Exercises: Gaussian elimination

A technology store is preparing three types of welcome packs for new customers. The first pack includes 1 mouse, 1 keyboard and 1 mouse pad for $35$ €. The second pack has 2 mice and 1 keyboard for $45$ €. The third includes 1 mouse, 2 keyboards and 1 mouse pad for $55$ €.

Let $x$ be the price of the mouse, $y$ the price of the keyboard and $z$ the price of the mouse pad.

1. a) Write the system of equations that represents the situation.

1. b) Write the augmented matrix $(A|B)$ of that system.

Solve the following system of linear equations using Gaussian elimination. Notice that the first unknown already has a coefficient of $1$, making it easier to use as a pivot:

$\begin{cases} x + y + z = 6 \\ x - y + 2z = 5 \\ x - y - z = -1 \end{cases}$

Given the following augmented matrix, which has already been partially transformed using Gaussian elimination, state which elementary row operation you would perform to obtain a zero in the position marked with the orange circle and write down the new row $F_3$.

$\begin{pmatrix} 1 & 2 & -1 & | & 3 \\ 0 & 1 & 3 & | & 5 \\ 0 & 2 & 4 & | & 8 \end{pmatrix}$

A student has applied Gaussian elimination to a system of 3 equations with 3 unknowns and reached the following echelon matrix:

$\begin{pmatrix} 1 & -1 & 2 & | & 4 \\ 0 & 1 & -1 & | & 1 \\ 0 & 0 & 3 & | & 6 \end{pmatrix}$

Finish solving the system by calculating the values of $x$, $y$ and $z$ using back substitution.

Classify the following systems of equations according to the number of solutions they have after observing the final result of applying Gaussian elimination in matrix form:

1. a) $\begin{pmatrix} 1 & 2 & 1 & | & 5 \\ 0 & 1 & -1 & | & 2 \\ 0 & 0 & 0 & | & 0 \end{pmatrix}$

1. b) $\begin{pmatrix} 1 & 0 & 3 & | & 4 \\ 0 & 2 & 1 & | & 1 \\ 0 & 0 & 0 & | & 7 \end{pmatrix}$

To solve the following system, the first step of the Gauss method is usually to make the coefficient of $x$ in the first equation equal to $1$ or $-1$. Which elementary transformation would you perform first to make the calculation easier? Justify your answer.

$\begin{cases} 3x - 2y + z = 10 \\ x + y - z = 2 \\ 2x + 3y + 2z = 15 \end{cases}$

Three friends go to a coffee shop. Luis pays $7$ € for $2$ coffees and $1$ toast. María pays $10$ € for $1$ coffee, $1$ toast and $1$ juice. Finally, Juan pays $11$ € for $2$ toasts and $1$ juice.

Use Gaussian elimination to determine the price of each item (coffee, toast and juice).

Apply Gaussian elimination to solve the following homogeneous system. Remember that this type of system always has at least the trivial solution:

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

Lucía is solving the following system of linear equations using Gaussian elimination:

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

To do so, she writes the associated augmented matrix:

$$\begin{pmatrix} 1 & 2 & -1 & | & 4 \\ 2 & 3 & 1 & | & 5 \\ 1 & -1 & 3 & | & 0 \end{pmatrix}$$

In the first stage of row reduction, she applies the operations $F_2 \leftarrow F_2 - 2F_1$ and $F_3 \leftarrow F_3 - F_1$, correctly obtaining:

$$\begin{pmatrix} 1 & 2 & -1 & | & 4 \\ 0 & -1 & 3 & | & -3 \\ 0 & -3 & 4 & | & -4 \end{pmatrix}$$

To get a zero in position $(3,2)$, Lucía decides to apply the operation $F_3 \leftarrow F_3 + \frac{3}{2}F_1$, arguing that since there is a $2$ in row 1, multiplying it by $3/2$ and adding it to the $-3$ in row 3 will cancel out that entry.

1. a) Explain with justification what conceptual error Lucía makes with this operation and why it ruins the row reduction process of Gaussian elimination.
2. b) State which row must be taken as the pivot at this step and write the correct elementary row operation that eliminates the entry in position $(3,2)$.
3. c) Perform that transformation, verify that the matrix is in row echelon form, and find the unique solution to the system using back-substitution.

Consider the following system of linear equations:

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

Two students discuss the best way to begin Gaussian elimination:

• Sergio proposes starting directly with the given order, using the coefficient $3$ in the first row as the pivot and performing the operations $F_2 \leftarrow 3F_2 - 2F_1$ and $F_3 \leftarrow 3F_3 - F_1$.

• Marta proposes swapping the first and third rows first ($F_1 \leftrightarrow F_3$) so that the first pivot is $1$, and from there creating zeros in the first column.

1. a) Compare both proposals: what practical advantages does Marta's strategy have over Sergio's regarding arithmetic difficulty and the risk of error with signs or fractions? Furthermore, justify whether the row swap operation changes the solution of the system.
2. b) Following Marta's strategy, write the augmented matrix after swapping the rows and apply the necessary elementary row operations to obtain the row echelon form.
3. c) From the row echelon matrix, classify the system and find its solution using back-substitution.

