Exercises: Problem solving using systems of equations

The sum of Salma's and her grandfather's ages is $72$ years. $6$ years ago, the grandfather's age was five times Salma's age. Set up a system of equations and find their current ages.

The difference between two numbers is $14$. If we add twice the smaller number to the larger number, the result is $53$. Find both numbers by setting up a system of equations.

In a stationer's, $3$ notebooks and $2$ pens cost $12$ euros, while $4$ notebooks and $1$ pen cost $13,50$ euros. What is the price of a notebook and of a pen?

The perimeter of a rectangle is $54 \text{ cm}$. It is known that the base is $3 \text{ cm}$ greater than twice the height. Determine the dimensions of the rectangle.

In a car park there are motorbikes and cars. In total there are $35$ vehicles and $114$ wheels (not counting spare wheels). Calculate how many motorbikes and how many cars are in the car park.

A money box contains $45$ coins that add up to a total of $14,40$ euros. If all the coins are $20 \text{ cents}$ and $50 \text{ cents}$, find out how many coins of each type there are.

In a multiple-choice exam of $30$ questions, $5$ points are awarded for each correct answer and $2$ points are deducted for each incorrect or blank answer. A student scored $94$ points. How many questions did they answer correctly?

A merchant wants to prepare $50 \text{ kg}$ of a coffee blend. He uses one coffee costing $6 \text{ €/kg}$ and another costing $8 \text{ €/kg}$. If he wants the price of the blend to be exactly $7,20 \text{ €/kg}$, how many kilograms of each type of coffee should he use?

At the ticket offices of the Museo del Prado in Madrid, a standard individual ticket costs $15\text{ \euro}$ and the reduced student fare costs $7,50\text{ \euro}$. A high school group along with their chaperones totals $24$ people, and they have paid exactly $210\text{ \euro}$ for all the tickets.

1. a) Define precisely what each unknown ($x$ and $y$) represents, stating their units.
2. b) Set up the system of two linear equations that describes the situation.
3. c) Solve the system to calculate how many tickets of each type were purchased.
4. d) Reason, without needing to solve a new system, why it would have been impossible for the total revenue from those $24$ people to be $215\text{ \euro}$ while keeping the same individual prices.

Hugo is trying to solve the following problem from his textbook: “Marta is currently three times as old as her brother Lucas. In $4$ years, Marta will be only twice as old as Lucas will be. How old are they now?”. To solve it, Hugo sets up this system, letting $x$ be Marta's current age and $y$ be Lucas's current age:

Subtracting $4$ from both sides of the second equation, Hugo gets $x = 2y$, which directly contradicts the first equation ($x = 3y$). Hugo then states that “the problem is poorly worded and has no logical solution in reality”.

1. a) Identify the conceptual error Hugo made when translating the condition “in $4$ years” into algebraic language.
2. b) Explain in your own words why his approach is incorrect and write the second equation properly.
3. c) Solve the corrected system and verify that the resulting ages satisfy both sentences of the original problem statement to the letter.

The sum of the digits of a two-digit number is $12$. If we reverse the order of its digits, we obtain a new number that is $18$ units greater than the original number. What is the number?

An architect designs a rectangular room whose area must be $108 \text{ m}^2$. Furthermore, to fit the budget for the skirting boards, the perimetre of the room must be exactly $42 \text{ m}$. Set up a system of equations and calculate the dimensions of the room.

If $2$ is added to both the numerator and the denominator of a fraction, the resulting fraction is equivalent to $\frac{2}{3}$. If, instead, $1$ is subtracted from both, the fraction is equivalent to $\frac{1}{2}$. Find the original fraction.

An investor divides a capital of $20\,000 \text{ €}$ between two investment funds. The first offers an annual yield of $4\%$ and the second of $6\%$. At the end of one year, the total profit from both funds amounts to $1\,050 \text{ €}$. How much money did she invest in each fund?

At a greengrocer's, $1 \text{ kg}$ of apples, $2 \text{ kg}$ of pears, and $1 \text{ kg}$ of bananas cost $8,50 \text{ €}$. In addition, $2 \text{ kg}$ of apples and $1 \text{ kg}$ of pears cost $6 \text{ €}$. If the price of $1 \text{ kg}$ of bananas is equal to the price of $1 \text{ kg}$ of apples plus $1 \text{ kg}$ of pears, what is the price of each type of fruit?

In a right-angled triangle, the hypotenuse measures $25 \text{ cm}$ and the sum of its legs is $35 \text{ cm}$. Set up a system of non-linear equations to determine the length of each leg.

A tank can be filled using two taps, A and B. If both are turned on at the same time, the tank fills in $6$ hours. Tap A alone takes $5$ hours less than tap B to fill the full tank. Set up a system of equations and find how long each tap would take to fill it individually.

In a cooperative oil mill in Jaén, $1\,000\text{ liters}$ of a special coupage (blend) of extra virgin olive oil are to be bottled by combining two native varieties: picual (with a production cost of $6,50\text{ \euro/l}$) and arbequina (with a cost of $8,50\text{ \euro/l}$). The cooperative requires the resulting average cost per liter of the blend to be exactly $7,90\text{ \euro/l}$.

1. a) Before carrying out algebraic calculations, make a reasoned estimate of which variety should contribute more liters to the tank (picual or arbequina?). Justify your answer looking solely at the distance between the unit prices and the desired final price.
2. b) Set up a system of two equations with two unknowns ($x$: liters of picual, $y$: liters of arbequina) and solve it using the method of your choice.
3. c) Check whether the obtained values match the estimate you made in section a) and interpret the result within the context of oil production.

The Physical Education department of a school purchases sports equipment from an official distributor in two successive orders:

• In the first order, they buy $3$ basketballs and $4$ volleyballs for a total amount of $146\text{ \euro}$.

