Exercises: Discussion of systems with a parameter

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

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

Determine the value of $k$ for the following system to be consistent with infinitely many solutions and interpret the result geometrically:

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

Calculate for which value of the parameter $a$ 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 in terms 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 $t$ for the pair $(2, -1)$ to be the solution of the system:

$ \left\{ \begin{array}{l} 3x + ty = 8 \\ tx - 5y = 9 \end{array} \right. $

Discuss the following system in terms of the parameter $b$:

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

Study the relative position of the lines $r: 5x + 2y = 1$ and $s: 10x + 4y = k$ in terms of the parameter $k$.

Determine the value of $c$ for the following homogeneous system to have infinitely many solutions:

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

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 in terms of the tax $p$.

Apply Gaussian elimination to discuss the following system in terms of the parameter $m$:

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

Three trajectories of surveillance drones in a plane are given by the equations of three lines. Determine the value of the parameter $a$ for the three drones to pass through the exact same point (i.e., for the system to have a unique solution).

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

Discuss the following system in terms of the parameter $k$ and indicate for which values the system represents two 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 chemical 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 in terms 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 distributes their capital across three investment funds. The profitability of the third fund depends on an interest rate $r$. The equations that model 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. $

Discuss the following system of linear equations in terms of the parameter $k$, using Gaussian elimination. For the case where it is consistent with 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 flow of vehicles per hour at three main intersections is modeled by a system of equations. Due to construction, 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 position of the three planes in three-dimensional space given by the following system, in terms of the parameter $a$.

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

Accompany the discussion with the following graphical representation of the geometric situation when the system is inconsistent and the planes intersect in pairs forming a triangular prism.

A logistics company distributes three types of goods using a fleet of trucks. The restrictions on weight, volume and delivery time generate the following system of equations, where $t$ is a parameter related to the maximum time allowed:

$ \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 in terms of the parameter $\alpha$. Pay special attention to the values that cancel out the leading coefficients during the Gaussian elimination process.

$ \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 between three species (predator, prey and plant) in a state of equilibrium is described by the following system, where $\beta$ represents the consumption rate of the 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 single equilibrium point, infinitely many or none.

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

$r: x + 2y = 3$

$s: 2x - y = 1$

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

Discuss the relative position of the three lines in terms of the parameter $m$. Observe the following diagram that illustrates the situation in which the three lines intersect at a single point (pencil of concurrent lines).

Discuss the following system of equations in terms of the values of the parameter $m$ and find the solution when it is consistent and indeterminate:

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