Exercises: Discussion of a system of two equations

Discuss, in terms of the parameter $m$, the following system of linear equations:

$x - 2y = 3$

$mx + 4y = -6$

Determine the value of the parameter $k$ for the following system to be inconsistent:

$3x + ky = 5$

$6x - 4y = 2$

Find the value of $a$ for the following system to have infinitely many solutions. Write the general solution in that case.

$2x - y = a$

$-4x + 2y = 8$

Discuss the relative position of the lines $r: px + 3y = 1$ and $s: 2x + py = 0$ in terms of the parameter $p$.

Discuss the following system of equations in terms of the parameter $t$:

$tx + y = t$

$x + ty = 1$

Find the values of the parameter $b$ for which the following system is consistent with a unique solution:

$x + by = 2$

$bx + y = 2$

Discuss the system dependent 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.

Two surveillance drones follow rectilinear trajectories 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 trajectories of the drones intersect at a single point, are parallel or coincide exactly.

Discuss, in terms of the parameter $k$, the following system of equations:

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

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

In a market study, the supply and demand curves of a technological product are given by a linear system dependent on a tax $t$:

$2x - ty = 3$

$tx + 2y = 4$

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

Discuss the following system and, in cases where it is consistent with infinitely many solutions, express the infinite 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 that model the mass balance of the process depend on the concentration $c$ of a specific reagent:

$cx + 4y = c^2$

$x + cy = 2$

Discuss the consistency of the system in terms of the concentration $c$.

Discuss the system in terms 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 in terms of the parameter $k$ to avoid possible short circuits.

Discuss the following system of linear equations in terms of the parameters $a$ and $b$:

$ax + y = b$

$x + ay = 2$

The trajectory of a projectile follows a parabola described by the equation $y = -x^2 + 4x$, while a defense laser beam follows the line $y = mx$. Discuss the number of intersection points between the projectile and the laser in terms of the parameter $m$ (which represents the gradient of the laser).

Discuss the following linear system in terms of the parameter $k$ (consider 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 dependent 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 in terms of the value of the resistance $R$.

Discuss the consistency of the following non-linear system in terms of the parameter $k$:

$x^2 - y^2 = k$

$x + y = 2$

Discuss the system in terms 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 following a circular trajectory modeled in the plane by the equation $x^2 + y^2 = 4$. A comet travels in a straight line according to the equation $x + y = c$, where $c$ is a parameter that depends on its initial position. Discuss the number of collision points between the satellite and the comet in terms of $c$.

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 is not consistent with a unique solution? Justify your answer mathematically.