Exercises: Solving systems and Cramer's rule

Two friends, Lucía and Marcos, are comparing their subscriptions to streaming platforms. Lucía pays a total of $16$ € per month for one "Cinema" plan and two "Series" plans. Marcos, for his part, has two "Cinema" plans and one "Series" plan for $17$ € per month. The figure summarises the plans each of them has.

1. a) Express the situation as a system of two equations with two unknowns (let $x$ be the monthly price of the Cinema plan and $y$ that of the Series plan).

1. b) Calculate the determinant of the coefficient matrix.

1. c) Solve the system using Cramer's Rule to find the monthly price of each plan.

A nutritionist designs a personalised diet for an athlete by combining three types of food ($A, B$ and $C$). The amounts of protein, carbohydrates and fats per unit of food are given by the following system of equations, where $x, y, z$ represent the units of each food required to reach the daily targets:

$\begin{cases} 2x + y + z = 7 \\ x + 2y + z = 8 \\ x + y + 2z = 9 \end{cases}$

1. a) Write down the coefficient matrix $A$ and the augmented matrix $A^*$.
2. b) Verify that the system is a Cramer system by calculating the determinant of $A$.
3. c) Find the value of the unknown $x$ applying exclusively Cramer's rule.

In an esports tournament, points are awarded for matches won ($x$), drawn ($y$) and for secondary objectives completed ($z$). A team obtained the following results over three matchdays:

• Matchday 1: $1$ win, $1$ draw and $1$ secondary objective = $10$ points.

• Matchday 2: $2$ wins and $1$ draw = $13$ points.

• Matchday 3: $1$ win and $2$ secondary objectives = $11$ points.

1. a) Set up the associated system of equations.
2. b) Solve it using Cramer's rule to determine how many points each category is worth.

Determine with reasons which of the following systems of equations is a Cramer system and, therefore, has a unique solution that can be found using determinants:

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

1. b) $\begin{cases} x + 2y = 5 \\ 3x - y = 1 \end{cases}$

1. c) $\begin{cases} x - y + z = 2 \\ 2x + y = 3 \end{cases}$

Justify your answer by analysing the number of equations, the number of unknowns and the value of the determinant of the coefficient matrix.

Given the system of linear equations:

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

Use Cramer's rule to calculate only the value of the unknown $z$. Clearly state the two determinants you had to calculate to obtain the result.

In a school cafeteria, a group of teachers has placed two orders:

• Order 1: $3$ coffees and $2$ toasts for a total of $7,10$ €.

• Order 2: $2$ coffees and $4$ toasts for a total of $7,40$ €.

Let $x$ be the price of a coffee and $y$ the price of a toast (in euros).

1. a) Express the problem in matrix form $A \cdot X = B$, stating the matrices $A$, $X$ and $B$.

1. b) Calculate the inverse matrix $A^{-1}$.

1. c) Although the system can be solved by matrix inversion, apply Cramer's rule to obtain the price of a coffee and of a toast, and check that they match those given by $X = A^{-1} \cdot B$.

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

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

1. a) Find the determinant of the coefficient matrix in terms of $k$.
2. b) Determine for which values of $k$ the system cannot be solved using Cramer's rule.
3. c) For $k = 2$, solve the system using this rule.

Geometrically, three planes in space can intersect at a single point. This point represents the unique solution of a system of three equations in three unknowns.

Given the system representing the position of three planes:

$\begin{cases} x + y + z = 2 \\ 2x + 3y + 5z = 11 \\ x - 5y + 6z = 29 \end{cases}$

Prove, using the determinant of the coefficient matrix, that the three planes intersect at a single point and find its coordinates $(x, y, z)$ using Cramer's rule.

Hugo is solving the following system of two linear equations with two unknowns using Cramer's rule:

$$\begin{cases} 3x - y = 7 \\ 2y + x = 7 \end{cases}$$

On his exam, he wrote the following working:

"I write the coefficient matrix $A = \begin{pmatrix} 3 & -1 \\ 2 & 1 \end{pmatrix}$. Its determinant is $|A| = 3\cdot 1 - (-1)\cdot 2 = 5 \ne 0$, so it is a Cramer's system.

I substitute the column of constants to find the unknown $x$:

$|A_x| = \begin{vmatrix} 7 & -1 \\ 7 & 1 \end{vmatrix} = 7 - (-7) = 14$.

Therefore, applying Cramer's rule: $x = \dfrac{|A|}{|A_x|} = \dfrac{5}{14}$".

