Exercises: Systems of equations

Solve the following system of linear equations using whichever method you consider most appropriate (substitution, equating expressions, or elimination) and state whether it is a system with a unique solution (SCD), infinitely many solutions (SCI), or no solution (SIN):

$\begin{cases} 3x - 2y = 1 \\ x + 4y = 11 \end{cases}$

Observe the graphical representation of the two lines that form the following system of equations. Determine, without doing any algebraic calculations, the solution of the system and check whether the system is consistent and independent (has a unique solution).

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

In systems of three equations with three unknowns, sometimes the structure allows us to solve it immediately. Find the values of $x$, $y$, and $z$ in the following triangular system using back-substitution (starting with the last unknown):

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

Apply the first step of Gaussian elimination to eliminate the unknown $x$ from the second and third rows in the following system. You do not need to finish solving it, just write down the resulting system after performing the appropriate elementary operations:

$\begin{cases} x + y + z = 6 \\ 2x - y + 3z = 9 \\ -x + 2y + z = 3 \end{cases}$

A non-linear system is one in which at least one of the equations is not of first degree. Find the points of intersection between the line and the parabola given by the following system:

$\begin{cases} y = x^2 \\ y = x + 2 \end{cases}$

Two friends, Lucía and Marcos, went to a café. Lucía paid $5,50$ € for 2 coffees and 1 orange juice. Marcos paid $6,00$ € for 1 coffee and 2 orange juices. Set up a system of equations and calculate the price of a coffee and of an orange juice.

Classify the following systems of equations without solving them, simply by comparing the ratios of their coefficients and constant terms:

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

1. b) $\begin{cases} 5x + y = 10 \\ 5x + y = 15 \end{cases}$

The perimeter of a rectangular venue for events is $50$ metres. It is known that the length of the venue is $5$ metres more than the width. Calculate the dimensions of the venue by setting up a system of two equations with two unknowns ($x$ for the length and $y$ for the width).

In a Mathematics exam, Claudia tries to classify the following system of linear equations without actually solving it:

Claudia writes on her sheet: “Since $\dfrac{3}{-2} = -1,5$ and $\dfrac{-6}{4} = -1,5$, the coefficients of the unknowns have the same ratio. This means that the two equations represent the same line and, therefore, the system is consistent dependent, with infinitely many solutions”.

1. a) Find the error in Claudia's reasoning and explain which essential element she forgot to compare.
2. b) Rigorously classify the system by comparing the full ratios of the coefficients: $\dfrac{a}{a'}$, $\dfrac{b}{b'}$, and $\dfrac{c}{c'}$.
3. c) Explain what the actual geometric position of both lines in the Cartesian plane is and how many solutions the system has.

Consider the following two systems of linear equations:

System A:

System B:

1. a) For each system, analyze the arrangement of its coefficients and unknowns and justify which algebraic method (substitution, equating, or elimination) allows you to reach the solution most quickly and with the least risk of working with complex fractions.
2. b) Solve System B using exclusively the method you selected as optimal in the previous section, and verify that the found solution simultaneously satisfies both equations.

Solve the following system of linear equations with fractional coefficients, simplifying the expressions beforehand:

A mountain rescue team uses an antenna to locate hikers. The antenna's radio signal, located at the origin of coordinates, defines a circular coverage area given by the equation $x^2 + y^2 = 25$. A rescue helicopter moves along a straight path described by the equation $y - x = 1$. The figure shows the coverage area and the path (each square of the scale equals 1 unit on both axes). Calculate the intersection points where the helicopter enters and leaves the antenna's coverage range by solving the system:

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

To organise an end-of-year trip, Year 12 students are selling two types of T-shirts: the "Eco" model for $12$ € and the "Premium" model for $18$ €. By the end of the week they have sold a total of $150$ T-shirts and raised $2220$ €. Set up a system of equations and calculate how many T-shirts of each type they have sold.

A rectangular plot intended for school allotments needs to be fenced. It is known that the total perimeter of the plot is $50$ metres and its area is $150$ $m^2$. Find the dimensions of the plot by setting up and solving a system of non-linear equations.

Solve the following system of non-linear equations using the substitution method:

In a city-centre café, three friends order different combinations of items.

• The first has a coffee, a soft drink, and a slice of cake for $7,50$ €.

• The second has two coffees and a soft drink for $6,00$ €.

• The third has a coffee and two slices of cake for $8,50$ €.

Calculate the price of each item by setting up a system of three equations and solving it using Gaussian elimination.

Determine the consistency of the following system of equations depending on the value of the parameter $k$. For what value of $k$ is the system consistent with infinitely many solutions?

An eco-friendly courier company in Valencia delivers packages daily using two types of vehicles: electric vans and cargo bikes. In total, the active fleet today consists of $35$ vehicles. The daily maintenance and recharging cost is $120\text{ €}$ for each electric van and $25\text{ €}$ for each cargo bike. The total maintenance cost for the day was exactly $1825\text{ €}$.

1. a) Estimate before calculating: If all vehicles were cargo bikes, what would the total cost be? And what if all were electric vans? From these two extremes and the actual cost ($1825\text{ €}$), argue whether one should expect there to be more vans or more bikes in the fleet.
2. b) Set up the linear system of two equations with two unknowns associated with the problem and solve it using the method you consider most appropriate.
3. c) Check whether the exact number of vehicles obtained confirms your prediction from part (a) and reflect on the usefulness of bounding the result before operating.

Design a linear system of two equations with two unknowns, $x$ and $y$, that simultaneously satisfies the following three requirements:

1. The system must be inconsistent.

2. The first equation must be presented in simplified form starting from the fractional expression $\dfrac{x-1}{2} + \dfrac{y+3}{4} = 1$.

