Exercises: Solutions of a system

Two friends went to a video game store. The first bought 2 action games and 1 strategy game for $80$ €. The second bought 1 action game and 3 strategy games for $90$ €. Check whether the pair $(x, y) = (30, 20)$, where $x$ is the price of an action game and $y$ is the price of a strategy game, is the solution to the system describing this situation.

Express the following system of linear equations in 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} 3x - 2y + z = 5 \\ x + 4y - 3z = -2 \\ 2x - y = 0 \end{cases} $

Look at the graphical representation of the two equations of a first-degree system with two unknowns. Based on the diagram, state the solution to the system and classify it according to its number of solutions (Consistent Determined, Consistent Undetermined or Inconsistent).

Given the following augmented matrix $M^*$ of a system of equations in row echelon form after applying Gaussian elimination, give reasons as to whether the system has a solution and, if so, whether it is unique:

$M^* = \begin{pmatrix} 1 & -1 & 3 & | & 4 \\ 0 & 2 & -1 & | & 1 \\ 0 & 0 & 0 & | & 2 \end{pmatrix}$

In a cafe, the prices of a coffee ($x$), a toast ($y$) and a juice ($z$) are related by the following system:

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

Determine whether the triple $(x, y, z) = (1, 2, 3)$ is a valid solution for the prices of these products.

Without solving, state whether the following two systems of equations are equivalent and justify your answer by comparing their solutions or using elementary row operations:

System 1: $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$

System 2: $\begin{cases} x + y = 5 \\ 2x = 6 \end{cases}$

Find the value of parameter $k$ such that the pair $(x, y) = (2, -1)$ is a solution to the following system of linear equations:

$ \begin{cases} kx + 3y = 1 \\ x + ky = 0 \end{cases} $

Classify the following system of equations according to its geometric interpretation in the plane. To do so, analyze the relationship between the coefficients of the unknowns and the constant terms:

$\begin{cases} 2x - 4y = 6 \\ -x + 2y = -3 \end{cases}$

Justify your answer by stating what relative position the lines represented have.

Lucía is solving the following system of linear equations for a class assignment:

To check whether the triple $(x, y, z) = (2, 1, 0)$ is a solution, Lucía substitutes these values only into the first equation and obtains $2 + 1 - 0 = 3$. Since the equality holds, she states in her notebook that $(2, 1, 0)$ is the unique solution of the system "because there are three equations with three unknowns and the verification was exact".

1. a) Point out and explain in detail the two conceptual errors made by Lucía in her reasoning.
2. b) Rigorously check whether the triple $(2, 1, 0)$ is or is not a solution of the complete system.
3. c) Without needing to row-reduce the matrix, show that the system cannot be consistent independent by adding the first equation to the second and comparing the resulting equation with the third equation.

To analyze the viability of two business telephone rates, a linear system of two equations with two unknowns ($x$ and $y$, where both variables represent costs in euros) needs to be constructed.

1. a) Write a linear equation with non-zero coefficients that has the pair $(x, y) = (20, 15)$ as one of its solutions.
2. b) Add a second linear equation with different coefficients from the previous one, such that the joint system is consistent independent (unique solution) and its only solution is still $(20, 15)$. Justify analytically and geometrically why you are sure that the solution is unique.
3. c) Modify your second equation so that the new system becomes inconsistent. Explain what geometric relationship the two lines now have and what it means for the system to be inconsistent in terms of common points.

A 2nd-year Bachillerato student is analyzing the relative position of three planes in space. The system of equations that defines that position is the following:

1. a) Determine the value of $k$ for which the three planes intersect in a line. How is the system classified in this case according to the number of its solutions?

1. b) If $k = 0$, what is the relative position of the planes? Justify your answer using the rank of the coefficient matrix and of the augmented matrix.

Discuss the consistency of the following system of linear equations depending on the real parameter $m$:

Indicate in which cases the system is Consistent Determined, Consistent Undetermined or Inconsistent. You do not need to solve it.

A digital marketing agency manages advertising for a brand on three platforms: Instagram ($x$), TikTok ($y$) and YouTube ($z$). During the last month, it is known that:

• The total spend across the three platforms was $2500$ €.

• Spending on Instagram was twice the spending on YouTube.

• The sum of the spending on Instagram and TikTok exceeded three times the spending on YouTube by $100$ €.

1. a) Set up a system of linear equations to determine how much was invested in each platform.

1. b) Solve the system using Gaussian elimination.

Given the system depending on the parameter $a$:

1. a) Calculate the value of $a$ so that the system is consistent and indeterminate.

1. b) For the value of $a$ found in the previous section, write the general solution of the system expressing the unknowns $x$ and $y$ in terms of $z$. Use $z = \lambda$.

Consider the matrix equation $A \cdot X = B$, where:

1. a) Justify that the matrix $A$ is regular (has an inverse) by calculating its determinant.

