Discussing a System of Two Equations
Two drones film the Carnival of La Vega, flying along straight paths that depend on a parameter $v$ related to the wind. Depending on the value of $v$, the paths cross at a single point or run parallel (in other systems, they could even coincide completely). To discuss the system is to answer that for all values of $v$ at once, without solving it value by value.
What it means to discuss a system
Each linear equation in two unknowns is a line, so the type of system is the relative position of two lines:
Determine the relative position of the lines $r: px + 3y = 1$ and $s: 2x + py = 0$ according to the values of the parameter $p$.
Determine the relative position of the lines $r: px + 3y = 1$ and $s: 2x + py = 0$ according to the values of the parameter $p$.
The determinant of the system
Instead of comparing fractions (which fail if some denominator is zero), we use the determinant of the coefficients.
If $\Delta=0$ and the proportion also extends to the constant terms, the system is consistent and dependent. If not, it is inconsistent. For a given value of the parameter, a system never gives both results at once.
Play with the parameter $m$ in $\begin{cases} x-2y=3 \\ mx+4y=-6 \end{cases}$ and watch when the lines coincide:
Discuss the following system of linear equations according to the values of the parameter $m$: $x - 2y = 3$ $mx + 4y = -6$
Discuss the following system of linear equations according to the values of the parameter $m$:
$x - 2y = 3$
$mx + 4y = -6$
Determine the value of the parameter $k$ for which the following system is inconsistent: $3x + ky = 5$ $6x - 4y = 2$
Determine the value of the parameter $k$ for which the following system is inconsistent:
$3x + ky = 5$
$6x - 4y = 2$
Discuss the following system of equations according to the values of the parameter $t$: $tx + y = t$ $x + ty = 1$
Discuss the following system of equations according to the values of the parameter $t$:
$tx + y = t$
$x + ty = 1$
Special values and cases
When $\Delta$ is a second-degree expression in the parameter, it vanishes at two values (or one, or none). Each value is studied separately.
Find the values of the parameter $b$ for which the following system has a unique solution: $x + by = 2$ $bx + y = 2$
Find the values of the parameter $b$ for which the following system has a unique solution:
$x + by = 2$
$bx + y = 2$
Discuss the system depending on the parameter $c$: $cx - 2y = 1$ $x - cy = 2$
Discuss the system depending on the parameter $c$:
$cx - 2y = 1$
$x - cy = 2$
Discuss the system according to the values of the parameter $m$: $x + y = m$ $m^2 x + y = m^2$
Discuss the system according to the values of the parameter $m$:
$x + y = m$
$m^2 x + y = m^2$
Solutions in the dependent and inconsistent cases
When the system is consistent and dependent, the two equations are the same line. You solve for one unknown in terms of a free parameter $\lambda$.
To find out whether some value makes a system inconsistent, check whether in any of the cases with $\Delta=0$ the equations contradict each other. If $\Delta$ is never zero, the system is always consistent and independent and the answer is "no." A typical case is a determinant of the form $4+t^2$: it is always positive.
Find the value of $a$ for which the following system has infinitely many solutions. Write down the general solution in this case. $2x - y = a$ $-4x + 2y = 8$
Find the value of $a$ for which the following system has infinitely many solutions. Write down the general solution in this case.
$2x - y = a$
$-4x + 2y = 8$
Given the following system of linear equations: $3x - 2y = 4$ $x + y = m$ Is there any value of the parameter $m$ for which the system is inconsistent? Justify your…
Given the following system of linear equations:
$3x - 2y = 4$
$x + y = m$
Is there any value of the parameter $m$ for which the system is inconsistent? Justify your answer mathematically.
Discuss the following system and, in cases where it has infinitely many solutions, express the solutions in terms of a parameter $\lambda$: $mx + (m+1)y = m$…
Discuss the following system and, in cases where it has infinitely many solutions, express the solutions in terms of a parameter $\lambda$:
$mx + (m+1)y = m$
$mx + 2my = 2$
Coefficients with expressions
Coefficients can be expressions like $k+1$, $k-1$, or even fractions and trigonometric functions. The method is the same: compute $\Delta$ and simplify it before setting it equal to zero.
Discuss the following system of equations according to the values of the parameter $k$: $(k+1)x + y = 2$ $kx + (k-1)y = 1$
Discuss the following system of equations according to the values of the parameter $k$:
$(k+1)x + y = 2$
$kx + (k-1)y = 1$
Discuss the following linear system according to the values of the parameter $k$ (given that $k \neq 0$ and $k \neq 1$): $\frac{x}{k} + \frac{y}{k-1} = 1$…
Discuss the following linear system according to the values of the parameter $k$ (given that $k \neq 0$ and $k \neq 1$):
$\frac{x}{k} + \frac{y}{k-1} = 1$
$kx + (k-1)y = 2$
Discuss the following linear system with parameter $\alpha \in [0, 2\pi]$: $(\sin \alpha) x + (\cos \alpha) y = 1$ $(\cos \alpha) x - (\sin \alpha) y = 0$ Is there any…
Discuss the following linear system with parameter $\alpha \in [0, 2\pi]$:
$(\sin \alpha) x + (\cos \alpha) y = 1$
$(\cos \alpha) x - (\sin \alpha) y = 0$
Is there any value of $\alpha$ for which the system does not have a unique solution? Justify your answer mathematically.
Discussion in real-world problems
The parameter can be a tax, the wind, or a concentration. The discussion is identical, but the conclusion is written in the language of the situation.
