Discussing systems with one parameter
Wilkin has a corner store (colmado) and sells two products whose prices depend on a value $m$ that changes every week. Depending on $m$, sometimes he can find the prices, sometimes he gets infinitely many possibilities, and sometimes the data contradict each other. Discussing a system with a parameter means deciding, without solving it for each value, in which cases each of these things happens.
Two equations: comparing ratios of coefficients
A system of two equations with two unknowns is a pair of lines. There are three possibilities:
| Position of the lines | Type of system | Solutions |
|---|---|---|
| They intersect at one point | Consistent and independent (CI) | one |
| They coincide | Consistent and dependent (CD) | infinitely many |
| They are parallel | Inconsistent (IN) | none |
Move the slider and watch when the lines $3x+2y=5$ and $ax+4y=9$ become parallel.
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. $
Determine the value of $k$ for which the following system has infinitely many solutions and interpret the result geometrically:…
Determine the value of $k$ for which the following system has infinitely many solutions and interpret the result geometrically:
$ \left\{ \begin{array}{l} kx + 2y = 4 \\ 3x + 6y = 12 \end{array} \right. $
Find the value of the parameter $a$ for which the system is inconsistent: $ \left\{ \begin{array}{l} x + ay = 7 \\ 2x - 4y = 3 \end{array} \right. $
Find the value of the parameter $a$ for which the system is inconsistent:
$ \left\{ \begin{array}{l} x + ay = 7 \\ 2x - 4y = 3 \end{array} \right. $
When the parameter is in several coefficients
If the parameter appears in more than one coefficient, avoid ratios (you could divide by zero). Use the cross product instead: the coefficients are proportional when
$$a\,b' = a'\,b.$$This gives an equation in the parameter, and its solutions are the "special" values. For each one, you check separately whether the system is IN or CD.
Discuss the following system according to the values of the parameter $b$: $ \left\{ \begin{array}{l} x - 2y = 1 \\ bx + 4y = -2 \end{array} \right. $
Discuss the following system according to the values of the parameter $b$:
$ \left\{ \begin{array}{l} x - 2y = 1 \\ bx + 4y = -2 \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 $p$.
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. $
Finding the parameter so that a condition holds
Sometimes you are not asked to discuss, but to make a specific pair a solution. Then you substitute $x$ and $y$ into each equation and obtain equations for the parameter.
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. $
Three lines in the plane
When there are three equations and two unknowns, each equation is a line and the question is whether all three pass through the same point.
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. $
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).
Three equations with three unknowns: Gaussian elimination with a parameter
Gaussian elimination consists of eliminating unknowns to leave the system in echelon form. With a parameter, the last equation ends up in the form $(\text{expression in the parameter})\cdot z = \text{number}$, and the discussion depends on whether that expression vanishes or not.
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 $r$.
Finer cases: factoring the coefficient
In the harder exercises, the coefficient in the last equation can be a polynomial in the parameter, such as $a^2-1$ or $m^2+m-2$. You factor it or solve the equation to find all the values that make it vanish, and each one is a separate case.
Discuss the consistency of the following system according to the values of the parameter $a$:…
Discuss the consistency of the following system according to the values of the parameter $a$:
$ \left\{ \begin{array}{l} ax + y + z = 1 \\ x + ay + z = 1 \\ x + y + az = 1 \end{array} \right. $
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. $
Discuss the following system according to the values of the parameter $\alpha$. Pay special attention to values that make the leading coefficients zero during Gaussian…
Discuss the following system according to the values of the parameter $\alpha$. Pay special attention to values that make the leading coefficients zero during Gaussian elimination.
$ \left\{ \begin{array}{l} x + \alpha y - z = 2 \\ 2x - y + \alpha z = 5 \\ x + 10y - 6z = 1 \end{array} \right. $
Solving in the consistent cases
Often, besides discussing, you are asked for the solution for a particular value of the parameter. You substitute that value into the echelon-form system and solve. If it is CD, you call one unknown $\lambda$ (or $t$) a free variable and solve for the rest in terms of it.
Find the values of the parameter $m$ for which the following system of equations is inconsistent, and solve it for $m = 2$:…
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. $
A logistics company distributes three types of goods using a fleet of lorries. The weight, volume, and delivery time constraints give rise to the following system of…
A logistics company distributes three types of goods using a fleet of lorries. The weight, volume, and delivery time constraints give rise to the following system of equations, where $t$ is a parameter related to the maximum permitted time:
$ \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 of equations according to the values of the parameter $m$ and find the solution when it is consistent and underdetermined (indeterminate):…
Discuss the following system of equations according to the values of the parameter $m$ and find the solution when it is consistent and underdetermined (indeterminate):
$ \left\{ \begin{array}{l} 3x - y + 2z = 1 \\ x + 4y + z = m \\ 2x - 5y + mz = 0 \end{array} \right. $
Contextual problems and relative position
When the problem comes from a context (traffic, ecosystem, mixtures), the work is the same: triangulate and study when the coefficient vanishes. At the end, interpret the result in the context: CI is a single possible situation, CD means infinitely many, and IN means the data are incompatible.
If the problem asks for the "relative position" of the equations, the answer comes straight from the discussion. With three unknowns, each equation represents a plane in space (like the walls and ceiling of a room), and the discussion tells how they are arranged:
| Type of system | Position of the three planes |
|---|---|
| CI | They meet at a single point |
| CD | They share a line (like the pages of an open book) or are the same plane |
| IN | There is no point common to all three: two or all three are parallel, or they intersect two at a time forming a triangular prism |
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 $m$.
Study the relative positions of the three planes in three-dimensional space given by the following system, depending on the parameter $a$:…
Study the relative positions of the three planes in three-dimensional space given by the following system, depending on the parameter $a$:
$ \left\{ \begin{array}{l} 2x - y + 3z = 1 \\ x + ay - z = 2 \\ 3x + 2y + 2z = a \end{array} \right. $
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.
Homogeneous systems
With two equations there are nontrivial solutions when the lines coincide. With three equations, when the last coefficient vanishes upon triangulating.
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 ($x=0, y=0, z=0$).
Back to the beginning: if the prices at Wilkin's corner store depend on $m$, discussing the system tells for which values of $m$ the prices can be found uniquely (CI), which leave infinitely many possibilities (CD), and which make the data contradict each other (IN).