Solve the following system of equations using the Gaussian elimination method:

$$ \begin{cases} x + y + z = 2 \\ 2x + 3y + 5z = 11 \\ x - 5y + 6z = 29 \end{cases} $$

1. a) Write the augmented matrix of the system and transform it, using row operations, into an echelon matrix.

1. b) From the echelon matrix, classify the system (consistent determinate, consistent indeterminate, or inconsistent) and justify your answer.

1. c) If the system is consistent, find its solution.

Apply Gaussian elimination to discuss and solve the following system of linear equations. If the system turns out to be consistent undetermined, write down the general solution in terms of a parameter $\lambda$:

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

A social media content creator analyses their monthly income from three platforms: TikTok, YouTube and Instagram. It is known that:

• In total, last month they earned $2500$ €.

• What they earned on YouTube is triple what they earned on TikTok.

• The sum of earnings on TikTok and Instagram exceeded earnings on YouTube by $100$ €.

Set up a system of three equations with three unknowns and use Gaussian elimination to find out how much money was generated on each platform.

A sports shop sells three types of balls: footballs, basketballs and volleyballs. A club made three different purchases:

• The first purchase was $2$ footballs, $1$ basketballs and $1$ volleyballs for $90$ €.

• The second purchase was $1$ football, $2$ basketballs and $3$ volleyballs for $120$ €.

• The third purchase was $3$ footballs and $3$ basketballs for $150$ €.

Use Gaussian elimination on the augmented matrix to determine the price of each type of ball.

Given the following system depending on a constant value $k$, apply Gaussian elimination to transform it into a row echelon system. Determine for which value of $k$ the system is inconsistent:

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

Three friends, Nerea, Borja and Celia, went to a cafe.

• The first time, Nerea had $2$ coffees and $1$ toast, paying $5$ €.

• The second time, Borja had $1$ coffee, $1$ toast and $1$ juice, paying $6$ €.

• The third time, Celia had $3$ coffees and $2$ juices, paying $11$ €.

Write down the augmented matrix associated with the system and use Gaussian elimination to check whether the data is sufficient to find the price of each product (that is, whether the system is consistent determined).

Analyse using Gaussian elimination the relative position of the three planes represented by the following equations. State whether they intersect at a point, in a line, or if they have no points in common:

$$ \pi_1: x + 2y - z = 1 $$ $$ \pi_2: 2x + y + 2z = 2 $$ $$ \pi_3: 3x + 3y + z = 3 $$

Solve the following system of four equations with three unknowns using Gaussian elimination. Remember to eliminate any rows that turn out to be a linear combination of the previous ones during row reduction:

$$ \begin{cases} x + y + z = 6 \\ x - y + z = 2 \\ 2x + 2z = 8 \\ 3x + y + 3z = 14 \end{cases} $$

During a metropolitan mobility campaign in Spain, a transport consortium sells three types of monthly passes at subsidized rates: Youth Pass at $15$ €, Standard Pass at $30$ €, and Senior Pass at $10$ €. At the end of the first month, the following commercial data are collected:

• A total of $100.000$ passes were sold across the three types combined.

• The number of Youth Passes sold exceeded the combined sum of Standard and Senior Passes by $20.000$ units.

• Total revenue from the sale of these transit passes amounted to $1.900.000$ €.

1. a) Assign an unknown to each pass type, translate the information into a system of three linear equations, and write its augmented matrix $(A|B)$.
2. b) Apply Gaussian elimination to the augmented matrix until transforming it into row echelon form.
3. c) Solve the echelon system using back-substitution, determine how many passes of each type were sold, and calculate what percentage of total sales corresponded to the Youth Pass.

On a day of high renewable generation in Spain, three peninsular electricity generation sources—wind energy ($x$), solar photovoltaic energy ($y$), and hydroelectric energy ($z$)—jointly produced a total of $180\text{ GWh}$. The system operator also notes two facts:

• The combined sum of solar and wind energy generated $140\text{ GWh}$.

• Wind energy contributed exactly twice as much energy as hydroelectric power.

1. a) Estimate before calculating: Reason, without solving the system algebraically, why hydroelectric production cannot exceed $45\text{ GWh}$ and deduce a reasonable interval where wind and solar must lie, knowing that together they sum to $140\text{ GWh}$.
2. b) Set up the associated system of linear equations, write the augmented matrix $(A|B)$, and apply Gaussian elimination to obtain the equivalent row echelon matrix.
3. c) Solve the system using back-substitution, find the values of $x$, $y$, $z$, and numerically check whether they satisfy the three conditions of the problem and agree with the estimate made in part (a).

Consider the following system of equations depending on the real parameter $k$:

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

Use Gaussian elimination to discuss the consistency of the system according to the different values of $k$. In the cases where the system is consistent and indeterminate, find the general solution.

A logistics company organizes freight transport among three cities: A, B, and C. The total fleet consists of 42 trucks of three models: light, medium, and heavy. It is known that:

• The number of heavy trucks is half the sum of the light and medium ones.

