Exercises: 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):

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

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:

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

Three friends, Alex, Bea and Unai, went to a video game shop to take advantage of the sales on monthly subscriptions to different platforms.

• Alex bought one type A subscription, one type B and one type C for $30$ €.

• Bea bought two type B subscriptions and one type C for $25$ €.

• Unai bought only one type C subscription for $9$ €.

Set up the system of equations and calculate the price of each type of subscription ($A$, $B$ and $C$). Is this system already in row-echelon form?

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.

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.

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.

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.

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.

Marcos applies the Gaussian elimination method to a system of three linear equations with unknowns $x$, $y$, $z$. After correctly performing the first elementary operations, he reaches the following echelon system:

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

Looking at the last row, Marcos states: "The system has no solution because we have lost an equation and it is impossible to calculate three unknowns with only two equations available". Meanwhile, his classmate Alba counters: "It does have a solution; moreover, since the last row is $0 = 0$, that forces $z = 0$, so the solution is unique". (SOMEONE ELSE'S ERROR)

1. a) Identify and explain in detail the conceptual error made by Marcos and the one made by Alba in light of the classification criteria of Gaussian elimination.
2. b) Rigorously classify the system indicating the number of non-zero pivots $r$, the number of unknowns $n$, and how many free parameters are required.
3. c) Determine the general expression of all the solutions of the system in terms of a real parameter $\lambda$.

Consider the following system of linear equations:

$$\begin{cases} 3x - 2y + 4z = 11 \\ x + y - z = 0 \\ 2x + 3y - z = 5 \end{cases}$$

To solve it using Gaussian elimination, two possible paths are proposed for the first step: (CHOOSE THE METHOD)

• Strategy A: Keep the order of the equations as they are and transform the second row using $3F_2 - F_1$ and the third using $3F_3 - 2F_1$.

• Strategy B: First swap the first and second equations ($F_1 \leftrightarrow F_2$) to use the coefficient $1$ of the unknown $x$ as a pivot.

1. a) Reasonably compare both strategies, analyzing which one reduces operational complexity and minimizes the risk of making calculation errors with large coefficients or fractions. Furthermore, justify whether permuting two equations modifies the solution set of the original system, relying on the properties of elementary Gaussian row operations.
2. b) Following the strategy you considered most convenient, apply Gaussian elimination step by step (clearly indicating the row operations used) until obtaining the triangular system, and calculate the unique solution $(x, y, z)$ using back-substitution.

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:

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

A young investor decides to diversify her savings of $12.000$ € into three different financial products: a savings account yielding $2\%$, an investment fund yielding $5\%$, and tech stocks with a return of $8\%$.

It is known that:

• The total annual interest earned is $600$ €.

• The amount invested in tech stocks is half the sum of the other two investments.

Set up a system of three equations with three unknowns and solve it using Gaussian elimination to find how much money she invested in each product.

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.

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):

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

In a café in central Madrid, a group of friends went for snacks.

• On the first day, for 2 coffees, 1 soft drink, and 1 slice of cake, they paid $11$ €.

• On the second day, for 1 coffee, 2 soft drinks, and 2 slices of cake, they paid $16$ €.

• On the third day, for 3 coffees and 3 slices of cake (no soft drinks), they paid $18$ €.

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.

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 $z$:

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

An artisan workshop manufactures three types of lamps: model Alpha, model Beta and model Gamma.

• To manufacture an Alpha lamp, 20 minutes of assembly, 10 of painting and 10 of packaging are needed.

• For a Beta lamp, 10 minutes of assembly, 20 of painting and 10 of packaging.

• For a Gamma lamp, 10 minutes of assembly, 10 of painting and 20 of packaging.

If the workshop dedicated a total of $100$ hours today to assembly, $100$ hours to painting and $100$ hours to packaging, how many lamps of each type did they manufacture?

Note: Remember to convert all time units to the same measure (minutes or hours) before setting up the system.

A secondary school purchases 80 annual educational software licenses distributed among three modalities: Basic (€10 each), Advanced (€20 each), and Professional (€50 each), with a total invested budget of €1.650. Due to school implementation criteria, the number of Basic licenses purchased must be exactly equal to the sum of the Advanced and Professional licenses combined. (ESTIMATE AND CHECK)

1. a) Estimate and reason: Before calculating the complete system:

• Directly deduce the exact value of Basic licenses ($x$) by relating the pedagogical condition to the total of 80 licenses.

• If 30 or more Professional licenses had been purchased, how much would just those licenses cost added to the cost of the Basic ones you have already deduced? Based on this spending bound, assess whether it is plausible that the number of Professional licenses is equal to or greater than 30.

1. b) Model the problem by setting up a system of three linear equations with three unknowns and use Gaussian elimination to triangularize the associated matrix and find how many licenses of each modality have been purchased.
2. c) Compare the numerical results obtained with the deductions and estimates from section a) and justify why the solution found is consistent in the school context.

A study by the National Statistics Institute (INE) on tourist spending analyzes a sample of 120 travelers staying overnight in a Spanish city over a weekend. The travelers stay in three types of establishments: four-star hotels ($x$), three-star hotels ($y$), and regulated tourist apartments ($z$). (MODEL)

The following realistic data are available:

• The municipal tourist tax per night is €3 per person in four-star hotels, €2 in three-star hotels, and €1,50 in tourist apartments. The total municipal revenue for that night generated by the group of 120 people was €280.