• In the second order, they buy $5$ basketballs and $4$ volleyballs for a total amount of $194\text{ \euro}$.

1. a) Set up the corresponding system of linear equations, letting $x$ be the price of a basketball and $y$ be the price of a volleyball.
2. b) Analyze the three common algebraic methods (substitution, elimination, and equating). Justify which one saves the most calculations and minimizes the risk of error when solving this specific system.
3. c) Solve the system using the method you selected in section b) and verify the result against the original amounts of both orders.

The areas of three of the faces of a rectangular prism (cuboid) that meet at the same vertex are $24 \text{ cm}^2$, $32 \text{ cm}^2$, and $48 \text{ cm}^2$. Set up a system of equations with the lengths of its edges and calculate the volume of the prism.

A number is made up of three digits that sum to $18$. The hundreds digit is equal to the sum of the tens digit and the units digit. If the order of the digits is reversed, the resulting number is $396$ units less than the original number. Determine what the number is.

A goldsmith has two gold alloys. The first contains $80\%$ gold and the second $60\%$ gold. He wants to melt parts of both to obtain $50 \text{ grams}$ of a new alloy with $75\%$ gold. How many grams of each alloy should he use?

The area of a rhombus is $120 \text{ cm}^2$ and the sum of its diagonals is $34 \text{ cm}$. Set up a system of non-linear equations to find the length of the diagonals and then calculate the perimetre of the rhombus.

Three workers, Alba, Bernat, and Carlos, have to paint a building. If Alba and Bernat work together, it takes them $12$ days. If Bernat and Carlos work together, it takes them $20$ days. If Alba and Carlos work together, it takes them $15$ days. Set up a system of equations to work out how long each one would take working alone. (Hint: use as unknowns the fraction of the job that each one completes in a day).

A farmer wants to fence a rectangular piece of land. He has $100 \text{ m}$ of fencing for the perimetre. Knowing that one of the sides of the land must measure at least $15 \text{ m}$ and that the area of the land must be strictly greater than $600 \text{ m}^2$, set up a system of inequalities for the measurements of the base and height, and determine between which values the measurement of the longer side can range.

A cyclist leaves city A for city B at a constant speed. One hour later, a motorcyclist sets off from A in pursuit at a speed that is $15 \text{ km/h}$ higher than that of the cyclist, catching up with him just as he arrives at B. If the distance between A and B is $60 \text{ km}$, set up a system of equations to find the speeds of both and the time the cyclist took to complete the journey.

A municipal mobility officer in a Spanish provincial capital designs a campaign to promote public transportation. To balance the budget of an average user, they determine that an ideal solution is for a person to purchase exactly $x = 12$ single tickets and $y = 4$ combined monthly passes per year.

1. a) Write the complete statement of a realistic household budget problem that combines the total number of transit passes purchased per year ($16$ passes in total) and a pre-set total annual expenditure (assigning a price of $1,50\text{ \euro}$ to the single ticket and $35\text{ \euro}$ to the monthly pass), so that its formulation leads unambiguously to this solution.
2. b) Write the system of equations associated with the problem you have written and analytically verify that its unique solution is, indeed, $(12, 4)$.
3. c) Imagine replacing the total expenditure condition with this one: “three times the number of single tickets plus three times the number of monthly passes equals $48$”. Reason why with this new condition it is impossible to determine whether the user bought $(12, 4)$ or a different combination, explaining what happens mathematically to the system.

Two vehicles depart simultaneously at $09:00\text{ h}$ to cover the route between Madrid and Zaragoza ($315\text{ km}$ along the A-2 motorway) traveling in opposite directions: a car sets off from Madrid toward Zaragoza and a delivery van sets off from Zaragoza toward Madrid. Due to the cargo and the vehicle type, the car's average speed is $15\text{ km/h}$ faster than that of the van. Both vehicles pass each other on the motorway after exactly $1\text{ hour and } 30\text{ minutes}$ ($1,5\text{ hours}$) of continuous travel.

1. a) Define the unknowns, precisely stating their units of measurement.
2. b) Using the physical relationship $d = v \cdot t$, set up a linear system of two equations with two unknowns that models the difference in speeds and the sum of distances covered up to the moment they pass each other.
3. c) Solve the system to determine the average speed of each vehicle.
4. d) Interpret the result: reason whether both drivers have respected the generic legal speed limit on motorways in Spain ($120\text{ km/h}$) and calculate at what distance from Madrid the encounter took place.

Topic challenge. How can you optimize energy consumption and space in your home?

1. Real data and tariffs: Check the Red Eléctrica de España (REE) website for today's average kWh price and look online for a commercial tariff with a fixed monthly fee. Model the monthly cost $y$ based on the kWh consumed $x$ using two linear equations with two unknowns.

2. Linear solving and GeoGebra: Set up the resulting linear system. Solve it algebraically (using elimination, substitution, or equating) and plot it in GeoGebra to find the consumption threshold where both options break even.

3. Solar space (non-linear system): Use a tape measure to measure the available outline on a window or balcony in your home to obtain its real perimeter ($2x + 2y = p$). If the solar panel requires a collection surface area of $x \cdot y = 1,2$ $\text{m}^2$, set up and solve the non-linear system to determine the possible dimensions ($x$, $y$).

4. Final report: Write a short report that includes the data table, GeoGebra screenshots, the solutions to both systems, and your reasoned recommendation for saving at home.

Assessment criteria:

• Correct mathematical formulation and resolution of the linear and non-linear systems.

• Rigorous use of GeoGebra for graphical representation and interpretation of solutions.

• Accuracy in collecting real data (measurements and web) and clarity in the report's conclusions.