Exercises: Systems of equations with two and three unknowns. Classification of systems

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

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

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

Given the lines $r$ and $s$ represented in the following graph, write the system of equations defining their intersection and classify it according to its geometry.

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 $B$:

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

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

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

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

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

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.

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.

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.

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.

Discuss the consistency of the following system depending on the values of the real parameter $m$, using the Rouché-Frobenius theorem:

$\begin{cases} mx + y + z = 1 \\ x + my + z = m \\ x + y + mz = m^2 \end{cases}$

State for which values of $m$ the system is Consistent Independent, Consistent Dependent or Inconsistent.

Given the equations of three planes in space:

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

$\pi_2: 2x + 3y + z = 3$

$\pi_3: kx + 10y + 4z = 11$

1. a) Determine the value of $k$ so that the three planes intersect in a line (underdetermined consistent system).
2. b) For the found value of $k$, obtain the equation of said line.
3. c) What is the relative position of the planes if $k = 5$?

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.

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.

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 $a = -2$.

Write a system of three equations with three unknowns that satisfies simultaneously:

1. It is Consistent Dependent.

2. One of its solutions is $(1, 0, 2)$.

3. The coefficient matrix is of the form:

   $A = \begin{pmatrix} 1 & -1 & 1 \\ 2 & 1 & a \\ 3 & 0 & b \end{pmatrix}$

Calculate the required values of $a$ and $b$, and propose consistent constant terms. Justify your construction using ranks.

Discuss the following system of linear equations depending on the various values of the real parameters $a$ and $b$. Solve it only in the case where the system is consistent dependent (infinitely many solutions):

$$\begin{cases} ax + y + z = b \\ x + ay + z = 1 \\ x + y + az = 1 \end{cases}$$

A biotechnology laboratory is developing a new serum by combining three compounds: $A$, $B$ and $C$. The cost per gram of each compound depends on an import factor $k$. The system governing the required quantities $x, y, z$ (in grams) to stabilise 100 ml of serum is:

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

1. a) Determine for which values of $k$ the laboratory cannot determine a unique quantity of each compound to manufacture the serum.
2. b) If $k = -2$, is it possible to manufacture the serum following these proportions? Justify your answer based on the consistency of the system.

Consider the three planes in space whose equations are:

$\pi_1: x + ay + z = 1$

$\pi_2: ax + y + (a-1)z = a$

$\pi_3: x + ay + (a+1)z = b$

Study the relative position of the three planes depending on the values of the parameters $a$ and $b$. For which values of the parameters do the three planes intersect in a single line?

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.

Discuss the consistency of the following system of 4 equations with 3 unknowns depending on the values of the parameter $k$:

$$\begin{cases} x + y + z = 3 \\ 2x + y - z = 2 \\ x - y + kz = 1 \\ 3x + 2y + (k-1)z = 5 \end{cases}$$

In the cases where it is consistent, determine whether the solution is unique.

Three courier companies ($P_1, P_2, P_3$) share the market of a city. Each month, 20% of the customers of $P_1$ switch to $P_2$ and 10% to $P_3$. Of the customers of $P_2$, 10% switch to $P_1$ and 10% to $P_3$. Of the customers of $P_3$, 10% switch to $P_1$ and 20% to $P_2$.

If the current market shares are $x, y, z$ and we want the market shares to remain identical next month (equilibrium state), set up a system of equations. Show that this system is consistent dependent and calculate the relationship between the market shares at equilibrium.

Given the system of equations:

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

1. a) Add an equation of the form $ax + by + cz = d$ so that the resulting system is consistent and independent (unique solution) and its solution is $(1, 1, 0)$.
2. b) Add an equation so that the system is inconsistent.
3. c) Is it possible to add an equation so that the system is consistent and dependent (infinitely many solutions) but with a different solution set from that of the original system? Justify your answer.

Determine the values of the parameters $a$ and $b$ for which the following system has infinitely many solutions depending on a single parameter:

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

For the values found, write the solution in parametric form.