Exercises: Discussion of systems with one parameter

Discuss the following system of equations according to the values of the parameter $m$:

$ \left\{ \begin{array}{l} 2x - 3y = 5 \\ 4x - 6y = m \end{array} \right. $

Determine the value of $k$ for which the following system has infinitely many solutions and interpret the result geometrically:

$ \left\{ \begin{array}{l} kx + 2y = 4 \\ 3x + 6y = 12 \end{array} \right. $

Find the value of the parameter $a$ for which the system is inconsistent:

$ \left\{ \begin{array}{l} x + ay = 7 \\ 2x - 4y = 3 \end{array} \right. $

Given the following system in echelon form, discuss its solutions according to the values of the parameter $p$:

$ \left\{ \begin{array}{l} x + y + z = 2 \\ 2y - z = 4 \\ (p - 3)z = 5 \end{array} \right. $

Find the value of the parameter $t$ such that the pair $(2, -1)$ is the solution to the system:

$ \left\{ \begin{array}{l} 3x + ty = 4 \\ tx - 5y = 9 \end{array} \right. $

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

$ \left\{ \begin{array}{l} x - 2y = 1 \\ bx + 4y = -2 \end{array} \right. $

Determine the relative position of the lines $r: 5x + 2y = 1$ and $s: 10x + 4y = k$ according to the values of the parameter $k$.

Determine the value of $c$ for which the following homogeneous system has infinitely many solutions:

$ \left\{ \begin{array}{l} cx + y = 0 \\ x + cy = 0 \end{array} \right. $

When studying the homogeneous system $\begin{cases} x - 2y + z = 0 \\ 2x + ky - 2z = 0 \\ x + 4y - 3z = 0 \end{cases}$ in terms of the real parameter $k$, Lucía states out loud: "If we find a value of $k$ for which a row vanishes upon row reduction, the system will become inconsistent because it will have no solutions."\na) Without performing algebraic calculations, identify Lucía's conceptual error and explain why a homogeneous system can never be inconsistent, stating which solution is always guaranteed.\nb) Apply Gaussian elimination to the system's matrix to determine the value of $k$ that makes the system admit solutions other than the trivial one. For that value, find a non-zero particular solution.

To discuss the linear system depending on the parameter $m \in \mathbb{R}$:\n$\begin{cases} (m - 2)x + y + z = 2 \\ x - 2y + 3z = 1 \\ 2x + y - z = 0 \end{cases}$\ntwo students propose two different paths to begin Gaussian elimination:\n- Strategy A (Álvaro): Keep the original order of the equations and take $(m - 2)$ as the first pivot, dividing the first row by $(m - 2)$ or multiplying the rows below by that expression.\n- Strategy B (Berta): Rearrange the equations by swapping the first and second rows ($F_1 \leftrightarrow F_2$), so that the first pivot is the coefficient $1$ of the unknown $x$.\na) Compare both strategies: what algebraic risks and unnecessary branching does Álvaro's strategy introduce that are avoided with Berta's? Justify which one is preferable.\nb) Following Strategy B, row reduce the system and find the value of $m$ for which the system ceases to be consistent independent (that is, it does not have a unique solution).

In an electronic components factory, the production cost of two types of microchips is given by a system of linear equations. If $p$ represents a variable tax on the first microchip, the system is:

$ \left\{ \begin{array}{l} (p+1)x + 3y = 25 \\ 2x + py = 10 \end{array} \right. $

Discuss the system according to the tax $p$.

Apply Gaussian elimination to discuss the following system according to the values of the parameter $m$:

$ \left\{ \begin{array}{l} x + y + z = 1 \\ 2x - y + 3z = 4 \\ 3x + 2y + mz = 5 \end{array} \right. $

Three straight streets in a new neighbourhood are given by the equations of three lines. Determine the value of the parameter $a$ so that all three streets pass through the exact same point (that is, so that the system has a unique solution and they converge at a single square).

$ \left\{ \begin{array}{l} x + 2y = 5 \\ 3x - y = 1 \\ 2x + ay = 4 \end{array} \right. $

Discuss the following system according to the values of the parameter $k$ and state for which values the system represents intersecting, parallel, or coincident lines. Solve it for $k = 0$.