• The total number of tourists staying in hotels (adding four-star and three-star) exceeds four times the number of those who stayed in tourist apartments by 20 people.

1. a) Set up the system of three linear equations with three unknowns that precisely translates the three conditions of the problem.
2. b) Apply Gaussian elimination to triangularize the system (indicating at each step the elementary row operations) and find how many tourists chose each accommodation modality.
3. c) If the City Council decides to increase the municipal tax for tourist apartments to €2,50 per night while keeping hotel taxes unchanged, by how many euros and by what exact percentage would the revenue generated by this group of 120 tourists increase?

A small technology company is managing three artificial intelligence projects: Alpha, Beta, and Gamma. The total budget allocated for the three is $450.000$ €. 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 $30.000$ €.

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

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.

An agricultural cooperative mixes three types of olive oil: Extra Virgin ($x$), Virgin ($y$), and Lampante ($z$) to produce a batch of $1000$ litres. Extra Virgin costs $8$ €/litre, Virgin costs $6$ €/litre, and Lampante costs $3$ €/litre. The total cost of the mixture was $5800$ €.

Due to quality regulations, the quantity of Virgin oil must be exactly three times the difference between Extra Virgin and Lampante.

Determine using Gaussian elimination how many litres of each type of oil the mixture contains.

Find a three-digit number knowing that the sum of its digits is $12$. If the hundreds digit and the units digit are swapped, the number decreases by $198$ units. Finally, we know that twice the tens digit is equal to the sum of the other two digits plus $6$.

Set up the corresponding system and solve it using Gauss elimination.

An investor has $60.000$ € which they divide among three financial products with annual returns of $2\%$, $4\%$, and $6\%$ respectively. At the end of the year, the total profits were $2.600$ €. If the capital invested at $6\%$ is the arithmetic mean of the other two invested capitals, how much money was allocated to each product?

Solve by applying Gaussian elimination.

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.

A furniture factory produces three models of shelving units: Basic, Standard, and Luxury. Each model must pass through three departments: Joinery, Varnishing, and Assembly. The time (in hours) required for each unit is shown in the following table:

Department	Basic	Standard	Luxury
Joinery	$1$	$2$	$3$
Varnishing	$1$	$1$	$2$
Assembly	$0,5$	$1$	$1$

If the factory has $400$ hours of Joinery, $250$ hours of Varnishing, and $160$ hours of Assembly available weekly, how many shelving units of each type should it manufacture to use up exactly the available hours? Use Gaussian elimination to solve the resulting system.

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.

A team of data analysts needs to reconstruct an original system of three linear equations with three unknowns $(x, y, z)$ whose computer record suffered a partial deletion. Only the following certified data remain from the calculation report: (INVERSE PROBLEM)

1. The first equation of the original system was $E_1: x - y + 2z = 1$.

2. The first elementary operation performed on the second row was $F_2 \leftarrow E_2 - 2E_1$, obtaining the reduced equation $F_2: y - 3z = 4$.

3. The system did not have a unique solution: it turned out to be an underdetermined consistent system with infinitely many solutions, one of them being the particular triple $(x, y, z) = (7, 10, 2)$.

1. a) Reasonably deduce the complete algebraic expression of the original second equation ($E_2$) before applying the Gaussian transformation.
2. b) Design an original third equation ($E_3$) in such a way that, when applying elementary operations to eliminate the unknown $x$ and the unknown $y$, the last row vanishes completely ($0x + 0y + 0z = 0$) and the triple $(7, 10, 2)$ satisfies that equation.
3. c) Apply Gaussian elimination to the system of three equations you reconstructed to check that it is reduced to echelon form with two pivots, and express all its solutions in terms of a real parameter $\lambda$.
4. d) Without repeating the entire triangularization process, explain what single change you should make to the constant term of your equation $E_3$ so that the resulting system becomes inconsistent, justifying what contradiction would appear in the last row.

Consider the following system of linear equations depending on the real parameter $k$: (SOMEONE ELSE'S ERROR)

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

When triangularizing the system using Gaussian elimination by applying the operations $F_2 \leftarrow F_2 - 2F_1$, $F_3 \leftarrow F_3 - F_1$, and finally $F_3 \leftarrow F_3 - F_2$, the following equality is obtained in the third row:

$$(k^2 + 2k - 3)z = k - 1$$

Hugo examines this last equation and states out loud:

"Since $k^2 + 2k - 3$ vanishes for $k = 1$ and for $k = -3$, the coefficient of $z$ disappears for both values. Consequently, whether $k = 1$ or $k = -3$, the system has infinitely many solutions because in both cases we are left with a free variable".

1. a) Locate and reason through the fundamental error in Hugo's deduction. Explain why it is essential to compare the behavior of the left-hand side with that of the right-hand side (constant term) when applying Gaussian elimination.
2. b) Rigorously discuss and classify the system for all possible values of $k \in \mathbb{R}$ (distinguishing between consistent independent, consistent dependent, and inconsistent).
3. c) Solve the system for $k = 1$ by expressing its solutions in terms of a parameter $\lambda$, and obtain the general expression of the unknown $z$ in terms of $k$ when the system is consistent independent.