Discussing systems with a parameter
In a factory, the cost of two microchips depends on a variable tax $p$ that appears in the coefficients of the system. You already know how to discuss systems of two equations by comparing the ratios of the coefficients. Now you are going one step further: systems of three equations, discussed using Gauss's method, and the special case of homogeneous systems.
Discussing by triangulating with Gauss
Given the following system in echelon form, discuss its solutions according to the values of the parameter $p$:…
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. $
Apply Gaussian elimination to discuss the following system according to the values of the parameter $m$:…
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. $
Homogeneous systems: they always have the trivial solution
Determine the value of $c$ for which the following homogeneous system has infinitely many solutions:…
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. $
In a chemistry experiment, three substances are mixed. The equilibrium conditions of the experiment depend on a catalyst whose concentration is $c$. The mathematical…
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 ($x=0, y=0, z=0$).
Geometric interpretation: lines and planes
Each equation with two unknowns is a line; with three unknowns, a plane in space. Discussing the system means deciding how those lines or planes are positioned relative to one another.
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…
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.…
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. $
Discussions with two parameters or subtler cases
In the hardest exercises, the pivot that vanishes is not an isolated value: you have to factorise an expression in the parameter, or two parameters appear at once. The strategy is the same, with more care when triangulating.
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…
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…
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 $m$.
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…
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.
Back to the start: If the costs of the microchips depend on the parameter $p$, discussing the system lets us identify for which values of the tax there is a single fixed solution (determinate), room for infinitely many possible combinations (indeterminate), or contradictory costs that are impossible to satisfy (inconsistent).