Discussing Systems with a Parameter
At an electronic components factory in Guadalajara, the cost of two microchips depends on a variable tax $p$ that appears in the coefficients of the system $(p+1)x+3y=25$, $2x+py=10$. For which values of $p$is there a unique solution? For which values is there none? To discuss a system is to answer those questions without solving it once for every value of the parameter.
2×2 Systems with a Parameter
A parameter is a letter that represents a fixed but unknown number inside the coefficients. Each value of the parameter gives a different system.
With two unknowns, each equation is a line. You already know there are three cases: the lines intersect at one point, they are parallel, or they coincide.
Move the slider and watch what happens to the second line when $k$ changes.
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. $
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 Value of the Parameter That Is Asked For
Many problems do not ask you to discuss everything, but for one specific value of the parameter. The method changes depending on what is asked:
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. $
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. $
Discussing with Gaussian Elimination
With three equations the procedure is to triangularize with Gaussian elimination, just as without a parameter. The difference is that at some step a coefficient containing the parameter appears, and you have to analyze what happens when it equals zero.
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. $
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. $
Homogeneous Systems
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
An equation with two unknowns is a line, and with three unknowns it is a plane in space. Discussing a system means deciding how those objects are positioned relative to one another.
Three lines in the plane can pass through the same point or form a triangle:
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).
Finer Cases: Pivots That Factor
In the hard exercises the pivot is an expression like $a^2-1$ or $m^2+m-6$, and you have to factor it to see all the critical values. Sometimes the system is symmetric and it is useful to add the three equations or subtract them from one another.
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 $r$.
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. $
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. $
Back to the beginning: in the microchip system, $D=(p+1)p-6=p^2+p-6=(p+3)(p-2)$. If $p\neq2$ and $p\neq-3$ there is a unique solution. If $p=2$ we get $3x+3y=25$ and $2x+2y=10$, which contradict each other. If $p=-3$ we get $-2x+3y=25$ and $2x-3y=10$, which added together give $0=35$. In both cases the system is inconsistent: there are never infinitely many solutions.