Exercises: 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 $m = 1$?

In the analysis of a financial market, it has been determined that the price equilibrium of three assets is given by the following system:

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

Discuss the consistency of the system according to the values of the parameter $p$ (price of the third asset). State when the system is consistent with a unique solution (SCD).

Discuss the following homogeneous system depending on the parameter $a$:

$$\begin{cases} x + ay = 0 \\ ax + 9y = 0 \end{cases}$$

Remember that homogeneous systems are always consistent. State for which values of $a$ the system has only the trivial solution $(0,0)$ and for which it has infinitely many solutions.

It is known that the following system of linear equations has a unique solution for a specific value of the parameter $t$:

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

1. a) Calculate the determinant of the matrix of coefficients in terms of $t$.
2. b) State for which values of $t$ the system is consistent with a unique solution.
3. c) Study what happens if $t = -1$.

A nutritionist designs a diet with two foods. The protein and carbohydrate content depends on an enrichment factor $k$ according to the system:

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

1. a) Find the value of $k$ that makes the system consistent with infinitely many solutions.
2. b) For that value of $k$, is there any solution where the food quantities $(x, y)$ are negative? Briefly justify your answer.

Consider the system of three equations with three unknowns:

$$\begin{cases} x + y + z = 0 \\ 2x + 3y + 2z = 0 \\ x + y + (b^2-3)z = 0 \end{cases}$$

1. a) Classify the system in terms of the parameter $b$.
2. b) Is there any value of $b$ for which the system is inconsistent? Justify your answer based on the theory of homogeneous systems.

David is solving a university entrance exam problem with the following homogeneous system depending on the real parameter $m$:

$$\begin{cases} x + y + z = 0 \\ 2x + my + 2z = 0 \\ 3x + 2y + z = 0 \end{cases}$$

David calculates the determinant of the coefficient matrix and obtains $|A| = 4 - 2m$. After setting it to zero, he deduces that the root is $m = 2$ and writes on his exam:

“For $m \ne 2$, the determinant is non-zero, so the system is Consistent Determined and the only solution is the trivial one $(x, y, z) = (0, 0, 0)$. For $m = 2$, the determinant is zero; therefore, the rank of the coefficient matrix decreases, does not match the number of unknowns, and the system becomes Inconsistent, with no solutions at all.”

1. a) Identify the conceptual error made by David in his conclusion for $m = 2$. Give a reasoned explanation of why a homogeneous system can never be inconsistent, relying on the Rouché-Capelli Theorem and the existence of the trivial solution.
2. b) Correct the complete discussion of the system for the case $m = 2$, calculating the rank of the coefficient matrix $A$ and the augmented matrix $A^*$.
3. c) Solve the system for $m = 2$ and express its infinitely many solutions in parametric form using the real parameter $\lambda$.

A system of three linear equations with three unknowns ($x, y, z$) depending on two real parameters, $a$ and $k$, is to be designed with the following structure:

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

1. a) For which specific value of $a$ are the first two equations identical? Prove that, for that value of $a$, the system can never be inconsistent, regardless of the real number chosen for $k$.
2. b) Calculate the determinant of the coefficient matrix $|A|$ in terms of $a$, factor it, and determine which two values of $a$ make the system cease to be consistent determined.
3. c) If we set $a = 2$, find what value the constant term $k$ must necessarily take for the system to have a solution (to be consistent indeterminate) and what happens if $k$ takes any other different value.

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

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

1. a) Discuss the system according to the different values of the parameter $a$ using the Rouché-Fröbenius Theorem.

1. b) Solve the system for the case in which it is consistent with infinitely many solutions and write the solution in parametric form.

1. c) Is there any value of $a$ for which the system is inconsistent? Justify your answer.

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.

Analyse the consistency of the system according to the values of the parameter $t$ and solve it when possible:

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

In your analysis, pay special attention to the case $t = -2$ and explain in detail what happens to the ranks of the matrix of coefficients and the augmented matrix.

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?