3. The line corresponding to the second equation must pass through the point $(-2, 5)$ and contain no fractions in its coefficients once simplified.

Show step by step the derivation of the second equation, analyze the relationship between the slopes of both lines, and analytically justify, using the ratio of their coefficients, why the constructed system has no solution at all.

Solve the following system of nonlinear equations. Keep in mind that you will need to simplify the algebraic expressions before applying the substitution method and check the validity of all the solutions obtained (remember that denominators cannot be zero, that is, $x\neq y$ and $x\neq -y$):

$\begin{cases} \displaystyle \frac{x+y}{x-y} + \frac{x-y}{x+y} = \frac{10}{3} \\ x^2 + y^2 = 20 \end{cases}$

Analyse and solve the following system of linear equations in terms of the real parameter $k$ using Gaussian elimination:

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

Determine for which values of $k$ the system is consistent with a unique solution, consistent with infinitely many solutions, or inconsistent. Explain the geometric interpretation of each case (planes intersecting at a point, along a line, or having no common points).

An investment fund management firm for young entrepreneurs distributes capital of $300.000$ € among three tech projects: Applied AI, Green Energy, and Cybersecurity.

It is known that:

1. The total annual profit was $23.500$ €.

2. The return on investment for each project was $5\%$ for AI, $10\%$ for Green Energy, and $8\%$ for Cybersecurity.

3. The capital invested in Cybersecurity is equal to the sum of the capital invested in AI and half of that invested in Green Energy.

Set up a system of three equations with three unknowns and use Gaussian elimination to calculate how much money was allocated to each project.

Find the values of $x$ and $y$ that satisfy the following system of logarithmic equations. Remember to apply logarithmic laws to convert the system into an algebraic one and check that the solutions do not invalidate the arguments of the original logarithms:

$\begin{cases} \log(x) + \log(y) = 2 \\ \log(x^2) - \log(y) = 1 \end{cases}$

An architect is designing a rectangular play area for a school. The total perimeter of the compound is $54$ metres. For safety reasons, a diagonal fence must be installed to divide the rectangle into two triangles. If the length of this diagonal is $15$ metres more than the difference between the length and the width of the rectangle, calculate the dimensions of the play area by setting up a system of quadratic equations.

Solve the following system of exponential equations by using an appropriate substitution ($u = 3^x$, $v = 2^y$):

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

Express the final values of $x$ and $y$ using logarithms if they are not integers.

An organic chemistry laboratory wants to prepare $10$ litres of a special solution with an alcohol concentration of $35\%$. To do so, three different mixtures are available:

• Mixture A: $20\%$ alcohol.

• Mixture B: $30\%$ alcohol.

• Mixture C: $50\%$ alcohol.

The laboratory manager imposes the condition that the volume of Mixture C used must be twice the volume of Mixture A used. How many litres of each mixture are needed to obtain the desired combination?

Given the supply function ($O$) and demand function ($D$) of a financial product in the market:

$D(x) = \displaystyle \frac{40}{x}$

$O(x) = x - 3$

Where $x$ is the price of the asset and $y$ is the quantity of assets.

1. Set up the system of equations to find the market equilibrium point (where supply equals demand).

2. Solve the system analytically.

3. Does the negative solution make economic sense? Justify your answer based on the nature of the variables.

An olive oil cooperative in Úbeda (Jaén) has packaged its monthly production of $12.000\text{ L}$ into three types of containers: $5\text{ L}$ jugs, $2\text{ L}$ bottles, and $0,5\text{ L}$ bottles. The following production data are available:

• In total, $6.750$ containers have been used.

• The number of $0,5\text{ L}$ bottles used is three times the number of $5\text{ L}$ jugs.

1. a) Set up a system of three linear equations with three unknowns that represents the information, clearly stating what quantity each unknown represents.
2. b) Solve the system using Gaussian elimination and interpret how many containers of each type have been produced.
3. c) During an internal audit, a hypothesis is proposed: if the packaging cost were $0,80\text{ €}$ per $5\text{ L}$ jug, $0,40\text{ €}$ per $2\text{ L}$ bottle, and $0,20\text{ €}$ per $0,5\text{ L}$ bottle, would it have been possible to achieve a total packaging cost of $2.800\text{ €}$ while maintaining the total number of $6.750$ containers and the condition that the $0,5\text{ L}$ bottles triple the $5\text{ L}$ jugs? Set up this new $3 \times 3$ system, analyze its consistency, and interpret the meaning of the result in the business context.

Mateo is trying to solve a market equilibrium problem for a handmade footwear workshop in Elche (Alicante). The daily production $x$ (in tens of pairs) and the unit selling price $y$ (in tens of euros) satisfy the following non-linear system:

This is the solution that Mateo presents:

1. Solve for $y$ in the first equation: $y = 2x - 1$.

2. Substitute into the second: $x^2 - 2x(2x - 1) + 3(2x - 1)^2 = 19$.

3. Expand the products: $x^2 - 4x^2 - 2x + 3(4x^2 + 1) = 19$.

4. Combine like terms: $x^2 - 4x^2 - 2x + 12x^2 + 3 = 19 \implies 9x^2 - 2x - 16 = 0$.

5. Apply the quadratic formula, obtain roots with inexact radicals, and conclude that no viable integer production levels or prices exist for the workshop.

1. a) Identify and explain precisely the two algebraic errors made by Mateo in step 3 when expanding the parentheses and remarkable products.
2. b) Redo the algebraic expansion starting from step 2 until you obtain the correct quadratic equation in the unknown $x$.
3. c) Solve the system by uniquely pairing each value of $x$ with its corresponding value of $y$.
4. d) Interpret the solutions obtained: argue whether all mathematical solutions make sense in the real economic context of the business and determine the final equilibrium quantity of pairs and unit price.