Systems with a Parameter
The Lakeside High snack stand has two receipts from the same lunch period, but the second one is smudged. Receipt A says 1 pretzel and 3 lemonades cost $\$4$. Receipt B says 2 pretzels and $k$ lemonades cost $\$5$, and the number $k$ is unreadable. Can the treasurer always find the price of a pretzel $x$ and a lemonade $y$? That depends on $k$:
$$\begin{cases} x + 3y = 4 \\ 2x + ky = 5 \end{cases}$$A letter like $k$ in a system is a parameter: a number that is fixed but not yet known. In this lesson you learn to discuss a system, which means deciding for each value of the parameter whether the system has one solution, infinitely many, or none.
Two equations: comparing coefficients
You already know the three possible outcomes for a linear system from Integrated Math I. Each has a name and a picture.
| Graph of the two lines | Number of solutions | Name |
|---|---|---|
| The lines cross | exactly one | consistent and independent |
| The lines are the same line | infinitely many | consistent and dependent |
| The lines are parallel | none | inconsistent |
With a parameter, the coefficients decide which picture you get. Write the system as $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$. Eliminating $y$ (multiply the first equation by $b_2$, the second by $b_1$, and subtract) gives
$$(a_1b_2 - a_2b_1)\,x = c_1b_2 - c_2b_1.$$Eliminating $x$ in the same way gives $(a_1b_2 - a_2b_1)\,y = a_1c_2 - a_2c_1$. Everything depends on the number $D = a_1b_2 - a_2b_1$.
Slide the parameter and watch the second line. At $a = 6$ the lines have the same slope, so they never meet. For any other value they cross once.
Discuss the following system of equations in terms of the parameter $m$: $ \left\{ \begin{array}{l} 2x - 3y = 5 \\ 4x - 6y = m \end{array} \right. $
Discuss the following system of equations in terms of the parameter $m$:
$ \left\{ \begin{array}{l} 2x - 3y = 5 \\ 4x - 6y = m \end{array} \right. $
Determine the value of $k$ for the following system to be consistent with infinitely many solutions and interpret the result geometrically:…
Determine the value of $k$ for the following system to be consistent with infinitely many solutions and interpret the result geometrically:
$ \left\{ \begin{array}{l} kx + 2y = 4 \\ 3x + 6y = 12 \end{array} \right. $
Calculate for which value of the parameter $a$ the system is inconsistent: $ \left\{ \begin{array}{l} x + ay = 7 \\ 2x - 4y = 3 \end{array} \right. $
Calculate for which value of the parameter $a$ the system is inconsistent:
$ \left\{ \begin{array}{l} x + ay = 7 \\ 2x - 4y = 3 \end{array} \right. $
When the special values come from a quadratic
Often $D$ is not linear in the parameter. In that case set $D = 0$, factor, and treat each root as its own case.
The picture above matches the next example. Away from the special values you get one solution, and each special value gets its own verdict.
To solve the system for one particular value of the parameter, substitute it and use substitution or elimination from Integrated Math I.
Discuss the following system in terms of the parameter $b$: $ \left\{ \begin{array}{l} x - 2y = 1 \\ bx + 4y = -2 \end{array} \right. $
Discuss the following system in terms 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 in terms of the tax $p$.
Discuss the following system in terms of the parameter $k$ and indicate for which values the system represents two intersecting, parallel or coincident lines. Solve it…
Discuss the following system in terms of the parameter $k$ and indicate for which values the system represents two 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. $
Using a known solution to find the parameter
Sometimes you are told that a point is a solution and asked for the parameter. A point is a solution only if it satisfies every equation, so substitute its coordinates into the equations.
Find the value of the parameter $t$ for the pair $(2, -1)$ to be the solution of the system: $ \left\{ \begin{array}{l} 3x + ty = 8 \\ tx - 5y = 9 \end{array} \right. $
Find the value of the parameter $t$ for the pair $(2, -1)$ to be the solution of the system:
$ \left\{ \begin{array}{l} 3x + ty = 8 \\ tx - 5y = 9 \end{array} \right. $
Three equations: Gaussian elimination with a parameter
With three unknowns you use Gaussian elimination. You use row operations to turn the system into echelon form, where each equation starts one unknown later than the one above it. Then you read off the last equation.
The allowed row operations are: swap two equations, multiply an equation by a nonzero number, and add a multiple of one equation to another. With a parameter there is one extra rule: do not multiply or divide an equation by an expression that contains the parameter, because it might equal $0$.
Given the following system in echelon form, discuss its solutions in terms of the parameter $p$:…
Given the following system in echelon form, discuss its solutions in terms 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 in terms of the parameter $m$:…
Apply Gaussian elimination to discuss the following system in terms 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 distributes their capital across three investment funds. The profitability of the third fund depends on an interest rate $r$. The equations that model the…
An investor distributes their capital across three investment funds. The profitability of the third fund depends on an interest rate $r$. The equations that model 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 $r$.
Pivots that factor, and choosing the first row
In harder problems the last coefficient is a polynomial in the parameter, such as $a^2 - 1$. Factor it. Every root is a separate case.
Discuss the consistency of the following system in terms of the parameter $a$:…
Discuss the consistency of the following system in terms of the parameter $a$:
$ \left\{ \begin{array}{l} ax + y + z = 1 \\ x + ay + z = 1 \\ x + y + az = 1 \end{array} \right. $
In the analysis of an urban traffic network, the flow of vehicles per hour at three main intersections is modeled by a system of equations. Due to construction, one of…
In the analysis of an urban traffic network, the flow of vehicles per hour at three main intersections is modeled by a system of equations. Due to construction, 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 $m$.
