Gaussian Elimination

In Gaussian elimination, the objective is to transform a system of equations into a row-echelon (triangular) form so that it is much easier to solve. Observe the following system, which is already in row-echelon form, and find the value of the unknowns $x$, $y$ and $z$ using back-substitution (from bottom to top):

$\begin{cases} x + y + z = 9 \\ 2y - z = 5 \\ 3z = 9 \end{cases}$

In a local esports tournament, teams are awarded points based on achieved objectives: towers destroyed ($x$), dragons slain ($y$) and players eliminated ($z$). We know that:

1. A team's total number of actions ($x + y + z$) was $20$.

2. They destroyed the same number of towers as dragons slain ($x - y = 0$).

3. They captured $4$ dragons ($y = 4$).

Use this data to form a row-echelon system and determine how many actions of each type the team performed.

At the end of Gaussian elimination, a student arrives at the following coefficient matrix (where the last column is the constant term):

$\left(\begin{array}{ccc|c} 1 & 2 & -1 & 4 \\ 0 & 1 & 3 & 5 \\ 0 & 0 & 0 & 0 \end{array}\right)$

1. a) Write down the system of equations associated with this matrix.
2. b) Based on the last row, indicate whether the system is Consistent with a unique solution, Consistent with infinitely many solutions, or Inconsistent. Reason your answer.

Sometimes, to apply Gaussian elimination more efficiently, it is useful to swap the rows of the system. Given the following system:

$\begin{cases} 3x - 2y + z = 10 \\ x + y + z = 6 \\ 2y - 3z = -4 \end{cases}$

1. a) Explain why it would be convenient to swap the first equation ($E_1$) with the second ($E_2$) before starting operations.
2. b) Write down the resulting system after the swap and perform the first step of Gaussian elimination to eliminate $x$ from the new second equation.

Solve the following system of linear equations using Gaussian elimination. To make your work easier, notice that the variable $x$ has already been eliminated from the second equation:

$\begin{cases} x - y + z = 2 \\ y + 2z = 8 \\ 2x + y - z = 1 \end{cases}$

Solve the following system of equations step by step using Gaussian elimination:

$\begin{cases} x + y + z = 6 \\ x - y + z = 2 \\ x + y - z = 0 \end{cases}$

Hint: Subtract the first equation from the second and third equations to eliminate $x$ in a single step.

Solve the following system of linear equations using Gaussian elimination. Remember to organise the steps to obtain a triangular structure before calculating the values of the unknowns:

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

Consider the following system of linear equations where the coefficients are fractions:

$\begin{cases} \frac{x}{2} + \frac{y}{3} + z = 13 \\ x + \frac{y}{2} + \frac{z}{3} = 14 \\ \frac{x}{3} + y + \frac{z}{2} = 17 \end{cases}$

Transform the system to work with integer coefficients and use Gaussian elimination to find the values of $x$, $y$, and $z$. Explain step by step each elementary row operation performed on the matrix.

Determine whether the following system has a solution or not, applying the Gauss method until you obtain the last row:

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

Justify your answer by analyzing the relationship between the first and the second equation.

Given the following system of equations, use Gaussian elimination to analyse its nature based on the final result. If the system is consistent with a unique solution, find the values of $x$, $y$, and $z$:

$$\begin{cases} 3x - y + 2z = 7 \\ -x + 2y + z = 1 \\ 4x + y + 3z = 8 \end{cases}$$

Analyse and solve the following system. If the system turns out to have infinitely many solutions, express the general solution using a parameter $\lambda$ (or $t$):

$$\begin{cases} x - 3y + 2z = 4 \\ 2x + y - 3z = 1 \\ 5x - 8y + 3z = 13 \end{cases}$$

Find the value of the parameter $k$ for which the following system of equations is inconsistent (that is, has no solution):

$$\begin{cases} x + y + z = 2 \\ 2x + 3y + z = 3 \\ 3x + 4y + kz = 10 \end{cases}$$

Study and solve the following system of linear equations depending on the parameter $a$ using Gaussian elimination:

$\begin{cases} x + 2y - z = 3 \\ 2x + ay + z = 0 \\ 3x + 4y + (a-2)z = 3 \end{cases}$

Analyze the cases where the system has no solution and the case where it has infinitely many solutions. In the latter case, express the general solution.

Analyse the following system of linear equations depending on the values of the real parameter $k$, using Gaussian elimination:

$\begin{cases} x + y + kz = 1 \\ x + ky + z = k \\ kx + y + z = k^2 \end{cases}$

State for which values of $k$ the system is consistent with a unique solution, consistent with infinitely many solutions, or inconsistent. Solve it for the case where it has infinitely many solutions.

In a café in downtown Mexico City, a group of friends went for snacks.

• On the first day, for 2 coffees, 1 soft drink, and 1 slice of cake, they paid $220$ pesos.

• On the second day, for 1 coffee, 2 soft drinks, and 2 slices of cake, they paid $320$ pesos.

• On the third day, for 3 coffees and 3 slices of cake (no soft drinks), they paid $360$ pesos.

Determine the unit price of each item using Gaussian elimination. Is it possible for the system to have infinitely many solutions, or is the price unique? Justify your answer.

A logistics company uses three types of vans (Small, Medium, and Large) to deliver packages. Yesterday they completed three routes with the following characteristics:

• Route A: 2 small, 1 medium, and 3 large, carrying a total of $26$ containers.

• Route B: 1 small, 4 medium, and 1 large, carrying a total of $22$ containers.

• Route C: 3 small, 2 medium, and 2 large, carrying a total of $23$ containers.

Calculate the carrying capacity of each type of van using Gaussian elimination.

A small technology company is managing three artificial intelligence projects: Alpha, Beta, and Gamma. The total budget allocated for the three is $450,000$ pesos. It is known that the cost of project Gamma is equal to the sum of the cost of Alpha plus twice Beta. Furthermore, if we subtract the cost of Gamma from three times the cost of Alpha, we get exactly the cost of Beta plus $30,000$ pesos.

Set up a system of linear equations and use Gaussian elimination to determine the exact investment required for each project.