1. a) Identify and explain in detail the two conceptual errors made by Hugo in his solution.
2. b) Write the correct coefficient matrix $A$, calculate its determinant, and justify why the system is determinate consistent according to Cramer's criterion.
3. c) Correctly apply Cramer's rule to calculate the exact and complete solution $(x, y)$ of the system.

In a greenhouse farm in El Ejido (Almería), fertilizer solutions are prepared by combining two liquid fertilizers, $A$ and $B$. Fertilizer $A$ contains $10\,\%$ nitrogen and fertilizer $B$ contains $30\,\%$. The technical team needs to obtain exactly $100$ liters of a mixture containing $25\,\%$ total nitrogen. The situation is modeled by the system:

$$\begin{cases} x + y = 100 \\ 0,10x + 0,30y = 25 \end{cases}$$

where $x$ represents the volume in liters of fertilizer $A$ and $y$ the volume of fertilizer $B$.

1. a) Estimate with reasoning: Without solving the system, note that the desired final concentration ($25\,\%$) is much closer to that of fertilizer $B$ ($30\,\%$) than to that of $A$ ($10\,\%$). Do you expect the required volume of $B$ to be greater than or less than that of $A$? Will it exceed $50$ liters? Justify your qualitative answer before calculating.
2. b) Justify that the system is a Cramer's system by calculating the determinant of the coefficient matrix.
3. c) Apply Cramer's rule to calculate the exact liters of each fertilizer ($x$ and $y$) and check whether the numerical result matches your previous estimate.

Solve the following system of linear equations using Cramer's rule:

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

To do this, first identify the coefficient matrix $A$ and calculate its determinant. Then, obtain the determinants of the auxiliary matrices $A_x$, $A_y$ and $A_z$ to find the values of the unknowns.

A logistics company uses three types of drones ($D_1$, $D_2$ and $D_3$) to deliver parcels in a city. It is known that:

• In one hour, 2 $D_1$ drones, 1 $D_2$ drone and 3 $D_3$ drones can deliver a total of 28 parcels.

• If 1 $D_1$ drone, 3 $D_2$ drones and 2 $D_3$ drones operate, they deliver 25 parcels.

• With 3 $D_1$ drones, 2 $D_2$ drones and 1 $D_3$ drone, the capacity is 27 parcels.

Set up a system of equations where the unknowns are the hourly delivery capacity of each type of drone and solve it using Cramer's rule.

Given the system 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) Show that for $k = 2$ the system is a Cramer system.
2. b) Solve the system for this value of $k = 2$ using Cramer's method.

A data centre consumes energy from three sources: solar ($x$), wind ($y$) and hydro ($z$). The costs per MWh and total production must satisfy the following financial equilibrium:

$\begin{cases} x + y + z = 100 \\ 5x + 10y + 15z = 950 \\ 2x - y + 3z = 110 \end{cases}$

Calculate the amount of MWh required from each source using Cramer's rule. Justify whether the result obtained is consistent with the context (non-negative quantities).

Determine the inverse matrix of $A = \begin{pmatrix} 1 & -1 & 0 \\ 0 & 2 & 1 \\ 1 & 0 & 1 \end{pmatrix}$ and use it to solve the system of equations $AX = B$, where $B = \begin{pmatrix} 2 \\ 5 \\ 4 \end{pmatrix}$.

Relate this method to Cramer's rule, commenting on the role played by the determinant of the matrix in both procedures.

Three planes in space are given by the following equations:

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

$\pi_2 \equiv 2x - y + 3z = 9$

$\pi_3 \equiv 3x + 2y - z = -1$

Determine the relative position of the three planes by solving the system using Cramer's rule. If the system has a unique solution, what does that point represent geometrically with respect to the three planes?

In a chemistry laboratory, 10 litres of a solution with a specific concentration are to be made by mixing three products $A, B$ and $C$ whose alcohol concentrations are $10\%$, $20\%$ and $40\%$ respectively.

• The final mixture is required to have a concentration of $25\%$.

• The amount of product $C$ must be equal to the sum of the amounts of $A$ and $B$.

Set up the system of equations and use Cramer's rule to find how many litres of each product are needed.

Consider the following homogeneous system of linear equations:

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

1. a) Determine the value of $k$ for the system to be consistent with a unique solution.
2. b) For a value of $k$ that makes the system a Cramer system (for example, $k=1$), verify using Cramer's rule that the only solution is the trivial solution $(0, 0, 0)$.
3. c) Explain graphically what it means for the only solution to be the trivial solution in terms of planes passing through the origin.

