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Systems of linear equations · Systems

Systems with parameters

You have seen Rouché-Frobenius, Cramer and Gauss separately. Discussing a system with a parameter is where all three come together: you calculate $|A|$ to find the "special" values of the parameter, use Gauss or minors to find the ranks at those values, and Cramer to solve when the system is determinate. It is the longest type of exercise in the whole block, and one of the favourites in the university entrance exam (EvAU).

Before you start, make sure you know: Factorising polynomials (1st Bachillerato)
Before you start, make sure you know: The Gauss method (1st Bachillerato)

The discussion plan

Discussion outline depending on whether the determinant is non-zero (consistent determinate system) or zero (study the ranks separately). |A(k)| ≠ 0 SCD for that k |A(k)| = 0 study rg(A), rg(A*) separately
Practice★☆☆
On a streaming platform, the price of three types of subscriptions (Basic, Standard and Premium) varies according to a monthly promotion represented by the parameter…

On a streaming platform, the price of three types of subscriptions (Basic, Standard and Premium) varies according to a monthly promotion represented by the parameter $k$. The system of equations relating the number of users of each type to total revenue is:

$$\begin{cases} x + y + z = 10 \\ x + ky + z = 12 \\ x + y + kz = 14 \end{cases}$$

Discuss the consistency of the system according to the values of the real parameter $k$.

x + y + z = 10 x + ky + z = 12 x + y + kz = 14 k k X Y Z Solution
Solve

Special values: consistent indeterminate or inconsistent, never both

Bifurcation diagram for $|A|=0$: consistent indeterminate if the ranks coincide or inconsistent if they differ. special parameter value (|A| = 0) rg(A) = rg(A*) → SCI rg(A) ≠ rg(A*) → SI
Practice★☆☆
Determine for which value of the parameter $a$ the following system of linear equations is inconsistent: $$\begin{cases} x + 2y = 3 \\ ax + 4y = 1 \end{cases}$$ Justify…

Determine for which value of the parameter $a$ the following system of linear equations is inconsistent:

$$\begin{cases} x + 2y = 3 \\ ax + 4y = 1 \end{cases}$$

Justify your answer by analysing the proportionality of the coefficients and the constant terms.

Solve
Practice★★☆
Consider the following system of equations depending on the real parameter $k$: $\begin{cases} kx + y + z = 1 \\ x + ky + z = k \\ x + y + kz = k^2 \end{cases}$ a)…

Consider the following system of equations depending on the real parameter $k$:

$\begin{cases} kx + y + z = 1 \\ x + ky + z = k \\ x + y + kz = k^2 \end{cases}$

  1. a) Discuss the system according to the values of the parameter $k$.
  2. b) Solve the system in the case where it is consistent with infinitely many solutions.
Solve

Homogeneous systems with a parameter

Discussion outline for homogeneous systems with the trivial unique solution if $|A| \ne 0$ or infinitely many solutions if $|A| = 0$. A·X = 0 homogeneous system |A(m)| ≠ 0 only (0,0,0) |A(m)| = 0 infinitely many
Practice★☆☆
Consider the following homogeneous system of equations, where $m$ is a real parameter: $$\begin{cases} mx + y + z = 0 \\ x - my + z = 0 \\ 2x + y + z = 0 \end{cases}$$…

Consider the following homogeneous system of equations, where $m$ is a real parameter:

$$\begin{cases} mx + y + z = 0 \\ x - my + z = 0 \\ 2x + y + z = 0 \end{cases}$$
  1. a) Calculate the determinant of the coefficient matrix $A$.
  2. b) For which values of $m$ does the system have solutions other than the trivial solution $(0, 0, 0)$?
Solve

Solving the consistent indeterminate case

When the system is consistent and indeterminate, the general solution is expressed with $n-rg(A)$ parameters. In many problems, you will also be asked for a particular solution that satisfies some additional condition (for example, that the components add up to a given value).

Flow chart for obtaining a particular solution from the general solution of a consistent indeterminate system and an extra condition. general solution: (x(λ), y(λ), λ) extra condition → equation in λ → specific λ → substitute and obtain the particular triple
Practice★★★
Given the homogeneous system: $\begin{cases} ax + y + 3z = 0 \\ x + ay + 2z = 0 \\ x + y + az = 0 \end{cases}$ 1. Determine the values of $a$ for which the system admits…

Given the homogeneous system:

$\begin{cases} ax + y + 3z = 0 \\ x + ay + 2z = 0 \\ x + y + az = 0 \end{cases}$

  1. Determine the values of $a$ for which the system admits non-trivial solutions.

  2. For the largest value of $a$ found in the previous part, obtain the general solution and find a particular solution such that the sum of its components is $10$.

  3. If $a=0$, describe the solution set geometrically.

Solve
Practice★★★
Consider the following system of linear equations depending on the real parameter $k$: $\begin{cases} kx + y + z = k^2 \\ x + ky + z = k \\ x + y + kz = 1 \end{cases}$…

Consider the following system of linear equations depending on the real parameter $k$:

$\begin{cases} kx + y + z = k^2 \\ x + ky + z = k \\ x + y + kz = 1 \end{cases}$

You are required to:

  1. Discuss the system according to the different values of the parameter $k$.

  2. Solve the system in the cases where it is consistent with infinitely many solutions, expressing the solution in parametric form.

  3. If the system is consistent with a unique solution, is there any value of $k$ for which the solution is $(x, y, z) = (1, 1, 1)$? Justify your answer.

Solve

Summary

Let's practice! 10 medium exercises in this section
★★☆ Consider the following system of equations depending on the real parameter $k$:…★★☆ Given the system of equations:…★★☆ A logistics company organises shipments of three types of containers: A, B, and C. The…
Let's practice! 11 hard exercises in this section
★★★ Consider the following system of linear equations depending on the real parameter $k$:…★★★ Three planes in space $\mathbb{R}^3$ are given by the following equations, where $a$ is a…★★★ A last-mile logistics company uses three types of vehicles: electric vans ($F$), cargo…
See all the exercises in this section