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

$$A = \begin{pmatrix} 1 & -1 & k \\ 2 & -k & 4 \\ 1 & -1 & 2 \end{pmatrix}, \quad B = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$$

1. a) Determine the values of $k$ for which matrix $A$ does not have an inverse.

1. b) Discuss the system of equations $A \cdot X = B$ according to the values of $k$.

1. c) Solve the system for $k = 3$.

Given the system of linear equations:

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

1. a) Is there any value of $a$ for which the system is consistent with a unique solution? Justify your answer based on the rank of the matrix of coefficients.

1. b) Find the value of $a$ that makes the system consistent with infinitely many solutions.

1. c) For the value of $a$ found in the previous section, give a particular solution where $z = 0$.

An agricultural cooperative produces three types of olive oil (Extra Virgin, Virgin and Lampante). The quantities produced in tonnes ($x, y, z$) are limited by olive availability and pressing time, following this model that depends on a parameter $p$ related to annual rainfall:

$$\begin{cases} x + y + z = 100 \\ 2x + py + 2z = 200 \\ px + 2y + 2z = 150 \end{cases}$$

1. a) Discuss the viability of production (consistency of the system) in terms of $p$.

1. b) If in a year of extreme drought the parameter is $p = 2$, what can you state about the cooperative's production based on the analysis of the system?

Two students, Carmen and Mateo, are about to discuss the following system depending on the real parameter $k$:

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

Carmen states: “The most reliable and orderly approach is to calculate the determinant $|A|$, find its roots, and apply the Rouché-Capelli Theorem by comparing ranks using minors.”

Mateo replies: “It is much more straightforward to use Gaussian elimination on the augmented matrix from the beginning, dividing the rows by whatever expressions with $k$ are needed to produce zeros quickly.”

1. a) Analyze Mateo's proposal: what mandatory algebraic precautions must be taken when performing Gaussian row operations on a matrix with parameters when multiplying or dividing a row by an expression containing $k$? Why is Carmen's strategy usually preferred to avoid discussion errors in Bachillerato?
2. b) Follow Carmen's proposal: calculate $|A|$, factor it, and provide a reasoned discussion of the type of system obtained according to the different values of $k$.
3. c) For the particular case $k = 1$, classify the system and solve it, justifying whether you consider Cramer's Rule or Gaussian elimination to be more advantageous in this specific numerical case.

An express courier company delivers packages across three fare zones in an autonomous community: Urban ($x$), Metropolitan ($y$), and Provincial ($z$), where the variables represent the average fuel cost (in euros) per delivery in each zone. In order to audit the impact of fuel prices, data from three workdays are recorded:

• Day 1: 10 urban, 10 metropolitan, and 10 provincial deliveries resulted in a total cost of $120$ euros.

• Day 2: 10 urban, 20 metropolitan, and 10 provincial deliveries generated a total cost of $160$ euros.

• Day 3: 20 urban, 30 metropolitan, and $k$ provincial deliveries totaled a cost of $G$ euros, where $k$ is an expected technical coefficient and $G$ is the recorded cost.

1. a) Formulate the system of linear equations that models the fuel expenditure for the three days in terms of the unit costs $x, y, z$.
2. b) Study for what value of the technical parameter $k$ the coefficient matrix has a zero determinant. If this value of $k$ occurs, what would the cost $G$ have to be exactly so that the records of the three days are not contradictory to one another (that is, so that the system is not inconsistent)?
3. c) During a monitoring week, $k = 25$ was set and an expenditure of $G = 305$ euros was recorded on the third day. Classify the system using the Rouché-Capelli Theorem and calculate the unit fuel cost ($x, y, z$) corresponding to each type of delivery.

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 $a = 0$.

A digital marketing company analyses the impact of three advertising campaigns ($x, y, z$). The management costs and the return on investment are linked by a technological efficiency factor $k$ according to the following model:

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

1. a) Discuss the viability of the model (consistency of the system) depending on the factor $k$.
2. b) Is there any value of $k$ for which the system has no solution? If so, interpret what it would mean for the company.
3. c) Determine the values of $k$ for which the investment in the three campaigns is the same ($x = y = z$).