• For every 3 medium trucks, there are 4 light ones.

• Due to a new environmental regulation, if the number of heavy trucks were doubled and the number of light trucks were halved, the fleet would increase by 6 units.

Set up the corresponding system of equations and use Gaussian elimination to determine how many trucks of each model the company has.

Analyse the following system of four equations in three unknowns using Gaussian elimination:

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

The figure shows the augmented matrix of the system, with the columns for $x$, $y$, $z$ and the column of constant terms $b$.

Determine, with justification, whether the system has a solution. If it is consistent, find all its solutions. If it is inconsistent, identify which equation or equations cause the inconsistency.

A nutritionist designs a special diet based on three foods: Shake A, Bar B and Nut C. Each serving must provide specific amounts of carbohydrates, proteins and fats. The table of nutrients per serving is as follows:

Food	Carbohydrates (g)	Proteins (g)	Fats (g)
Shake A	20	10	5
Bar B	30	5	10
Nut C	10	15	20

The total daily diet is required to contain 250 g of carbohydrates, 110 g of proteins and 130 g of fats.

1. a) Set up the system of equations.
2. b) Solve it using Gaussian elimination to find the number of daily servings of each food.
3. c) Is it possible to meet the target if the patient has a nut allergy (C=0)? Justify your answer based on the solution to the system.

Discuss and solve the following homogeneous system depending on the parameter $a$:

$\begin{cases} ax + y + z = 0 \\ x - ay + z = 0 \\ 2x + y + az = 0 \end{cases}$

State for which values of $a$ the system admits non-trivial solutions ($x=0, y=0, z=0$) and calculate them for these values using Gaussian elimination.

An investor divides 100 000 € between three financial products: a savings account at 1% per year, government bonds at 3% per year and an investment in tech stocks at 8% per year.

At the end of the first year, total profits were 3 900 €. The investor notes that if they had invested twice as much in bonds as in the savings account, while keeping the investment in stocks the same and leaving the rest of the 100 000 € in the savings account, their profit would have been 4 100 €.

Determine using Gaussian elimination the amount invested in each product.

Given the system of equations:

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

1. a) Add a third equation of the form $ax + by + cz = d$ so that the resulting system is consistent determined with solution $(1, 2, 1)$.
2. b) Add a third equation so that the system is inconsistent.
3. c) Add a third equation so that it is consistent undetermined.

In each case, explain your reasoning based on the row structure of the augmented matrix.

Solve the following system of 4 equations with 4 unknowns using Gaussian elimination:

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

Describe step-by-step the elementary row operations used to bring the matrix to triangular form.

Upon applying Gaussian elimination to the augmented matrix $(A|B)$ of a system of three linear equations in three variables ($x, y, z$), a student obtains the following row echelon matrix:

$$(A^*_{\text{escalonada}}) = \begin{pmatrix} 1 & -2 & 3 & | & 5 \\ 0 & 1 & -1 & | & 2 \\ 0 & 0 & a - 3 & | & b + 1 \end{pmatrix}$$

where $a$ and $b$ are real parameters.

1. a) Discuss with justification the consistency of the system according to the different real values of $a$ and $b$. In the case where the system is consistent dependent, determine its general solution expressing all variables in terms of a real parameter $\lambda$.
2. b) Determine what condition $a$ and $b$ must satisfy for the system to be consistent independent and have its unique solution with $z = 4$. Provide a specific numerical example of values of $a$ and $b$ that satisfy this condition.
3. c) It is known that the row echelon matrix was obtained by applying the following successive row operations to the initial matrix $(A|B)$: 1st $F_2 \leftarrow F_2 - 2F_1$; 2nd $F_3 \leftarrow F_3 - F_1$; 3rd $F_3 \leftarrow F_3 + 2F_2$ (already using the new row $F_2$). Knowing that the first row did not change and was $F_1 = (1, -2, 3 \mid 5)$, reconstruct inversely and with justification what the original equation occupying the third row ($F_3$) of the starting system was for the case where $a = 3$ and $b = -1$.

Consider the following homogeneous system of linear equations depending on the real parameter $k$:

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

During a review session, two students make categorical statements about this system:

• Elena states: "For some values of $k$, the system will be inconsistent because when reducing to echelon form, a contradiction of the form $0 = c$ with $c \ne 0$ will appear."

• Carlos refutes: "Elena is wrong: no homogeneous system can ever be inconsistent. Therefore, for any value of $k$, the only solution the system has is the trivial solution $(0, 0, 0)$."

1. a) Critically analyze both statements: explain on a theoretical basis why Elena's statement is false and what conceptual error Carlos makes when generalizing the uniqueness of the trivial solution.
2. b) Write the coefficient matrix of the system and apply Gaussian elimination to obtain an equivalent row echelon matrix in terms of $k$.
3. c) Discuss the system for all real values of $k$, indicating precisely for which values there exist solutions other than the trivial solution.
4. d) For the positive integer value of $k$ that allows the system to admit non-trivial solutions, find the general solution in terms of a parameter $\lambda$ and find a particular solution in which $z = 2$.