Solutions of a system

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).

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.

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.

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.

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} $

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

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:

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

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.