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 $\lambda$.

Consider the following homogeneous system:

$$ \begin{cases} (m+1)x + y + z = 0 \\ x + (m+1)y + z = 0 \\ x + y + (m+1)z = 0 \end{cases} $$

1. a) Determine the values of the parameter $m$ for which the system has solutions other than the trivial one $(0, 0, 0)$.

1. b) For the values found in the previous part, describe the relative position of the three planes represented by the equations of the system.

Analyse the consistency of the following system of four equations with three unknowns according to the values of $m$:

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

For which value of $m$ does the system have a unique solution? Calculate it in that case.

Consider the matrix $A = \begin{pmatrix} 1 & 0 & -1 \\ 0 & a & 3 \\ 4 & 1 & -a \end{pmatrix}$.

1. a) Find the values of $a$ for which the system $A \cdot X = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$ has infinitely many solutions.
2. b) For $a = 2$, solve the matrix equation $A \cdot X = B$, where $B = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$, using the inverse matrix.

A team of researchers is studying the evolution of three populations of microorganisms in a controlled environment. The growth relationships depend on the temperature $T$ (measured on a normalised scale as parameter $t$):

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

1. a) Discuss the system for the different values of $t$.
2. b) Geometrically interpret what happens when $t = 1$ and when $t = -2$.
3. c) If it is known that the sum of the three populations is constant and equal to 1, what values can $t$ take?

It is known that the system of equations:

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

is consistent with infinitely many solutions.

1. a) Find the values of the parameters $a$ and $b$.
2. b) For the values found, write down the general solution to the system.
3. c) If we add a fourth equation $x + y + z = 2$, does the consistency of the system change? Reason your answer.

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

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

1. a) Estimate without calculating: Look carefully at the first and third equations when $a = 1$. How do they relate to each other? Now look at the second equation: for what value of $b$ would it be proportional to the other two? From these proportionality relationships, formulate a reasoned hypothesis about what type of system will be obtained when $a = 1$ and $b = 2$, what will happen when $a = 1$ and $b \ne 2$, and whether the value of $b$ will have any influence when $a \ne 1$.
2. b) Verify formally: Calculate the determinant $|A|$ in terms of $a$. Apply the Rouché-Capelli Theorem, thoroughly analyzing the ranks of $A$ and the augmented matrix $A^*$ to classify the system in all possible cases according to the values of $a$ and $b$. Compare the result with the estimation you made in the previous part.
3. c) For the values $a = 1$ and $b = 2$, determine the number of degrees of freedom of the solution, find the general solution in parametric form, and describe what geometric figure this set of solutions represents in three-dimensional space.

A financial modeling team analyzes the exchange relationships between three assets using the following system depending on the real parameters $\lambda$ and $\mu$:

$$\begin{cases} x + y + z = 1 \\ 2x + \lambda y + 2z = 2 \\ 3x + 3y + \lambda z = \mu \end{cases}$$

To validate the robustness of the economic model, three key scenarios must be investigated:

1. a) Universal consistency scenario: Prove analytically that there exists a value of $\lambda$ for which the system is consistent indeterminate completely independently of the value taken by $\mu$. Explain algebraically why the constant term $\mu$ is harmless in that specific case and find the rank of the augmented matrix $A^*$.
2. b) Critical sensitivity scenario: Determine the other value of $\lambda$ that makes $|A| = 0$ and find what condition $\mu$ must satisfy for the model to become inconsistent (with no possible solutions). If $\mu = 3$ is adopted in this scenario, solve the system by expressing the infinitely many solutions in terms of a single real parameter $t$.
3. c) Determined equilibrium scenario: If the parameters are calibrated to $\lambda = 1$ and $\mu = 0$, justify why the system admits a unique solution without the need to solve it completely, and calculate exclusively the value of the variable $z$ by applying Cramer's Rule directly.