), and Paper/Cardboard ($z$), measured in tonnes. Due to regional regulations on emissions, the daily processing balance is modeled using the following system of linear equations, where $\lambda$ represents a regulatory adjustment coefficient:

$$\begin{cases} x + y + z = 150 \\ 2x + 3y + z = 260 \\ 3x + 4y + \lambda z = 410 \end{cases}$$

1. Express the system in matrix form and discuss its consistency according to the different values of the regulatory parameter $\lambda \in \mathbb{R}$.

2. If the administration sets the coefficient to $\lambda = 2$, is it possible to know with certainty how many exact tonnes of each waste are processed per day? If on a day with $\lambda = 2$ the scale registers exactly $40$ tonnes of glass, determine the tonnes of plastic and paper that have been treated.

3. Is there any value of $\lambda$ for which the three regulatory conditions are mutually inconsistent? Justify your answer by analyzing the linear combination of the equations.

To discuss the system of equations depending on the real parameter $a$:

$$\begin{cases} x + ay + z = 1 \\ 2x + y - az = 2 \\ x - y + z = 0 \end{cases}$$

two students propose different strategies:

• Sergio proposes applying Gaussian elimination from the first step, triangulating the augmented matrix by performing row operations with the parameter $a$.

• Elena proposes first calculating the determinant of the coefficient matrix $|A|$, finding the critical values of $a$, and reserving the study by ranks or row reduction only for those isolated values.

• Analyze the drawbacks and computational risks involved in the path chosen by Sergio (what mandatory algebraic precautions does pivoting with parameters in Gaussian elimination require in order not to lose solutions or divide by zero?).

• Carry out Elena's strategy: calculate and factor $|A|$, and discuss the consistency of the system for all possible cases using the Rouché-Frobenius theorem.

• Justify which of the two methods is safer and more efficient in this particular case. For which value of $a$ could you solve the system using Cramer's rule with the least computational effort?

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 ($D$). The energy consumption per kilometer, maintenance cost, and payload capacity are defined by the following system, where $m$ represents an energy efficiency coefficient that varies depending on 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 for 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 if there is any value of $m$ for which it is possible to make deliveries 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 $x + y = z$.

Consider the following system of three equations with three unknowns depending on two real parameters $a$ and $b$:

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

1. Perform a reverse analysis: determine what value the parameter $a$ must necessarily take for the system to be consistent dependent regardless of the value that $b$ takes. Justify your reasoning by demonstrating the linear dependence of the rows of the coefficient matrix and the augmented matrix.

2. Find the exact relationship between $a$ and $b$ for the system to be inconsistent. Is there any combination of values of $a$ and $b$ for which the general solution depends on two free parameters (two degrees of freedom)? Prove why it is or is not possible based on the maximum rank of matrix $A$.

3. If $b = 3$ is fixed, determine for which value of $a$ the triple $(0, 0, 1)$ is part of the solution set, and classify the resulting system.

Consider the system of four equations with three unknowns depending on the real parameter $k$:

$$\begin{cases} x + y = 1 \\ y + z = 2 \\ x + z = k \\ x + 3y + kz = 5 \end{cases}$$

1. Estimate: Without calculating 4th-order determinants or row reducing the complete system, answer with conceptual justification:

a) Can this system be consistent dependent for any value of $k$? Justify your answer by analyzing the rank of the matrix formed solely by the first three equations.

b) Taking into account that the first three equations determine a unique solution $(x, y, z)$ as a linear function of $k$, and that substituting it into the fourth equation yields an expression in $k$, how many values of $k$ at most would you expect to make the system consistent?

1. Check:

a) Solve the subsystem formed by the first three equations to find $(x, y, z)$ in terms of $k$. Substitute the obtained expressions into the fourth equation and deduce the values of $k$ for which the complete system has a solution.

b) Set up the augmented matrix $A^*$ of dimension $4 \times 4$. Calculate its determinant $|A^*|$ with justification, factor it, and classify the complete system for all $k \in \mathbb{R}$ by applying the Rouché-Frobenius theorem.

1. Compare the results of part 2 with your estimates from part 1 and explain what geometric position the four planes occupy when $k = 2$.

Unit challenge. How does tax regulation affect the energy transition? Imagine that you are advising an energy consultancy firm and you must analyze the feasibility of the electricity generation mix.

1. Data collection: Go to the Red Eléctrica de España website (REE, Real-time demand and generation section) and collect the generation in MWh of three sources during a recent peak hour: wind ($x$), solar ($y$), and combined cycle ($z$), as well as the total demand.

2. Parametric modeling: Set up a $3 \times 3$ system. The first equation will represent the total coverage of demand ($x + y + z = D$). The second will model the observed proportion between renewables ($x - 1,5y = 0$). The third will evaluate the overall cost by assigning a tax parameter $k$ to gas emissions: $20x + 15y + kz = C$.

3. Algebraic analysis: Apply the Rouché-Capelli Theorem to classify the system according to the values of $k$. Solve it using Cramer's rule for the determinate consistent case and check the result using the Gaussian elimination method.

4. Simulation and report: Plot or solve the system in GeoGebra or in a spreadsheet by assigning a plausible real value to $k$ (for example, $k = 45,5$). Write a reasoned report (maximum one page) evaluating whether the results make physical and economic sense.

The following will be assessed:

• Rigor in the formal discussion with Rouché-Capelli and accuracy in algebraic solving (Gauss and Cramer).

• Correct extraction of data from REE and consistency in model formulation.

• Effective use of GeoGebra or computational software as a verification tool.

• Analytical quality and reasoned justification in the report's conclusions.