Exercises: Word problems on systems

In a hardware shop, a customer buys $4$ screws and $3$ nuts for a total of $2.10$ €. Another customer buys $2$ screws and $5$ nuts for $1.75$ €. Set up a system of linear equations and calculate the price of each item.

A seaside hotel has double and single rooms. In total, there are $50$ rooms and $87$ beds. Assuming that all rooms are fully occupied, how many rooms of each type does the hotel have?

The sum of two integers is $54$ and their difference is $12$. Set up a system of equations and calculate the value of both numbers.

On a farm school, there are chickens and cows. Counting the heads of the animals gives a total of $45$; counting the legs gives $140$. How many animals of each type are there on the farm?

A father's current age is three times that of his son. In $10$ years' time, the father's age will be exactly twice that of the son. Determine their current ages.

The perimeter of a rectangular plot is $130$ metres. The length of the plot is known to be $15$ metres greater than its width. Set up a system of equations and determine the dimensions of the plot.

A mathematics exam consists of $30$ questions. For each correct answer, $5$ points are added, and for each incorrect or unanswered question, $2$ points are deducted. A student scored $94$ points in total. How many questions did the student answer correctly?

In a winery, two types of wine are mixed—one at $4$ €/litre and another at $5.50$ €/litre—to obtain a barrel of $300$ litres of mixture that will be sold at $4.60$ €/litre. How many litres of each type of wine were used to make the mixture?

In an age problem, the following situation is posed: "Seán is currently three times as old as his nephew Oisín. In $6$ years, Seán's age will be twice what Oisín's will be then". To solve it, Aoife writes the following system of equations on her exam, where $x$ represents Seán's current age and $y$ Oisín's:

1. a) Identify the conceptual error that Aoife made in the second equation and explain in your own words why that expression does not faithfully translate the statement.
2. b) Correct the formulation by writing the proper system and solve it using the substitution method.
3. c) Check that the obtained ages verify point by point the two time conditions of the statement.

An oil mill in Kilkenny wants to prepare $100$ litres of an oil blend to market it at a final price of $6.80$ €/l. For this, it has extra virgin olive oil (EVOO) at $8.00$ €/l and virgin olive oil at $5.00$ €/l.

1. a) Before doing any exact calculations, make a qualitative and justified estimate: will more litres of extra virgin or virgin be needed to achieve that average price? Support your conjecture by comparing the price differences.
2. b) Set up a system of two linear equations with two unknowns ($x$ litres of extra virgin and $y$ litres of virgin) that models the mixture.
3. c) Solve the system and compare the solution obtained with the estimate you made in section a) to assess whether the result is consistent.

A sports shop sells footballs, basketballs and volleyballs. The price of a football is $25$ €, a basketball is $30$ € and a volleyball is $20$ €. A club has bought $40$ balls for a total of $1030$ €. Knowing that the number of footballs bought is equal to the sum of the basketballs and volleyballs, set up a system of three equations and determine how many balls of each sport were purchased.

A family allocates a portion of their monthly income to mortgage payments and another portion to food. Last month, the sum of these two expenses was $1200$ €. This month, the mortgage payment rose by $5\%$ and the food expenditure decreased by $10\%$, making the total expenditure on these two items $1170$ €. Set up a system of equations to calculate how much they spent on each item last month.

The dimensions of a rectangular playing field are such that its area is $6000$ m² and the length of its diagonal is $130$ m. Set up a non-linear system of equations to calculate the length and width of the field.

The sum of the three digits of a number is $12$. The tens digit is the arithmetic mean of the other two. If the order of the digits is reversed, the resulting number is $198$ units greater than the original. Set up a system of equations and find the number.

Two taps fill a tank in $6$ hours when opened together. If the first tap takes $5$ hours less than the second to fill the tank individually, set up a system of equations (which will include algebraic fractions) to determine how long each tap would take to fill it individually.

A chemistry laboratory has three solutions of the same acid with concentrations of $10\%$, $20\%$ and $40\%$. They want to obtain $100$ ml of a new solution at $25\%$. If the same volume of the $10\%$ solution is used as the $20\%$ solution, what volume of each solution should be mixed?

A cyclist makes a return trip between two towns separated by a distance $d$. On the outward journey, with a tailwind, they ride at a constant speed of $30$ km/h. On the return journey, with a headwind, they ride at $20$ km/h. If the total time taken for the complete return trip is $2.5$ hours, set up a system of equations and find the distance between the two towns and the time taken on each leg of the journey.

A company distributes its annual profits among its three partners A, B and C. Partner A receives twice as much as partner B, and partner C receives $4000$ € more than partner B. If the total profit to be distributed is $52000$ €, set up a system of three linear equations and calculate the amount each partner receives.

A small artisanal bakery in Galway analyzes its weekly electricity bill broken down into the three periods of a time-of-use tariff in Ireland: peak ($0.24$ €/kWh), mid-peak ($0.15$ €/kWh), and off-peak ($0.10$ €/kWh). The meter recorded a total consumption of $1200$ kWh and the billed amount for the energy term amounted to $167$ €. To maximize savings through night baking, the bakery organized its production so that the off-peak period consumption was exactly twice the sum of the peak and mid-peak consumptions combined.

1. a) Set up a system of three linear equations with three unknowns that gathers all the information.
2. b) Solve the system by applying the Gaussian elimination method in an orderly manner.
3. c) Determine how many kWh were consumed in each period and evaluate with reasoning whether the bakery managed to concentrate most of its spending in the cheapest period.

