Systems of Equations with Two and Three Unknowns. Classification

Classify the following systems of equations with two unknowns according to the number of solutions they have (Consistent Independent System, Consistent Dependent System or Inconsistent System) after attempting to solve them using your preferred method:

1. a) $\begin{cases} 2x + 3y = 8 \\ 4x + 6y = 15 \end{cases}$

1. b) $\begin{cases} x - y = 2 \\ 2x + y = 7 \end{cases}$

1. c) $\begin{cases} 3x + y = 5 \\ 6x + 2y = 10 \end{cases}$

In a gaming peripherals shop, three friends buy different products.

• Marcos buys 2 mice and 1 keyboard for $2\,200$ pesos.

• Julia buys 1 mouse and 2 keyboards for $2\,600$ pesos.

Set up a system of linear equations to determine the price of each item and classify it according to the number of solutions you expect to obtain.

Check whether the ordered triple $(x, y, z) = (1, 2, -1)$ is a solution to the following system of equations:

$\begin{cases} 3x - y + z = 0 \\ x + y + z = 2 \\ 2x + 3y - z = 9 \end{cases}$

Solve the following system of equations with three unknowns which is already in row-echelon form:

$\begin{cases} 2x + y - z = 5 \\ \quad \quad 3y + 2z = 7 \\ \quad \quad \quad \quad 4z = 8 \end{cases}$

Geometrically, a linear equation with three unknowns represents a plane in space. Look at the diagram and describe what type of system (Consistent Independent, Consistent Dependent or Inconsistent) would represent the intersection of two planes that are parallel to each other.

Express the following system of linear equations in its matrix form $A \times X = B$, clearly identifying the coefficient matrix $A$, the matrix of unknowns $X$ and the matrix of constant terms $B$:

$\begin{cases} x - 2y + 3z = 0 \\ 2x + y - z = 4 \\ 3x - y + 2z = -1 \end{cases}$

An Esports team has won a prize of 12 pesos 000 to be divided among its three members: the captain (A), the strategist (B) and the sniper (C). It is known that:

• The sum of what B and C receive is equal to what A receives.

• If the strategist (B) received 1 peso 000 more, they would have exactly the same as the sniper (C).

1. a) Set up a system of three equations with three unknowns to determine how much each person receives.
2. b) Classify the system and solve it.

Classify the following system of linear equations according to the number of solutions, using Gaussian elimination:

$\begin{cases} x + 2y - z = 3 \\ 2x - y + z = 1 \\ 5x + 5y - 2z = 10 \end{cases}$

If it is consistent, find its solution or solutions. If it has infinitely many solutions, write the solution in parametric form.

Study the consistency of the following system of 4 equations with 3 unknowns:

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

If the system is consistent, find the unique solution. Otherwise, state which equation causes the inconsistency.

Dos drones de vigilancia, $D_1$ y $D_2$, se desplazan por un plano siguiendo trayectorias rectilíneas dadas por las siguientes ecuaciones:

Drone 1: $2x - 3y = 5$

Drone 2: $x + ky = 2$

Se pide:

1. a) Determina el valor del parámetro $k$ para que los drones choquen en un punto (el sistema sea compatible determinado).

1. b) Is there any value of $k$ for which the drones follow parallel trajectories and never meet? Justify your answer using ranks.

1. c) Graph the situation if $k = -1.5$.

You can do it on paper and send a photo with your answer.

Without performing complex calculations, give reasons why the following homogeneous system is Consistent Dependent by analysing the relationship between its equations:

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

Consider the homogeneous system depending on the parameter $a$:

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

1. a) Classify the system depending on the values of $a$.
2. b) Determine for which values of $a$ the system admits non-trivial solutions ($x=0, y=0, z=0$).
3. c) Solve the system for $a = -2$.

Consider the homogeneous system depending on the parameter $m$:

$$\begin{cases} mx + y + z = 0 \\ x + my + z = 0 \\ mx + my + 2z = 0 \end{cases}$$

1. a) Determine the value of $m$ for which the system admits non-trivial solutions.
2. b) For said value of $m$, find the general solution and give a non-trivial particular solution.