Solving Systems and Cramer's Rule

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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 $X = A^{-1} \times B$.

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.

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.