Exercises: Discussion of a system of two equations

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

$x - 2y = 3$

$mx + 4y = -6$

Determine the value of the parameter $k$ for which the following system is inconsistent:

$3x + ky = 5$

$6x - 4y = 2$

Find the value of $a$ for which the following system has infinitely many solutions. Write down the general solution in this case.

$2x - y = a$

$-4x + 2y = 8$

Determine the relative position of the lines $r: px + 3y = 1$ and $s: 2x + py = 0$ according to the values of the parameter $p$.

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

$tx + y = t$

$x + ty = 1$

Find the values of the parameter $b$ for which the following system has a unique solution:

$x + by = 2$

$bx + y = 2$

Discuss the system depending on the parameter $c$:

$cx - 2y = 1$

$x - cy = 2$

Given the following system of linear equations:

$3x - 2y = 4$

$x + y = m$

Is there any value of the parameter $m$ for which the system is inconsistent? Justify your answer mathematically.

To classify the system $\begin{cases} 3x - ky = 6 \\ 6x - 4y = 12 \end{cases}$ in terms of the parameter $k \in \mathbb{R}$, Marcos writes on his exam:

"I equate the ratios of the coefficients: $\dfrac{3}{6} = \dfrac{-k}{-4} \implies \dfrac{1}{2} = \dfrac{k}{4} \implies k = 2$. Since the ratio of the constant terms is $\dfrac{6}{12} = \dfrac{1}{2}$, all three ratios are identical and the system is consistent dependent (coincident lines) if $k=2$. Therefore, if $k \ne 2$, since the ratios of the coefficients no longer match that of the constant terms, the system has no solution (it is inconsistent; parallel lines)".

1. a) Point out the conceptual error that Marcos makes in his deduction for $k \ne 2$ and explain the geometric difference he ignored between intersecting lines and parallel lines.
2. b) Correct the analysis and write the complete discussion of the system for all real values of $k$.

We wish to design a linear system of two equations with two unknowns ($x$ and $y$) that depends on a real parameter $a$ and a fixed constant $c$:

$\begin{cases} 2x + ay = 4 \\ ax + 8y = c \end{cases}$

1. a) State the inverse problem: determine which specific positive numerical value the constant $c$ must have so that there exists at least one value of $a$ making both lines coincident (consistent dependent system).
2. b) With that value of $c$ fixed, find the two critical values of $a$ and fully discuss the system (consistent independent, consistent dependent, or inconsistent) for all possible values of $a \in \mathbb{R}$, interpreting geometrically the relative position of the lines in each case.

Two surveillance drones follow straight paths described by the following system, where $v$ is a parameter related to wind speed:

$vx + 2y = 4$

$2x + vy = v$

Discuss for which values of $v$ the paths of the drones intersect at a single point, are parallel, or coincide completely.

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

$(k+1)x + y = 2$

$kx + (k-1)y = 1$

In a market study, the supply and demand curves for a technology product are given by a linear system depending on a tax $t$:

$2x - ty = 3$

$tx + 2y = 4$

Discuss the consistency of the system according to the values of $t$. Can it be the case that no equilibrium point exists (that is, that the system is inconsistent) if $t$ is a real number?

Discuss the following system and, in cases where it has infinitely many solutions, express the solutions in terms of a parameter $\lambda$:

$mx + (m+1)y = m$

$mx + 2my = 2$

Determine for which values of the parameter $a$ the following system has a unique solution $(x, y)$ such that the point of intersection of both lines lies exactly on the y-axis ($x=0$):

$ax + 3y = 6$

$2x - ay = a$

In a chemical plant, two liquid solutions are mixed. The equations modelling the mass balance of the process depend on the concentration $c$ of a specific reactant:

$cx + 4y = c^2$

$x + cy = 2$

Discuss the consistency of the system according to the concentration $c$.

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

$x + y = m$

$m^2 x + y = m^2$

Two high-voltage cables are modelled in a plane by the lines $r: 2x + (k-2)y = k$ and $s: (k-2)x + 2y = 2$. Discuss the relative position of the cables according to the values of the parameter $k$ to avoid potential short circuits.

Consider the system of linear equations $\begin{cases} x + my = 3 \\ mx + 9y = 9 \end{cases}$, where $m$ is a real parameter.

1. a) Estimate: Before performing formal algebraic transformations, observe the coefficients carefully:

• Estimate which two opposite values of $m$ will make the slopes of both lines coincide.

• Conjecture with reasoning whether for both signs of $m$ the lines will coincide at all points or whether for one of them they will turn out to be distinct parallel lines, looking at the sign of the constant terms.

1. b) Check: Discuss the system rigorously for all $m \in \mathbb{R}$ by studying the ratios $\dfrac{a_1}{a_2}$, $\dfrac{b_1}{b_2}$, and $\dfrac{c_1}{c_2}$. Classify it as consistent independent, consistent dependent, or inconsistent.
2. c) Evaluate: Compare the formal result with your initial estimate from part a). Did your conjecture about the coincidence of the lines hold true? Justify why the algebraic equality confirms or corrects visual intuition.