$ \left\{ \begin{array}{l} kx + (k-1)y = 2 \\ 3x + ky = k+1 \end{array} \right. $

In a chemistry experiment, three substances are mixed. The equilibrium conditions of the experiment depend on a catalyst whose concentration is $c$. The mathematical model is the following homogeneous system:

$ \left\{ \begin{array}{l} x + y - z = 0 \\ 2x - cy + 3z = 0 \\ cx + 2y - 2z = 0 \end{array} \right. $

Determine for which values of the concentration $c$ the system has solutions other than the trivial one ($x=0, y=0, z=0$).

Discuss the consistency of the following system according to the values of the parameter $a$:

$ \left\{ \begin{array}{l} ax + y + z = 1 \\ x + ay + z = 1 \\ x + y + az = 1 \end{array} \right. $

An investor divides their capital among three investment funds. The return on the third fund depends on an interest rate $r$. The equations modelling the profits are:

$ \left\{ \begin{array}{l} x + 2y + z = 100 \\ 2x + y + 3z = 150 \\ x - y + rz = 50 \end{array} \right. $

Discuss the system according to the values of the rate $r$.

Find the values of the parameter $m$ for which the following system of equations is inconsistent, and solve it for $m = 2$:

$ \left\{ \begin{array}{l} x - y + 2z = 3 \\ 2x + my - z = 1 \\ 3x - 2y + z = m \end{array} \right. $

Consider the equations of two planes in three-dimensional space:\n$\pi_1: x + 2y - z = 2$\n$\pi_2: 2x - y + z = 1$\nYour task is to construct the equation of a third plane $\pi_3: ax + by + cz = d$ to fulfill two independent goals:\na) Choose specific numerical values for $a, b, c$, and $d$ so that the system formed by the three planes is consistent independent (the three planes intersect at a single point). Prove by Gaussian elimination that your proposed system has a unique solution and calculate the coordinates of that point.\nb) Propose new values for $a, b, c$, and $d$ so that the system is inconsistent (the planes share no common point simultaneously), imposing the condition that the plane $\pi_3$ is not parallel to $\pi_1$ nor to $\pi_2$ (that is, the vector $(a, b, c)$ cannot be proportional to $(1, 2, -1)$ nor to $(2, -1, 1)$). Justify with Gauss why the system has no solution and geometrically interpret the arrangement of the three planes in space (triangular prism).

A network of three AEMET weather stations in the Sierra de Guadarrama ($E_1, E_2, E_3$) monitors accumulated water during a rain event. Let $x, y, z$ be the liters per square meter ($\text{l/m}^2$) recorded at $E_1, E_2$, and $E_3$, respectively. The following data are known:\n1. The total accumulated precipitation across the three stations sums to $90\text{ l/m}^2$.\n2. Station $E_3$ measured exactly twice the precipitation of station $E_1$.\n3. The rain gauge at station $E_2$ suffered wind-induced deviations, so the control algorithm applies the weighted relationship $x + ky + 2z = 150$, where $k \in \mathbb{R}$ is an environmental correction factor.\na) Write the system of three linear equations in the unknowns $x, y, z$ associated with the problem.\nb) Apply Gaussian elimination to row reduce the system and discuss it in terms of the parameter $k$. Is there any value of $k$ for which the data are contradictory (inconsistent system)?\nc) Determine for which value of the factor $k$ the system is consistent dependent, and explain what physical meaning this indeterminacy has for the meteorological team. Finally, calculate the values of $x, y, z$ assuming a standard calibration factor ($k = 1$).

Discuss the following system of linear equations according to the values of the parameter $k$, using Gaussian elimination. For the case where it has infinitely many solutions, find the general solution expressed in terms of a parameter $\lambda$.

$ \left\{ \begin{array}{l} x + 2y - z = 1 \\ -x + y + 3z = 2 \\ 2x + y + kz = -1 \end{array} \right. $

In the analysis of an urban traffic network, the hourly flow of vehicles through three main intersections is modelled using a system of equations. Due to roadworks, one of the roads has a variable capacity $m$. The system governing the flow is:

$ \left\{ \begin{array}{l} mx + y + z = m \\ x + my + z = m \\ x + y + mz = m \end{array} \right. $

Discuss the flow of the network (consistency of the system) for all real values of $m$.

Study the relative positions of the three planes in three-dimensional space given by the following system, depending on the parameter $a$:

$ \left\{ \begin{array}{l} 2x - y + 3z = 1 \\ x + ay - z = 2 \\ 3x + 2y + 2z = a \end{array} \right. $

A logistics company distributes three types of goods using a fleet of lorries. The weight, volume, and delivery time constraints give rise to the following system of equations, where $t$ is a parameter related to the maximum permitted time:

$ \left\{ \begin{array}{l} x + y + tz = 1 \\ tx + y + z = t \\ x + ty + z = t^2 \end{array} \right. $

Discuss the system according to the values of $t$ and solve for the case $t = -2$.