1. b) Find the matrix of unknowns $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ by solving the associated system using the inverse matrix $X = A^{-1} \cdot B$.

A recycling centre processes three types of plastic: PET ($x$), HDPE ($y$) and PVC ($z$). Three shipments have been received with the following compositions in tonnes:

• Shipment 1: $1$ of PET, $2$ of HDPE and $1$ of PVC.

• Shipment 2: $2$ of PET, $3$ of HDPE and $0$ of PVC.

• Shipment 3: $0$ of PET, $1$ of HDPE and $2$ of PVC.

If the centre needs to obtain a total of $10$ tonnes of PET, $17$ of HDPE and $10$ of PVC, is it possible to achieve this by combining these three shipments? Set up the system and discuss whether it has a unique solution using Rouché–Frobenius theorem.

It is known that the following two systems are equivalent (they have the same solution):

System 1: $\begin{cases} x + y = 5 \\ 2x - y = 1 \end{cases}$

System 2: $\begin{cases} ax + y = 7 \\ x + by = 5 \end{cases}$

Find the values of the parameters $a$ and $b$. To do this, first solve System 1 and use the values obtained to substitute into System 2.

Consider the following homogeneous system:

1. a) For which values of $a$ does the system admit only the trivial solution $(0, 0, 0)$?

1. b) Find a value of $a$ for which the system is Consistent Indeterminate and calculate a non-trivial solution for that value.

In a parcel delivery logistics hub in Madrid, three types of urban shipments are managed: Standard ($x$), Express ($y$), and Fragile ($z$). During three consecutive shifts, the following deliveries and total costs are recorded:

• Morning shift: 2 standard shipments, 1 express, and 1 fragile, with a billed cost of $50$ €.

• Afternoon shift: 4 standard shipments, 2 express, and 2 fragile, with a billed cost of $100$ €.

• Night shift: 1 standard shipment, 2 express, and no fragile, with a billed cost of $40$ €.

1. a) Set up the system of three equations with three unknowns that models the billing of the three shifts.
2. b) Look at the equations corresponding to the morning and afternoon shifts. Explain which elementary transformation allows passing from one to the other and reason why one of the two equations is redundant.
3. c) Classify the system formed by the three shifts (consistent independent, consistent dependent, or inconsistent). Based on this classification, is it possible to determine the exact, individual price of each type of shipment? Justify your answer and state how many degrees of freedom the set of possible solutions has.

An agricultural cooperative in Jaén prepares jugs of a blend of two extra virgin olive oil varieties: Picual ($x$), with a production cost of $8$ €/L, and Hojiblanca ($y$), with a cost of $10$ €/L. They wish to obtain a final blend of $100$ litres with an exact average cost of $8,50$ €/L.

1. a) Before performing any analytical calculation, make a reasoned estimate of whether there will be more litres of Picual or Hojiblanca in the jug. Base your prediction on the proximity of the desired average price to the individual costs of each variety.
2. b) Set up the system of two linear equations that allows calculating the required litres of each type of oil and classify it according to the number of solutions by interpreting the relative position of the associated lines.
3. c) Solve the system and check whether the result matches your initial estimate. Would the blend be feasible if the cooperative set an average cost of $11$ €/L while keeping the same oils? Reason analytically what would happen to the values of the unknowns and their real economic meaning.

In a digital marketing agency, they are analysing the impact of three advertising campaigns ($x$, $y$, $z$). The performance of the campaigns depends on a seasonal market factor represented by the parameter $k$. The system of equations describing the return on investment is as follows:

1. a) Discuss the feasibility of the solutions of the system (its consistency) depending on the values that the market parameter $k$ can take, using the Rouché-Frobenius theorem.

1. b) In the case where the system is consistent with infinitely many solutions, obtain the general expression of the solutions. What relationship must exist between campaigns $x$ and $y$ in this scenario?

1. c) If the market factor is $k = 0$, determine whether it is possible to obtain a positive return in all campaigns simultaneously.

Consider the coefficient matrix $A$ and the matrix of constant terms $B$ of a cost system for a transport company:

1. a) Determine for which values of the parameter $a$ the system does not admit a unique solution.

1. b) For the values found in the previous section, study whether the system is consistent with infinitely many solutions or inconsistent. Interpret the result from the point of view of resource management: is there any cost configuration that is impossible to balance?

1. c) Solve the system for $a = 2$ exclusively using the inverse matrix method $X = A^{-1} \cdot B$.

A group of 2nd year Bachillerato students has designed a note-sharing app. They have three types of subscriptions: Basic ($x$), Premium ($y$), and Pro ($z$). At the end of the first month, the following data are collected:

1. The total number of users is 300.

2. If the number of Premium users were doubled, it would equal the sum of Basic and Pro users.

3. The total revenue was 2300 €. The monthly prices are: Basic (2 €), Premium (5 €), and Pro (10 €).

1. a) Set up a system of linear equations to determine the number of users of each type.

