Discussing a system of two equations
Two drones fly along straight paths that depend on a parameter $v$ related to the wind. Depending on the value of $v$, their paths cross at a single point or run parallel (in other systems, they could even coincide exactly). To discuss a system is to answer that question for all possible values of the parameter, without solving it value by value.
What "discussing" a system means
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 $m$:
$x - 2y = 3$
$mx + 4y = -6$
Classifying by comparing coefficients
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.
$2x - y = a$
$-4x + 2y = 8$
Typical cases: inconsistent, infinitely many solutions, unique solution
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:
$x + by = 2$
$bx + y = 2$
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 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 $k$:
$(k+1)x + y = 2$
$kx + (k-1)y = 1$
Discussion in real-life problems
The parameter is not always a loose letter: it can represent a tax, a concentration or the gradient of a path. The mathematical discussion is identical, but the result is interpreted in the real situation.
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?
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 $b$:
$ax + y = b$
$x + ay = 2$
Back to the start: To classify the drones' paths according to the wind $v$, compare the ratios of the coefficients of their lines: $\dfrac{v}{2}=\dfrac{2}{v}$ only if $v=\pm2$. For $v\ne\pm2$ the paths cross at one point (unique solution); for $v=2$ and $v=-2$ the ratio of the constant terms is different, so they run parallel (no solution). In this system they never coincide (there is no case with infinitely many solutions).