Exercises: Solving systems and Cramer's rule

Two friends, Lucía and Marcos, are comparing their subscriptions to streaming platforms. Lucía pays a total of $16$ pesos per month for one "Cinema" plan and two "Series" plans. Marcos, for his part, has two "Cinema" plans and one "Series" plan for $17$ pesos 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 $7.10$ pesos.

• Order 2: $2$ coffees and $4$ toasts for a total of $7.40$ pesos.

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

1. a) Express the problem in matrix form $A \times X = B$, 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 $X = A^{-1} \times B$.

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.

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 center 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 liters 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 liters 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.

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 $A \times X + B = 2B$, 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: Aluminum ($Al$), Titanium ($Ti$) and Carbon Fiber ($Fc$). The requirements per unit are detailed in the following table:

Material	$C_1$	$C_2$	$C_3$
Aluminum ($kg$)	$2$	$1$	$3$
Titanium ($kg$)	$1$	$2$	$1$
Fiber ($kg$)	$3$	$3$	$k$

1. a) If a total of $150\,kg$ of Aluminum 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 Fiber ($S_{Fc}$) and the structural constant $k$.
3. c) For $k = 4$ and a Fiber 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 divides $200\,000$ pesos among three financial products: cryptocurrencies, tech stocks and government bonds.

• After one year, the cryptocurrencies have risen by $20\%$, the stocks have fallen by $5\%$ and the bonds have yielded $3\%$. The total profit was $10\,800$ pesos.

• The investor notes that if they had invested in tech stocks the sum of the amounts invested in cryptocurrencies and bonds, their total profit would have been $5\,000$ pesos (maintaining the same yields).

1. a) Set up a system of linear equations to determine how much was invested in each product.
2. b) Solve the system using the inverse matrix method $X = A^{-1} \times B$.
3. c) What would happen if the yield on the 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.