For the drones: $\Delta=v\cdot v-2\cdot2=v^2-4$, which vanishes at $v=2$ and $v=-2$. For $v\ne\pm2$ they cross at one point. At $v=2$ we get $2x+2y=4$ and $2x+2y=2$ (parallel); at $v=-2$ we get $-2x+2y=4$ and $2x-2y=-2$, which added together give $0=2$ (also parallel). In no case do they coincide. Back to the start: the drones cross at a single point if $v\ne\pm2$ and fly parallel if $v=\pm2$.
Two surveillance drones follow straight paths described by the following system, where $v$ is a parameter related to wind speed: $vx + 2y = 4$ $2x + vy = v$ Discuss for…
Two surveillance drones follow straight paths described by the following system, where $v$ is a parameter related to wind speed:
$vx + 2y = 4$
$2x + vy = v$
Discuss for which values of $v$ the paths of the drones intersect at a single point, are parallel, or coincide completely.
In a market study, the supply and demand curves for a technology product are given by a linear system depending on a tax $t$: $2x - ty = 3$ $tx + 2y = 4$ Discuss the…
In a market study, the supply and demand curves for a technology product are given by a linear system depending on a tax $t$:
$2x - ty = 3$
$tx + 2y = 4$
Discuss the consistency of the system according to the values of $t$. Can it be the case that no equilibrium point exists (that is, that the system is inconsistent) if $t$ is a real number?
In a chemical plant, two liquid solutions are mixed. The equations modelling the mass balance of the process depend on the concentration $c$ of a specific reactant:…
In a chemical plant, two liquid solutions are mixed. The equations modelling the mass balance of the process depend on the concentration $c$ of a specific reactant:
$cx + 4y = c^2$
$x + cy = 2$
Discuss the consistency of the system according to the concentration $c$.
Two parameters
If the system has two parameters, for example $a$ and $b$, you work in two phases: first $\Delta$ (which sometimes depends on only one of them), and then, for each value that makes it zero, you study the second parameter.
Discuss the following system of linear equations according to the values of the parameters $a$ and $b$: $ax + y = b$ $x + ay = 2$
Discuss the following system of linear equations according to the values of the parameters $a$ and $b$:
$ax + y = b$
$x + ay = 2$
In an electrical circuit, the currents $I_1$ and $I_2$ satisfy the following system depending on a variable resistance $R$: $R I_1 + (R+1) I_2 = 2R$…
In an electrical circuit, the currents $I_1$ and $I_2$ satisfy the following system depending on a variable resistance $R$:
$R I_1 + (R+1) I_2 = 2R$
$(R-1) I_1 + R I_2 = R^2$
Discuss the existence and uniqueness of the currents depending on the value of the resistance $R$.
Discuss the system according to the values of the parameters $m$ and $n$: $mx + ny = 1$ $nx + my = 1$ Determine the exact conditions on $m$ and $n$ for the system to be…
Discuss the system according to the values of the parameters $m$ and $n$:
$mx + ny = 1$
$nx + my = 1$
Determine the exact conditions on $m$ and $n$ for the system to be inconsistent.
Conditions on the solution and lines
Sometimes you are asked for more than a classification: for example, that the intersection point lie on an axis. If the point is on the y-axis, substitute $x=0$ into both equations and require that they give the same $y$.
For the relative position of two lines use what you already know: $\Delta\ne0$ intersecting, $\Delta=0$ with proportional equations coincident, $\Delta=0$ with non-proportional equations parallel.
Determine for which values of the parameter $a$ the following system has a unique solution $(x, y)$ such that the point of intersection of both lines lies exactly on the…
Determine for which values of the parameter $a$ the following system has a unique solution $(x, y)$ such that the point of intersection of both lines lies exactly on the y-axis ($x=0$):
$ax + 3y = 6$
$2x - ay = a$
Two high-voltage cables are modelled in a plane by the lines $r: 2x + (k-2)y = k$ and $s: (k-2)x + 2y = 2$. Discuss the relative position of the cables according to the…
Two high-voltage cables are modelled in a plane by the lines $r: 2x + (k-2)y = k$ and $s: (k-2)x + 2y = 2$. Discuss the relative position of the cables according to the values of the parameter $k$ to avoid potential short circuits.
A line and a curve with a parameter
When one equation is of second degree, the number of solutions is given by the discriminant of the equation that results from substituting. With $\Delta_c=b^2-4ac$: positive, two points; zero, one (tangent line); negative, none.
The path of a projectile follows a parabola described by the equation $y = -x^2 + 4x$, while a defence laser beam follows the line $y = mx$. Discuss the number of…
The path of a projectile follows a parabola described by the equation $y = -x^2 + 4x$, while a defence laser beam follows the line $y = mx$. Discuss the number of intersection points between the projectile and the laser beam according to the parameter $m$ (which represents the slope of the laser beam).
Discuss the consistency of the following non-linear system according to the values of the parameter $k$: $x^2 - y^2 = k$ $x + y = 2$
Discuss the consistency of the following non-linear system according to the values of the parameter $k$:
$x^2 - y^2 = k$
$x + y = 2$
A satellite orbits along a circular path modelled in the plane by the equation $x^2 + y^2 = 4$. A comet moves in a straight line according to the equation $x + y = c$,…
A satellite orbits along a circular path modelled in the plane by the equation $x^2 + y^2 = 4$. A comet moves in a straight line according to the equation $x + y = c$, where $c$ is a parameter depending on its initial position. Discuss the number of collision points between the satellite and the comet according to $c$.