Exercises: Systems of equations with two and three unknowns. Classification of systems

Classify the following systems of equations with two unknowns according to the number of solutions they have (Consistent Independent System, Consistent Dependent System or Inconsistent System) after attempting to solve them using your preferred method:

1. a) $\begin{cases} 2x + 3y = 8 \\ 4x + 6y = 15 \end{cases}$

1. b) $\begin{cases} x - y = 2 \\ 2x + y = 7 \end{cases}$

1. c) $\begin{cases} 3x + y = 5 \\ 6x + 2y = 10 \end{cases}$

Solve the following system of equations with three unknowns which is already in row-echelon form:

$\begin{cases} 2x + y - z = 5 \\ \quad \quad 3y + 2z = 7 \\ \quad \quad \quad \quad 4z = 8 \end{cases}$

Given the lines $r$ and $s$ represented in the following graph, write the system of equations defining their intersection and classify it according to its geometry.

Express the following system of linear equations in its matrix form $A \cdot X = B$, clearly identifying the coefficient matrix $A$, the matrix of unknowns $X$ and the matrix of constant terms $B$:

$\begin{cases} x - 2y + 3z = 0 \\ 2x + y - z = 4 \\ 3x - y + 2z = -1 \end{cases}$

Without performing complex calculations, give reasons why the following homogeneous system is Consistent Dependent by analysing the relationship between its equations:

$\begin{cases} x + y + z = 0 \\ 2x + 2y + 2z = 0 \\ 3x - y + 2z = 0 \end{cases}$

Check whether the ordered triple $(x, y, z) = (1, 2, -1)$ is a solution to the following system of equations:

$\begin{cases} 3x - y + z = 0 \\ x + y + z = 2 \\ 2x + 3y - z = 9 \end{cases}$

Geometrically, a linear equation with three unknowns represents a plane in space. Look at the diagram and describe what type of system (Consistent Independent, Consistent Dependent or Inconsistent) would represent the intersection of two planes that are parallel to each other.

In a gaming peripherals shop, three friends buy different products.

• Marcos buys 2 mice and 1 keyboard for $110$ €.

• Julia buys 1 mouse and 2 keyboards for $130$ €.

Set up a system of linear equations to determine the price of each item and classify it according to the number of solutions you expect to obtain.

In a continuous assessment exam, Carlos classifies the following homogeneous system:

$$\begin{cases} x - 2y + z = 0 \\ 2x - 4y + 2z = 0 \\ -x + 2y - z = 0 \end{cases}$$

After observing the proportionality relationships between the equations, Carlos writes in his solution: “Since the second row is twice the first and the third is the first with changed signs, row echelon form yields $0 = 0$. Since the unknowns vanish completely in that equality, the system has no solution and therefore it is an Inconsistent System (SI).”

1. a) Explain with reasoning the two conceptual errors Carlos makes: one regarding the general definition of homogeneous systems and another regarding the interpretation of a zero row of the type $0x + 0y + 0z = 0$.
2. b) Correct the exercise: write the coefficient matrix $A$, determine the correct classification of the system (SCD, SCI, or SI), and justify how many solutions it has.

Design and write a concrete example for each of the following situations, justifying your deduction using the relationship between the coefficients or the structure of the augmented matrix:

1. a) A system of two linear equations with two unknowns ($x$ and $y$) that is inconsistent (SI), such that the first equation must be $2x - 3y = 5$.
2. b) A system of two linear equations with two unknowns ($x$ and $y$) that is consistent undetermined (SCI) and admits among its infinite solutions the particular solution $(x, y) = (1, -2)$.
3. c) A homogeneous system of two linear equations with three unknowns ($x, y, z$). Justify, based on the number of unknowns and the maximum possible rank of the coefficient matrix, why it is impossible for said system to be consistent determined (SCD).

Dos drones de vigilancia, $D_1$ y $D_2$, se desplazan por un plano siguiendo trayectorias rectilíneas dadas por las siguientes ecuaciones:

Drone 1: $2x - 3y = 5$

Drone 2: $x + ky = 2$

Se pide:

1. a) Determina el valor del parámetro $k$ para que los drones choquen en un punto (el sistema sea compatible determinado).

1. b) Is there any value of $k$ for which the drones follow parallel trajectories and never meet? Justify your answer using ranks.

1. c) Graph the situation if $k = -1,5$.

You can do it on paper and send a photo with your answer.

Classify the following system of linear equations according to the number of solutions, using Gaussian elimination:

$\begin{cases} x + 2y - z = 3 \\ 2x - y + z = 1 \\ 5x + 5y - 2z = 10 \end{cases}$

If it is consistent, find its solution or solutions. If it has infinitely many solutions, write the solution in parametric form.

