Exercises: Word problems on systems

In a hardware shop, a customer buys $4$ screws and $3$ nuts for a total of $126$ pesos. Another customer buys $2$ screws and $5$ nuts for $105$ pesos. 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 $130$ meters. The length of the plot is known to be $15$ meters 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 $240$ pesos/liter and another at $330$ pesos/liter—to obtain a barrel of $300$ liters of mixture that will be sold at $276$ pesos/liter. How many liters of each type of wine were used to make the mixture?

A sporting goods store sells soccer balls, basketballs, and volleyballs. The price of a soccer ball is $1\,500$ pesos, a basketball is $1\,800$ pesos, and a volleyball is $1\,200$ pesos. A club bought $40$ balls for a total of $61\,800$ pesos. Knowing that the number of soccer balls 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 $72\,000$ pesos. This month, the mortgage payment rose by $5\%$ and the food expenditure decreased by $10\%$, making the total expenditure on these two items $70\,200$ pesos. 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 $240\,000$ pesos more than partner B. If the total profit to be distributed is $3\,120\,000$ pesos, set up a system of three linear equations and calculate the amount each partner receives.

A company manufactures three computer models: basic, intermediate and advanced. The total daily production is $400$ units. The manufacturing cost is $18\,000$ pesos for the basic model, $30\,000$ pesos for the intermediate model and $48\,000$ pesos for the advanced model, with a total daily cost of $11\,700\,000$ pesos. 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 $400$ meters 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.