Discussing systems with a parameter
A Galway electronics firm prices two microchips, and the cost depends on a variable rate $p$ that appears in the coefficients of the system. You already know how to solve systems; now the coefficients contain a parameter, so the number of solutions can change. First we discuss systems of two equations by comparing ratios of coefficients, then systems of three equations by elimination, and the special case of homogeneous systems.
Two equations: comparing ratios
Two lines $a_1x+b_1y=c_1$ and $a_2x+b_2y=c_2$ either cross at one point, are parallel (no solution) or are the same line (infinitely many solutions). The slopes tell you which: lines with different slopes must cross.
Discuss the following system of equations according to the values of the parameter $m$: $ \left\{ \begin{array}{l} 2x - 3y = 5 \\ 4x - 6y = m \end{array} \right. $
Discuss the following system of equations according to the values of the parameter $m$:
$ \left\{ \begin{array}{l} 2x - 3y = 5 \\ 4x - 6y = m \end{array} \right. $
Find the value of the parameter $t$ such that the pair $(2, -1)$ is the solution to the system:…
Find the value of the parameter $t$ such that the pair $(2, -1)$ is the solution to the system:
$ \left\{ \begin{array}{l} 3x + ty = 4 \\ tx - 5y = 9 \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…
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 according to the tax $p$.
Discussing by eliminating variables
Augmented matrix and row operations
To save writing the letters, we can record the system as an augmented matrix: each row holds the coefficients of $x, y, z$ and then, after a bar, the constant term. For example, $x+y+z=1$ becomes the row $(1\ 1\ 1 \mid 1)$.
A row operation such as $F_2 \leftarrow F_2 - 2F_1$ means: subtract 2 times row 1 from row 2. It is exactly the same as eliminating a variable between two equations, so it does not change the solutions.
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. $
An investor divides their capital among three investment funds. The return on the third fund depends on an interest rate $r$. The equations modelling the profits are:…
An investor divides their capital among three investment funds. The return on the third fund depends on an interest rate $r$. The equations modelling 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 $r$.
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. $
When studying the homogeneous system $\begin{cases} x - 2y + z = 0 \\ 2x + ky - 2z = 0 \\ x + 4y - 3z = 0 \end{cases}$ in terms of the real parameter $k$, Aoife states…
When studying the homogeneous system $\begin{cases} x - 2y + z = 0 \\ 2x + ky - 2z = 0 \\ x + 4y - 3z = 0 \end{cases}$ in terms of the real parameter $k$, Aoife states out loud: "If we find a value of $k$ for which a row vanishes upon row reduction, the system will become inconsistent because it will have no solutions."\na) Without performing algebraic calculations, identify Aoife's conceptual error and explain why a homogeneous system can never be inconsistent, stating which solution is always guaranteed.\nb) Apply Gaussian elimination to the system's matrix to determine the value of $k$ that makes the system admit solutions other than the trivial one. For that value, find a non-zero particular solution.
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.
Determine the relative position of the lines $r: 5x + 2y = 1$ and $s: 10x + 4y = k$ according to the values of the parameter $k$.
Determine the relative position of the lines $r: 5x + 2y = 1$ and $s: 10x + 4y = k$ according to the values of the parameter $k$.
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. $
Consider the equations of two planes in three-dimensional space:\n$\pi_1: x + 2y - z = 2$\n$\pi_2: 2x - y + z = 1$\nYour task is to construct the equation of a third…
Consider the equations of two planes in three-dimensional space:\n$\pi_1: x + 2y - z = 2$\n$\pi_2: 2x - y + z = 1$\nYour task is to construct the equation of a third plane $\pi_3: ax + by + cz = d$ to fulfill two independent goals:\na) Choose specific numerical values for $a, b, c$, and $d$ so that the system formed by the three planes is consistent independent (the three planes intersect at a single point). Prove by Gaussian elimination that your proposed system has a unique solution and calculate the coordinates of that point.\nb) Propose new values for $a, b, c$, and $d$ so that the system is inconsistent (the planes share no common point simultaneously), imposing the condition that the plane $\pi_3$ is not parallel to $\pi_1$ nor to $\pi_2$ (that is, the vector $(a, b, c)$ cannot be proportional to $(1, 2, -1)$ nor to $(2, -1, 1)$). Justify with Gauss why the system has no solution and geometrically interpret the arrangement of the three planes in space (triangular prism).
Consider the three lines in the plane given by the equations: $r: x + 2y = 3$ $s: 2x - y = 1$ $t: mx + (m+1)y = 4$ Determine the relative position of the three lines…
Consider the three lines in the plane given by the equations:
$r: x + 2y = 3$
$s: 2x - y = 1$
$t: mx + (m+1)y = 4$
Determine the relative position of the three lines according to the values of the parameter $m$. Observe the following diagram illustrating the situation where the three lines intersect at a single point (concurrent lines).
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.
To discuss the linear system depending on the parameter $m \in \mathbb{R}$:\n$\begin{cases} (m - 2)x + y + z = 2 \\ x - 2y + 3z = 1 \\ 2x + y - z = 0 \end{cases}$\ntwo…
To discuss the linear system depending on the parameter $m \in \mathbb{R}$:\n$\begin{cases} (m - 2)x + y + z = 2 \\ x - 2y + 3z = 1 \\ 2x + y - z = 0 \end{cases}$\ntwo students propose two different paths to begin Gaussian elimination:\n- Strategy A (Seán): Keep the original order of the equations and take $(m - 2)$ as the first pivot, dividing the first row by $(m - 2)$ or multiplying the rows below by that expression.\n- Strategy B (Aoife): Rearrange the equations by swapping the first and second rows ($F_1 \leftrightarrow F_2$), so that the first pivot is the coefficient $1$ of the unknown $x$.\na) Compare both strategies: what algebraic risks and unnecessary branching does Seán's strategy introduce that are avoided with Aoife's? Justify which one is preferable.\nb) Following Strategy B, row reduce the system and find the value of $m$ for which the system ceases to be consistent independent (that is, it does not have a unique solution).
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 $m$.
Back to the start: if the microchip costs depend on the rate $p$, discussing the system tells you for which values of $p$ there is a single solution, infinitely many, or none (impossible costs).