A telecommunications company analyzes data traffic among three core nodes ($N_1$, $N_2$, and $N_3$). The average latencies $x$, $y$, and $z$ (in milliseconds) satisfy the following system:

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

To urgently configure the network firewall, the technical support team only needs to know the value of latency $z$, without requiring the values of $x$ or $y$.

1. a) Verify that this is a Cramer's system by calculating the determinant of the coefficient matrix $A$.
2. b) Choose the method: Compare the strategy of solving the entire system using Gaussian elimination versus using Cramer's rule for this specific purpose. Explain why the presence of zero coefficients in the equations and the need to calculate a single unknown make Cramer's rule computationally more direct here.
3. c) Apply Cramer's rule to calculate the exact value of $z$ by evaluating only the strictly necessary determinants.

A winery with the La Rioja designation of origin markets three types of wine: Crianza ($x$), Reserva ($y$), and Gran Reserva ($z$). During a national distribution campaign, sales to three commercial chains yield the following data:

• The Madrid distributor purchases $100$ bottles of Crianza, $50$ of Reserva, and $20$ of Gran Reserva for a total amount of €$1940$.

• The Barcelona distributor buys $60$ bottles of Crianza, $80$ of Reserva, and $40$ of Gran Reserva for a total amount of €$2480$.

• The Bilbao distributor buys $40$ bottles of Crianza, $40$ of Reserva, and $60$ of Gran Reserva with an invoice of €$2200$.

1. a) Set up the system of three linear equations in three unknowns describing the sales revenue and simplify each equation by dividing it by the corresponding numerical common factor.
2. b) Prove that the simplified system is a Cramer's system by calculating the determinant of the coefficient matrix. What does this result guarantee regarding the unit prices of the bottles?
3. c) Solve the system using Cramer's rule to determine the price per bottle in euros for each type of wine ($x$, $y$, $z$).

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

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

1. a) Discuss the system depending on the different values of $k$ using the Rouché-Frobenius theorem.
2. b) In the case where the system is consistent dependent, obtain the general solution expressed in parametric form. What does this solution set represent geometrically in space $\mathbb{R}^3$?
3. c) Solve the system for $k = 0$ using Cramer's rule.

Given the matrices $A = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 1 \\ m & 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 1 \\ 0 & 1 \\ 1 & 0 \end{pmatrix}$:

1. a) Determine the values of the real parameter $m$ for which the matrix $A$ is invertible.
2. b) For $m = 0$, solve the matrix equation $A \cdot X + B = 2B$, where $X$ is a matrix of appropriate dimensions.
3. c) Set up a system of equations to calculate the first column of the matrix $X$ and solve it using Cramer's rule.

An aerospace logistics company designs three types of cargo containers ($C_1, C_2, C_3$) for a mission to Mars. Each container requires a specific quantity of three critical materials: Aluminium ($Al$), Titanium ($Ti$) and Carbon Fibre ($Fc$). The requirements per unit are detailed in the following table:

Material	$C_1$	$C_2$	$C_3$
Aluminium ($kg$)	$2$	$1$	$3$
Titanium ($kg$)	$1$	$2$	$1$
Fibre ($kg$)	$3$	$3$	$k$

1. a) If a total of $150\,kg$ of Aluminium and $120\,kg$ of Titanium is available, set up a system of equations to determine how many containers of each type can be manufactured if all stock is consumed.
2. b) Discuss the existence of solutions depending on the available stock of Carbon Fibre ($S_{Fc}$) and the structural constant $k$.
3. c) For $k = 4$ and a Fibre stock of $270\,kg$, is it possible to manufacture a combination of containers such that the number of units of $C_1$ is double that of $C_2$? Justify your answer by solving using Gaussian elimination or Cramer's rule.

Consider the homogeneous system dependent on the parameter $a \in \mathbb{R}$:

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

1. a) Determine the values of $a$ for which the system admits solutions other than the trivial one.
2. b) For the values found in the previous section, describe the solution set (line or plane passing through the origin).
3. c) If $a = -2$, verify that the rank of the coefficient matrix is $2$. Write the general solution and identify how many free parameters are needed.

Consider the system of 4 equations in 3 unknowns:

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

1. a) Discuss the consistency of the system depending on the value of $a$.
2. b) Solve it in terms of $a$ using Cramer's Rule on an appropriate $3 \times 3$ subsystem.
3. c) Explain the geometric meaning of this system of 4 equations: How are the four planes positioned in space?