A rectangular communal plot in a village in County Galway has a perimeter of $140\text{ m}$. The town council is planning a reorganization of the land: if its length is decreased by $6\text{ m}$ and its width is simultaneously increased by $2\text{ m}$, the total area of the property undergoes no change.

1. a) Set up the system of two equations describing the problem, letting $x$ be the initial length and $y$ the initial width. Algebraically simplify the equation of the areas and explain why, despite the product of variables $x \times y$, the resulting system is linear.
2. b) Once the system is reduced to the form $\begin{cases} x + y = 70 \\ x - 3y = 6 \end{cases}$, compare two algebraic solving methods (for example, elimination versus substitution) and justify which one is more direct and less error-prone in this case.
3. c) Solve the system using the method you selected, find the original dimensions, and verify that the proposed modification keeps the area intact.

A company manufactures three computer models: basic, intermediate and advanced. The total daily production is $400$ units. The manufacturing cost is $300$ € for the basic model, $500$ € for the intermediate model and $800$ € for the advanced model, with a total daily cost of $195000$ €. If the company decides to increase production of the basic model by $20\%$ and reduce that of the advanced model by $10\%$, total production would increase by $15$ units. Set up a system of equations with three unknowns and determine the initial production of each model.

Determine the lengths of the sides of a right-angled triangle, knowing that its perimeter is $60$ cm and that the altitude to the hypotenuse is $12$ cm. Set up a non-linear system of equations with three unknowns (legs and hypotenuse) to solve it.

In a sports competition, gold, silver and bronze medals award $5$, $3$ and $1$ points respectively. A team has won a total of $25$ medals and $75$ points. Knowing that the number of silver medals exceeds the number of gold medals by $1$, set up a system of equations and find how many medals of each type the team won.

Two trains set off simultaneously from two stations A and B separated by $450$ km, travelling along parallel tracks in opposite directions. They pass each other after $3$ hours. If the train leaving A had left $1$ hour earlier than the train from B, they would have passed each other $306$ km from station A. Set up a system of equations to calculate the constant speeds of both trains.

An open-topped box is to be constructed from a rectangular sheet of cardboard by cutting squares of side $5$ cm from each corner and folding up the sides. The length of the original sheet is twice its width. If the volume of the resulting box must be $1500$ cm³, set up the necessary equation or system and determine the dimensions of the original sheet of cardboard.

A system of linear equations depending on a parameter $k$ models the distribution of vehicles on three alternative routes $x$, $y$, $z$:

$ \begin{cases} x + y + z = 100 \\ 2x + 3y + kz = 250 \\ x + 2y + 3z = 160 \end{cases} $

Determine for which values of the parameter $k$ the model has a unique solution, infinitely many solutions, or no solution, and interpret what it physically means for the system to be inconsistent.

A farmer wants to fence off a rectangular field bordering a straight river, so no fence is needed along the riverbank. He has exactly $400$ metres of fencing. Knowing that the enclosed area must be $20000$ m², set up a system of equations to find the dimensions of the field and discuss with justification whether there is a unique solution to the problem.

An alloy is composed of gold, silver and copper. The total weight of a piece of this alloy is $120$ grams. It is known that the combined weight of gold and silver exceeds the weight of copper by $40$ grams. When submerged in water, the piece loses $10$ grams of its apparent weight due to upthrust. Given that the weight loss in water for pure gold is $1/20$ of its weight, for silver $1/10$, and for copper $1/8$, set up a system of equations and calculate how many grams of each metal the piece contains.

A regional commission must distribute European funds among three environmental innovation projects: $A$, $B$, and $C$. It is known that the final amount each one must receive is €$100000$ for $A$, €$150000$ for $B$, and €$250000$ for $C$. The regulations already impose two fixed linear relationships among the allocated grants ($x$, $y$, $z$ in euros):

1) The overall combined budget is €$500000$ ($x + y + z = 500000$).

2) The allocation for $C$ must equal the sum of the allocations for $A$ and $B$ ($x + y - z = 0$).

1. a) Design a third independent linear equation with practical meaning in the distribution that relates the amounts for $A$ and $B$ so that the complete system has exactly $(100000, 150000, 250000)$ as its unique solution. Justify using Gaussian elimination or ranks why the resulting system is consistent and independent (unique solution).
2. b) Write a different third equation that turns the system into consistent and dependent (infinitely many solutions). Interpret what economic meaning this situation would have for the commission's decision margin.
3. c) Set up a third equation that makes the system inconsistent and explain algebraically why that condition would prevent executing the distribution.

A last-mile urban delivery company operates with a total fleet of $24$ electric vans of three cargo capacities: small ($1\text{ m}^3$), medium ($2\text{ m}^3$), and large ($4\text{ m}^3$). The total simultaneous cargo volume of the entire fleet is $56\text{ m}^3$. Logistics management is testing an organizational criterion depending on a real parameter $k$: the number of small vans ($x$) plus $k$ times that of medium vans ($y$) must equal that of large vans ($z$), that is, $x + k\times y - z = 0$.

1. a) Write the system of three equations with three unknowns and apply Gaussian elimination to the augmented matrix to row-reduce it in terms of the parameter $k$.
2. b) Discuss with reasoning the number of solutions according to the values of $k$, indicating for which value the system is inconsistent and for which ones it is consistent with a unique solution.
3. c) In actual practice, the number of vehicles of each type must be a non-negative integer ($x, y, z \in \mathbb{N}$). Solve the system for $k = 1$ and for $k = -1$, and reason in detail which of the two cases is logistically feasible in reality and which one turns out to be physically impossible despite being mathematically consistent.