Discussing a System of Two Equations
Two drones fly over the Pyramid of the Sun in Teotihuacan. Their paths are straight lines that depend on a parameter $v$ that represents the wind. Depending on the value of $v$, the paths cross at a single point or run parallel (in other systems, they could even be the same line). On this page you will first learn to classify a system of two equations and to see it as lines. Then you will learn to answer that question for all wind values at once.
What it means to classify a system
Each linear equation with two unknowns is a line in the coordinate plane. Solving the system means looking for the points that lie on both lines at the same time. That is why there are only three possibilities:
If the two lines have different slopes, they always intersect at one point (SCD). To be parallel or coincident, the slopes have to be equal.
Classifying a numerical system
The fastest way to classify is to compare the coefficients of the two equations.
Now you can handle a case where a number changes: in the next system, the value of $m$ does not change the slopes.
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.
Extension: the determinant and parameters
Discussing a system with a parameter means classifying it (SCD, SCI, or SI) as a function of a value $m$, $k$, $a$… that appears in its coefficients. The result is not a number: it is a list of cases.
Dividing by coefficients that might be zero is awkward. There is a cleaner way: the determinant of the system.
The procedure is always the same: (1) compute $\Delta$ in terms of the parameter, (2) solve $\Delta=0$, (3) study each value you find, (4) write the conclusion with all the cases.
Move $m$ in the figure. The lines are $x-2y=3$ and $mx+4y=-6$. For what value of $m$ do they lie on top of each other?
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 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 $p$.
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 $t$:
$tx + y = t$
$x + ty = 1$
The special values: dependent or inconsistent?
Each value that makes $\Delta=0$ is studied separately. For a given value of the parameter, the system is SCI or SI, never both.
When the system is SCI, the solution is written with a free parameter.
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:
$3x + ky = 5$
$6x - 4y = 2$
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.
$2x - y = a$
$-4x + 2y = 8$
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 $\lambda$:
$mx + (m+1)y = m$
$mx + 2my = 2$
When it is independent: Cramer's rule
Sometimes the discussion asks for more than classifying: for example, that the solution satisfy a condition, such as lying on the $y$-axis. If $\Delta\neq0$, the solution is obtained directly with Cramer's rule:
To require a condition, such as "the solution lies on the $y$-axis" ($x=0$), set the formula for $x$ equal to $0$ and solve for the parameter. Then check that this value satisfies $\Delta\neq0$.
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 $c$:
$cx - 2y = 1$
$x - cy = 2$
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 ($x=0$):
$ax + 3y = 6$
$2x - ay = a$
Discussion in real-world problems
The parameter can represent a tax, a concentration, a resistance, or wind speed. The mathematical discussion is the same. What changes is that at the end you interpret the result: "the paths coincide if $v=\ldots$", "there is no equilibrium if $t=\ldots$".
When the parameter appears in a real-world context, ask yourself whether it only makes sense within a certain range. For example, if a concentration can't be negative, rule out those values in the conclusion. If $\Delta$ is an expression that is never zero (such as $t^2+4$), the system can't be inconsistent for any real $t$.
Back to the beginning: for the drones, the system is $vx+2y=4$, $2x+vy=v$, with $\Delta=v^2-4$. If $v\neq\pm2$, the paths cross at one point. For $v=2$ and $v=-2$ the ratio of the constant terms is different, so the paths are parallel (SI). In this system they never coincide (there is no SCI case).
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 and complicated coefficients
With two parameters the method doesn't change: compute $\Delta$, set it equal to zero, and study the cases. Now the condition $\Delta=0$ relates the two parameters, and for each case you also have to compare the constant terms.
The determinant works the same way if the coefficients are fractions or expressions such as $\sin\alpha$ and $\cos\alpha$.
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 $k$:
$(k+1)x + y = 2$
$kx + (k-1)y = 1$
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.
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 $b$:
$ax + y = b$
$x + ay = 2$
More practice with multiple parameters and non-numerical coefficients:
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 $k \neq 1$):
$\frac{x}{k} + \frac{y}{k-1} = 1$
$kx + (k-1)y = 2$
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.
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.
A line and a curve
When one of the equations is not linear, such as a parabola or a circle, the system can no longer be classified with $\Delta$. The method is substitution: solve for one unknown from the line, substitute it into the curve, and get a quadratic equation. The number of solutions is given by the discriminant $D=b^2-4ac$:
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 $c$.