1. b) A database error makes the total revenue figure (2300 €) questionable. If we let $m$ be the total revenue, discuss for which values of $m$ the system makes physical sense (non-negative and integer users).

1. c) Is it possible that infinitely many combinations of users satisfy the conditions if we did not know the price of the Pro subscription? Justify your answer based on the theory of degrees of freedom of a system.

Given the matrices $M$ and $N$:

$M = \begin{pmatrix} 1 & 0 & -1 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, \quad N = \begin{pmatrix} 2 & 1 \\ 0 & 3 \\ 1 & -1 \end{pmatrix}$

Consider the matrix equation $M \cdot X = N \cdot C$, where $C = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$.

1. a) Before solving, justify whether the system of equations associated with this equality is consistent and independent (unique solution) by analyzing the rank of matrix $M$.

1. b) Find the column matrices $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ that satisfy the equation.

1. c) If we modify the third row of matrix $M$ to $(1, 2, 3)$, how does the solution to the system change? Classify the resulting new system without fully solving it.

Analyse the following homogeneous system depending on the real parameter $\lambda$:

1. a) Determine the values of $\lambda$ for which the system admits only the trivial solution $(0, 0, 0)$.

1. b) For the values of $\lambda$ that make the system consistent undetermined, find the parametric solution. Do these solutions represent a line or a plane in space?

1. c) Give reasons why this system can never be inconsistent, regardless of the value of $\lambda$.

Three investment portfolios ($A$, $B$, $C$) are composed of shares from three technology sectors: Artificial Intelligence ($x$), Green Energy ($y$) and Biotechnology ($z$).

• Portfolio $A$ has one share from each sector.

• Portfolio $B$ has 2 AI shares, 1 Green Energy share and no Biotechnology shares.

• Portfolio $C$ has 1 AI share and $m$ Biotechnology shares.

It is known that the total market value of the portfolios is 100 €, 80 € and 60 € respectively.

1. a) Write down the system of equations associated with this.

1. b) For what number of Biotechnology shares in portfolio $C$ ($m$) does the system have no solution? What does this mean in financial terms?

1. c) If $m = 2$, is it possible for the price of an AI share to be double that of a Green Energy share? Check this by analysing the consistency of the augmented system with this new condition.

Study the relative position of the following three planes in space depending on the parameter $a$:

1. a) Classify the system of equations formed by the three planes.

1. b) Identify the values of $a$ for which the three planes intersect in a single line. Write down the equation of this line in parametric form.

1. c) Is there any value of $a$ for which the three planes have no points in common? Describe the geometric configuration in that case.

Consider the following system of equations with an extra unknown in the constant terms:

1. a) Discuss the consistency of the system depending on the parameters $a$ and $b$.

1. b) If $a = 1$, what condition must $b$ satisfy for the system to be consistent? What type of consistency would it have?

1. c) In the case $a = -2$, prove that the system is consistent if and only if $b = 0$. What is the geometric interpretation of the planes in this case when $b \neq 0$?

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

A group of students debates the most efficient strategy to classify the system according to the values of $a$:

• Marcos suggests applying the substitution method by solving for $x$ in the first equation.

• Sofía suggests row-reducing the augmented matrix using elementary row operations (Gaussian elimination).

• Daniel suggests calculating the inverse matrix of the coefficient matrix using determinants and adjugates.

1. a) Critically compare the three proposed methods. Justify why Gaussian elimination is conceptually superior in this case and explain the serious limitation of Daniel's method when the system is not consistent independent.
2. b) Apply Gaussian elimination to the augmented matrix to determine, with justification, for which value of $a$ the system is consistent dependent and for which value (if any) it is inconsistent.
3. c) For the case where the system is consistent dependent, find the general solution in terms of a parameter $\lambda \in \mathbb{R}$ and interpret geometrically what the equations of the system represent in three-dimensional space.

A financial analyst designs an investment portfolio distributed across three assets ($x$, $y$, $z$, expressed in thousands of euros). The model requires that the set of feasible solutions be a line in three-dimensional space described by the following parametric equations:

1. a) Construct a linear system of three equations in three variables ($x, y, z$) whose general solution is precisely that line. Ensure that none of the equations is a trivial identity (such as $0 = 0$) or an obvious repetition of another.
2. b) Write the augmented matrix of your system and show, by applying elementary row operations (Gaussian elimination), that the system is consistent dependent, justifying the appearance of a zero row $(0\;0\;0\mid 0)$ and the absence of inconsistent rows $(0\;0\;0\mid c)$ with $c \ne 0$.
3. c) Management adds a fourth condition given by $x + 5y + 3z = k$. Determine with reasoning for which value of the parameter $k$ the new system of four equations preserves exactly the same infinitely many solutions, and what happens geometrically to the consistency of the system if $k$ takes any other value.