Discuss the following system in terms of the parameter $\alpha$. Pay special attention to the values that cancel out the leading coefficients during the Gaussian…
Discuss the following system in terms of the parameter $\alpha$. Pay special attention to the values that cancel out the leading coefficients during the Gaussian elimination process.
$ \left\{ \begin{array}{l} x + \alpha y - z = 2 \\ 2x - y + \alpha z = 5 \\ x + 10y - 6z = 1 \end{array} \right. $
Writing the solutions of a consistent case
After the discussion you are often asked to solve the system for a chosen value. If the case has one solution, substitute the parameter and back-substitute. If it has infinitely many, name the free unknown with a new letter, such as $\lambda$ (lambda), and express the others in terms of it.
For $m \ne 5$ the system has one solution, $z = 0$, $y = 3$, $x = -1$. Check: $-1 + 3 + 0 = 2$, $-2 + 9 = 7$, $-3 + 12 = 9$. ✓
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. $
Discuss the following system of linear equations in terms of the parameter $k$, using Gaussian elimination. For the case where it is consistent with infinitely many…
Discuss the following system of linear equations in terms of the parameter $k$, using Gaussian elimination. For the case where it is consistent with 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 of equations in terms of the values of the parameter $m$ and find the solution when it is consistent and indeterminate:…
Discuss the following system of equations in terms of the values of the parameter $m$ and find the solution when it is consistent and indeterminate:
$ \left\{ \begin{array}{l} 3x - y + 2z = 1 \\ x + 4y + z = m \\ 2x - 5y + mz = -1 \end{array} \right. $
Homogeneous systems
Determine the value of $c$ for the following homogeneous system to have infinitely many solutions:…
Determine the value of $c$ for the following homogeneous system to have infinitely many solutions:
$ \left\{ \begin{array}{l} cx + y = 0 \\ x + cy = 0 \end{array} \right. $
In a chemical experiment, three substances are mixed. The equilibrium conditions of the experiment depend on a catalyst whose concentration is $c$. The mathematical…
In a chemical 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 ($x=0, y=0, z=0$).
Geometry: three lines in the plane
Each equation in $x$ and $y$ is a line. Discussing the system means describing how the lines sit relative to each other. Two lines cross, are parallel, or coincide, which is exactly the three pictures from the start of this lesson, and $D$ tells you which one. In the 2×2 case, for the lines $5x + 2y = 1$ and $10x + 4y = k$, we get $D = 20 - 20 = 0$, so the lines are always parallel or the same line. Only the constant $k$ decides which.
With three lines in the plane you have three equations and only two unknowns. Solve two of them, then test the third.
(Some problems call three lines through one point a pencil of concurrent lines.)
Study the relative position of the lines $r: 5x + 2y = 1$ and $s: 10x + 4y = k$ in terms of the parameter $k$.
Study the relative position of the lines $r: 5x + 2y = 1$ and $s: 10x + 4y = k$ in terms of the parameter $k$.
Three trajectories of surveillance drones in a plane are given by the equations of three lines. Determine the value of the parameter $a$ for the three drones to pass…
Three trajectories of surveillance drones in a plane are given by the equations of three lines. Determine the value of the parameter $a$ for the three drones to pass through the exact same point (i.e., for the system to have a unique solution).
$ \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: $r: x + 2y = 3$ $s: 2x - y = 1$ $t: mx + (m+1)y = 4$ Discuss the relative position of the three lines in…
Consider the three lines in the plane given by the equations:
$r: x + 2y = 3$
$s: 2x - y = 1$
$t: mx + (m+1)y = 4$
Discuss the relative position of the three lines in terms of the parameter $m$. Observe the following diagram that illustrates the situation in which the three lines intersect at a single point (pencil of concurrent lines).
Geometry: three planes in space
With three unknowns, each equation is a plane. A solution of the system is a point that lies on all three planes. Looking at the right pictures tells you what kind of discussion to expect.
| Picture | Solutions |
|---|---|
| The three planes meet at one point | exactly one |
| The three planes share a common line, like pages around a book's spine | infinitely many (one free variable) |
| All three planes are the same plane | infinitely many (two free variables) |
| Two planes are parallel and different, and the third cuts them | none |
| The planes cross pairwise in three parallel lines (a triangular prism) | none |
Study the relative position of the three planes in three-dimensional space given by the following system, in terms of the parameter $a$.…
Study the relative position of the three planes in three-dimensional space given by the following system, in terms of the parameter $a$.
$ \left\{ \begin{array}{l} 2x - y + 3z = 1 \\ x + ay - z = 2 \\ 3x + 2y + 2z = a \end{array} \right. $
Accompany the discussion with the following graphical representation of the geometric situation when the system is inconsistent and the planes intersect in pairs forming a triangular prism.
Back to the snack stand: the receipts give $x + 3y = 4$ and $2x + ky = 5$, with $D = k - 6$. For every $k \ne 6$ the treasurer can find one price for the pretzel and one for the lemonade. If $k = 6$, the second receipt would be exactly double Receipt A, which should cost $\$8$, but it says $\$5$. The receipts contradict each other and no prices fit.