An investor splits $10.000$ euros among three financial products: cryptocurrencies, tech stocks and government bonds. Let $x$, $y$ and $z$ be the euros invested in cryptocurrencies, stocks and bonds, respectively.

• After one year, cryptocurrencies have gone up by $20\%$, stocks have gone down by $5\%$ and bonds have yielded $3\%$. The total profit was $540$ euros.

• The investor notices that, if they had invested in tech stocks an amount equal to the sum of what was invested in cryptocurrencies and in bonds (leaving $x$ and $z$ as they are), their total profit would have been $250$ euros (keeping the same returns).

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

1. b) Solve the system using the inverse matrix method $X = A^{-1} \cdot B$.

1. c) What would happen if the return on bonds were $5\%$ instead of $3\%$? Analyze whether the system would still have a unique solution.

Given the system of equations:

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

1. a) Find the relationship that must exist between $a$ and $b$ for the system to be consistent with infinitely many solutions.
2. b) If $a = 1$ and $b = 3$, reason whether it is possible to find a solution to the system such that the sum of its components is zero ($x + y + z = 0$). If so, calculate it using Cramer's rule.
3. c) For $a=1$ and $b=0$, determine the rank of the augmented matrix and describe the relative position of the three planes.

Prove the following theoretical properties about systems of equations:

1. a) If $A$ is a square matrix of order 3 such that $\det(A) = 5$, and we consider the system $AX = B$. If we replace the column of constant terms $B$ with its double $2B$, what relationship exists between the solution of the original system and that of the new system? Justify your answer using Cramer's rule.
2. b) If a system of 3 equations in 3 unknowns has $(1, 0, 1)$ and $(0, 1, 2)$ as solutions, give a reasoned proof that the system must be consistent with infinitely many solutions and find a third solution distinct from the previous ones.
3. c) Is it possible for a system of linear equations to have exactly two distinct solutions? Reason your answer based on the structure of the general solution of a consistent system with infinitely many solutions.

Inverse problem: You want to design a system of three linear equations in three unknowns depending on a real parameter $m$, of the form $A(m) \cdot X = B$, that simultaneously satisfies the following specifications:

1. The system must be a Cramer's system for all real values of $m$ except for $m = 2$ and $m = -2$.

2. For any value of $m \in \mathbb{R} \setminus \{-2, 2\}$, the value of the unknown $z$ in the solution must be identically zero ($z = 0$).

3. For the particular value $m = 1$, the unique solution of the system must be exactly $(x, y, z) = (1, -1, 0)$.

1. a) With reasoning, design a coefficient matrix $A(m)$ and a vector of constants $B$ that satisfy these three conditions. Justify how you selected the elements of $A(m)$ so that its determinant vanishes only at $m = \pm 2$.
2. b) Prove, using the properties of determinants, that the auxiliary matrix $A_z$ of your construction satisfies $|A_z| = 0$ regardless of the value of $m$, guaranteeing that $z = 0$ whenever the system is a Cramer's system.
3. c) Substitute $m = 1$ into your system and calculate using determinants the values of $|A|$ and $|A_x|$ to verify that Cramer's rule indeed leads to $x = 1$.

During a Mathematics II class, Sara emphatically states:

"If in a system of three equations with three unknowns the determinant of the coefficient matrix is zero ($|A| = 0$) and, at the same time, the three determinants associated with the unknowns also vanish ($|A_x| = 0$, $|A_y| = 0$, and $|A_z| = 0$), the system can never be inconsistent; it must necessarily be indeterminate consistent".

1. a) Someone else's error and counterexample: Prove with reasoning that Sara's statement is FALSE. To do so, construct a simple numerical counterexample (a $3 \times 3$ system representing three non-coincident parallel planes) where $|A| = |A_x| = |A_y| = |A_z| = 0$ holds, and prove, applying the Rouché-Frobenius Theorem, that this system is inconsistent.
2. b) Explain algebraically why calculating only 3rd-order determinants is insufficient to determine the consistency of a system when the rank of the coefficient matrix is 1.
3. c) Adapting Cramer's rule to indeterminate systems: If a $3 \times 3$ system is indeterminate consistent with rank 2, describe the procedure for selecting a non-zero minor of order 2 and applying Cramer's rule to express the solutions in terms of a parameter. Illustrate your explanation by solving step by step the following system:

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