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

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.

$2x - y = a$

$-4x + 2y = 8$

To classify the system $\begin{cases} 3x - ky = 6 \\ 6x - 4y = 12 \end{cases}$ in terms of the parameter $k \in \mathbb{R}$, Cian 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 Cian 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 $k$.

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

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

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

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.

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?

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.

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 ($x=0$):

$ax + 3y = 6$

$2x - ay = a$

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

$ax + y = b$

$x + ay = 2$

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.

In the approach airspace of Dublin 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 $(0, 0)$.

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

$x^2 - y^2 = k$

$x + y = 2$

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 $c$.