Discuss the consistency of the following system depending on the values of the real parameter $m$, using the Rouché-Frobenius theorem:

$\begin{cases} mx + y + z = 1 \\ x + my + z = m \\ x + y + mz = m^2 \end{cases}$

State for which values of $m$ the system is Consistent Independent, Consistent Dependent or Inconsistent.

Given the equations of three planes in space:

$\pi_1: x + y + z = 2$

$\pi_2: 2x + 3y + z = 3$

$\pi_3: kx + 10y + 4z = 11$

1. a) Determine the value of $k$ so that the three planes intersect in a line (underdetermined consistent system).
2. b) For the found value of $k$, obtain the equation of said line.
3. c) What is the relative position of the planes if $k = 5$?

An Esports team has won a prize of €12 000 to be divided among its three members: the captain (A), the strategist (B) and the sniper (C). It is known that:

• The sum of what B and C receive is equal to what A receives.

• If the strategist (B) received €1 000 more, they would have exactly the same as the sniper (C).

1. a) Set up a system of three equations with three unknowns to determine how much each person receives.
2. b) Classify the system and solve it.

Study the consistency of the following system of 4 equations with 3 unknowns:

$\begin{cases} x + y + z = 6 \\ x - y + z = 2 \\ 2x + z = 5 \\ 3x + y + 2z = 11 \end{cases}$

If the system is consistent, find the unique solution. Otherwise, state which equation causes the inconsistency.

Consider the homogeneous system depending on the parameter $a$:

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

1. a) Classify the system depending on the values of $a$.
2. b) Determine for which values of $a$ the system admits non-trivial solutions ($x=0, y=0, z=0$).
3. c) Solve the system for $a = -2$.

Write a system of three equations with three unknowns that satisfies simultaneously:

1. It is Consistent Dependent.

2. One of its solutions is $(1, 0, 2)$.

3. The coefficient matrix is of the form:

   $A = \begin{pmatrix} 1 & -1 & 1 \\ 2 & 1 & a \\ 3 & 0 & b \end{pmatrix}$

Calculate the required values of $a$ and $b$, and propose consistent constant terms. Justify your construction using ranks.

To classify and solve the system:

$$\begin{cases} x + y + z = 6 \\ 2x - y + z = 3 \\ x - 2y = -3 \end{cases}$$

two students propose different strategies:

• Irene proposes writing the augmented matrix $A^* = (A|B)$ and applying the Gaussian elimination method until obtaining a row echelon system.

• David proposes solving for $x$ directly in the third equation ($x = 2y - 3$), substituting that expression into the first and second equations to reduce the problem to a system of two unknowns ($y, z$), and solving by ordinary substitution.

1. a) Reasonably compare both methods for this specific case: which one requires less algebraic cost and minimizes the risk of arithmetic error? In what way does the absence of the unknown $z$ in the third equation influence this choice?
2. b) Apply the strategy you considered most efficient to formally classify the system and obtain its solution.
3. c) If the initial system had 4 equations with 4 unknowns without any zero coefficients, would David's method maintain its competitiveness against the Gaussian matrix algorithm? Justify your answer in terms of order and scalability.

Based on data from the urban transport survey of the Instituto Nacional de Estadística (INE), a municipal transport company in Spain analyzes the monthly sales of three types of travel passes: Youth Pass ($x$), Standard Pass ($y$), and Senior Pass ($z$). The following data from a study period are known:

• A total of $12\,000$ transport passes were issued.

• The monthly fare for each pass is $20$ € for the Youth Pass, $40$ € for the Standard Pass, and $10$ € for the Senior Pass, reaching an overall revenue of $290\,000$ €.

• The number of youth passes sold was exactly twice the combined sum of the standard and senior passes.

1. a) Model the situation by setting up the corresponding system of three linear equations with three unknowns and write it in matrix form $A \cdot X = B$, accurately identifying what each matrix and vector represents.
2. b) Write the augmented matrix $A^* = (A|B)$ associated with the problem.
3. c) Row-reduce the augmented matrix using elementary row operations to classify the system according to the number of solutions (SCD, SCI, or SI). Based on this classification, interpret whether the data provided by the municipal company allows knowing with certainty the exact number of passes sold of each type or if the information is insufficient or contradictory.

Discuss the following system of linear equations depending on the various values of the real parameters $a$ and $b$. Solve it only in the case where the system is consistent dependent (infinitely many solutions):

$$\begin{cases} ax + y + z = b \\ x + ay + z = 1 \\ x + y + az = 1 \end{cases}$$

A biotechnology laboratory is developing a new serum by combining three compounds: $A$, $B$ and $C$. The cost per gram of each compound depends on an import factor $k$. The system governing the required quantities $x, y, z$ (in grams) to stabilise 100 ml of serum is:

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

1. a) Determine for which values of $k$ the laboratory cannot determine a unique quantity of each compound to manufacture the serum.
2. b) If $k = -2$, is it possible to manufacture the serum following these proportions? Justify your answer based on the consistency of the system.