Discuss the following system according to the values of the parameter $\alpha$. Pay special attention to values that make the leading coefficients zero during Gaussian elimination.

$ \left\{ \begin{array}{l} x + \alpha y - z = 2 \\ 2x - y + \alpha z = 5 \\ x + 10y - 6z = 1 \end{array} \right. $

In a population dynamics study, the interaction among three species (predator, prey, and plant) in an equilibrium state is described by the following system, where $\beta$ represents the consumption rate of prey by the predator:

$ \left\{ \begin{array}{l} \beta x + y + z = 4 \\ x + y + z = \beta + 2 \\ x - y + \beta z = 2 \end{array} \right. $

Determine the values of $\beta$ for which the ecosystem has a unique equilibrium point, infinitely many, or none.

Consider the three lines in the plane given by the equations:

$r: x + 2y = 3$

$s: 2x - y = 1$

$t: mx + (m+1)y = 4$

Determine the relative position of the three lines according to the values of the parameter $m$. Observe the following diagram illustrating the situation where the three lines intersect at a single point (concurrent lines).

Discuss the following system of equations according to the values of the parameter $m$ and find the solution when it is consistent and underdetermined (indeterminate):

$ \left\{ \begin{array}{l} 3x - y + 2z = 1 \\ x + 4y + z = m \\ 2x - 5y + mz = 0 \end{array} \right. $

Consider the following homogeneous system depending on the real parameter $a$, where each equation represents a plane containing the origin:\n$\begin{cases} x + y + z = 0 \\ 2x + 3y + az = 0 \\ 3x + 4y + (a^2 - 1)z = 0 \end{cases}$\na) Estimate before operating: Without carrying out Gaussian elimination or computing determinants, answer with reasoning:\n - Can there be any value of $a$ for which the three planes do not share any point? Why?\n - Knowing that the three planes are neither parallel nor coincident with each other, what are the only two possible geometric configurations regarding their common intersection?\nb) Calculate and check: Row reduce the system using elementary row operations while keeping the parameter $a$. Determine the two critical values of $a$ for which the rank of the matrix decreases and the system admits infinitely many solutions.\nc) Interpret: For each of the two critical values obtained in the previous part, find the general solution in terms of a parameter $\lambda \in \mathbb{R}$ and deduce the equation of the line in which the three planes intersect. Check whether the results agree with your previous analysis in part a).

In an assessment test, students are asked to discuss the relative positions of three planes according to the parameter $k \in \mathbb{R}$:\n$\pi_1: x + y + z = 1$\n$\pi_2: 2x + 3y + kz = 3$\n$\pi_3: x + ky + 3z = 1$\nMateo sets up the augmented matrix and performs the operations $F_2 \leftarrow F_2 - 2F_1$ and $F_3 \leftarrow F_3 - F_1$, obtaining:\n$\begin{pmatrix} 1 & 1 & 1 & | & 1 \\ 0 & 1 & k - 2 & | & 1 \\ 0 & k - 1 & 2 & | & 0 \end{pmatrix}$\nFrom this step, Mateo argues:\n"To make the $(3,2)$ entry zero, I would have to divide the third row by $(k - 1)$; therefore, if $k = 1$, the operation is invalid and the system becomes inconsistent (the planes form a prism).\nIf $k \neq 1$, I subtract from row 3 row 2 multiplied by $(k - 1)$, obtaining as the last pivot $2 - (k - 1)(k - 2) = -k^2 + 3k$. Since this pivot vanishes for $k = 0$ and $k = 3$, I conclude that for these two values the system is consistent dependent and the planes intersect along a line; for any other value of $k$, it is consistent independent and they intersect at a point."\nYour task is to evaluate Mateo's work:\na) Identify and explain the two serious conceptual errors in his solution: the one regarding the supposed inconsistency when an operational coefficient vanishes ($k = 1$), and the one made when classifying $k = 0$ and $k = 3$ without checking the constant term of the resulting row.\nb) Substitute $k = 1$ directly into Mateo's intermediate matrix and solve the resulting system using Gauss. What is the true relative position of the three planes for $k = 1$?\nc) Carry out the complete and rigorous discussion of the system for all $k \in \mathbb{R}$, stating with justification the type of system and the geometric position the planes take in each case.