In the environmental monitoring protocols of an AEMET station in an industrial area, two primary pollutant gases, $x$ and $y$ (expressed in $\mu\text{g/m}^3$), are monitored using two calibrated sensors whose readings depend on a standardised thermal index $t \in \mathbb{R}$ according to the model:

$\begin{cases} x + ty = 5 \\ tx + 9y = 15 \end{cases}$

1. a) Discuss the system for all values of the thermal index $t$. Identify for which values of $t$ the readings provide a unique and well-defined concentration for both gases (consistent independent system).
2. b) Determine for which ambient temperature the protocol produces infinitely many valid solutions (consistent dependent system) and explain what this means in practice for the station technicians.
3. c) Find out whether there is any value of $t$ for which the sensors give contradictory readings that are impossible to satisfy simultaneously (inconsistent system), and geometrically interpret the relationship between the lines associated with each sensor in that situation.

Discuss the following system of linear equations according to the values of the parameters $a$ and $b$:

$ax + y = b$

$x + ay = 2$

The path of a projectile follows a parabola described by the equation $y = -x^2 + 4x$, while a defence laser beam follows the line $y = mx$. Discuss the number of intersection points between the projectile and the laser beam according to the parameter $m$ (which represents the slope of the laser beam).

Discuss the following linear system according to the values of the parameter $k$ (given that $k \neq 0$ and $k \neq 1$):

$\frac{x}{k} + \frac{y}{k-1} = 1$

$kx + (k-1)y = 2$

In an electrical circuit, the currents $I_1$ and $I_2$ satisfy the following system depending on a variable resistance $R$:

$R I_1 + (R+1) I_2 = 2R$

$(R-1) I_1 + R I_2 = R^2$

Discuss the existence and uniqueness of the currents depending on the value of the resistance $R$.

Discuss the consistency of the following non-linear system according to the values of the parameter $k$:

$x^2 - y^2 = k$

$x + y = 2$

Discuss the system according to the values of the parameters $m$ and $n$:

$mx + ny = 1$

$nx + my = 1$

Determine the exact conditions on $m$ and $n$ for the system to be inconsistent.

A satellite orbits along a circular path modelled in the plane by the equation $x^2 + y^2 = 4$. A comet moves in a straight line according to the equation $x + y = c$, where $c$ is a parameter depending on its initial position. Discuss the number of collision points between the satellite and the comet according to $c$.

Discuss the following linear system with parameter $\alpha \in [0, 2\pi]$:

$(\sin \alpha) x + (\cos \alpha) y = 1$

$(\cos \alpha) x - (\sin \alpha) y = 0$

Is there any value of $\alpha$ for which the system does not have a unique solution? Justify your answer mathematically.

To classify the system $\begin{cases} \lambda x + 2y = 4 \\ (\lambda + 3)x + (\lambda + 1)y = 6 \end{cases}$ in terms of the parameter $\lambda \in \mathbb{R}$, two 1st-year Bachillerato students propose different approaches:

• Sara decides to equate the ratios between coefficients: $\dfrac{\lambda}{\lambda + 3} = \dfrac{2}{\lambda + 1} = \dfrac{4}{6}$.

• Álvaro prefers to transform the system by elimination (Gaussian elimination) to triangularise it before drawing conclusions.

1. a) Critically analyse Sara's method: what mathematical danger arises with the values $\lambda = -3$ and $\lambda = -1$? Does a denominator vanishing mean that the system is inconsistent at those values, or merely that this specific tool cannot be applied directly?
2. b) Discuss and classify the system for all $\lambda \in \mathbb{R}$, rigorously analysing what happens exactly in the particular cases $\lambda = -3$ and $\lambda = -1$ and at the values where the slopes are equal.
3. c) Evaluate both methods: justify which of the two is more reliable when parameters appear in multiple coefficients of the same unknown, and in which cases it pays off to use the ratio rule.

In the approach airspace of Adolfo Suárez Madrid-Barajas Airport, the control tower monitors two aircraft whose straight-line flight paths on the horizontal navigation plane are described by the system:

$\begin{cases} ax + y = 1 \\ x + ay = b \end{cases}$

where $a$ and $b$ are real parameters set by radio beacons and approach manoeuvres.

1. a) Solve the inverse problem by determining what exact algebraic relationship the parameters $a$ and $b$ must satisfy so that:

1. The flight paths intersect at a single point located exactly on the x-axis ($y = 0$).

2. There is a critical risk of continuous collision because the flight paths are identical (coincident lines).

3. The planes follow strictly parallel but separate paths (distinct parallel lines).

1. b) Carry out an exhaustive discussion of the system, classifying it (consistent independent, consistent dependent, or inconsistent) for any combination of values $(a, b) \in \mathbb{R}^2$.
2. c) Prove using rigorous algebraic reasoning whether it is possible to find any pair of parameters $(a, b)$ that makes the single intersection point of the two flight paths the coordinate origin $(0, 0)$.