Consider the three planes in space whose equations are:

$\pi_1: x + ay + z = 1$

$\pi_2: ax + y + (a-1)z = a$

$\pi_3: x + ay + (a+1)z = b$

Study the relative position of the three planes depending on the values of the parameters $a$ and $b$. For which values of the parameters do the three planes intersect in a single line?

Consider the homogeneous system depending on the parameter $m$:

$$\begin{cases} mx + y + z = 0 \\ x + my + z = 0 \\ mx + my + 2z = 0 \end{cases}$$

1. a) Determine the value of $m$ for which the system admits non-trivial solutions.
2. b) For said value of $m$, find the general solution and give a non-trivial particular solution.

Discuss the consistency of the following system of 4 equations with 3 unknowns depending on the values of the parameter $k$:

$$\begin{cases} x + y + z = 3 \\ 2x + y - z = 2 \\ x - y + kz = 1 \\ 3x + 2y + (k-1)z = 5 \end{cases}$$

In the cases where it is consistent, determine whether the solution is unique.

Three courier companies ($P_1, P_2, P_3$) share the market of a city. Each month, 20% of the customers of $P_1$ switch to $P_2$ and 10% to $P_3$. Of the customers of $P_2$, 10% switch to $P_1$ and 10% to $P_3$. Of the customers of $P_3$, 10% switch to $P_1$ and 20% to $P_2$.

If the current market shares are $x, y, z$ and we want the market shares to remain identical next month (equilibrium state), set up a system of equations. Show that this system is consistent dependent and calculate the relationship between the market shares at equilibrium.

Given the system of equations:

$$\begin{cases} x + 2y + z = 3 \\ x + y - 3z = 2 \end{cases}$$

1. a) Add an equation of the form $ax + by + cz = d$ so that the resulting system is consistent and independent (unique solution) and its solution is $(1, 1, 0)$.
2. b) Add an equation so that the system is inconsistent.
3. c) Is it possible to add an equation so that the system is consistent and dependent (infinitely many solutions) but with a different solution set from that of the original system? Justify your answer.

Determine the values of the parameters $a$ and $b$ for which the following system has infinitely many solutions depending on a single parameter:

$$\begin{cases} x + ay + 2z = 3 \\ x + (2a-1)y + (a+2)z = 2b + 3 \\ x + y + az = b \end{cases}$$

For the values found, write the solution in parametric form.

At an AEMET environmental monitoring station, the normalized theoretical records of three parameters ($x$: temperature, $y$: relative humidity, $z$: pressure) must satisfy the following system of linear equations, where $k$ represents a reading transmitted by an auxiliary sensor:

$$\begin{cases} x + 2y - z = 3 \\ 2x + 4y - 2z = k \\ -x - 2y + z = -3 \end{cases}$$

1. a) Estimation phase: Without performing formal row-reduction operations, analyze the coefficients of the unknowns and the constant terms of the three equations. Estimate and explain what type of system you expect to obtain under two scenarios: $k = 6$ and $k \neq 6$. Geometrically interpret what each equation represents (a plane in $\mathbb{R}^3$) and anticipate the mutual relative position of the three planes in each scenario.
2. b) Verification phase: Write the augmented matrix $A^* = (A|B)$ and apply the Gauss method using elementary linear combinations to formally classify the system as a function of the parameter $k$.
3. c) Comparison and critical assessment: Compare the algebraic deduction obtained in (b) with your previous estimate in (a). In the case $k = 6$, determine how many free parameters the solution set depends on and explain why the actual geometric configuration corresponds to coincident planes and not to planes intersecting in a line, qualifying the initial intuition about the system's degrees of freedom.

The analytical and theoretical construction of systems of three linear equations with three unknowns ($x, y, z$) is considered:

1. a) Design a linear system of three equations with three unknowns $A \cdot X = B$, requiring that all coefficients of matrix $A$ be non-zero real numbers, such that the overall system is inconsistent (SI), but with the condition that any subsystem formed by any two equations is consistent undetermined (SCI). Explicitly write the three equations, express the augmented matrix $A^* = (A|B)$, and prove that both conditions are rigorously met.
2. b) Interpret the relative position of the three planes represented by the equations of your system in three-dimensional space and describe the geometric shape they delimit between each other.
3. c) Now consider the associated homogeneous system $A \cdot X = 0$, using exactly the same coefficient matrix $A$ that you constructed in section (a). Reason out the following questions without performing additional algebraic calculations:

1. Why is it structurally impossible for this homogeneous system to be inconsistent?

2. Does it admit only the trivial solution $(0, 0, 0)$ or does it have infinitely many solutions? Justify your deduction by relating the linear dependence you gave to the rows of $A$ in section (a) to the rank of said matrix.