Exercises: Problems with systems

In a hardware store, 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 coastal hotel has double and single rooms. In total there are $50$ rooms and $87$ beds. Assuming 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 there are hens and cows. If the animals' heads are counted, there is a total of $45$; if the legs are counted, there are $140$. 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, the father's age will be exactly double that of the son. Determine the current ages of both.

The perimeter of a rectangular plot measures $130$ feet. It is known that the length of the plot is $15$ feet more than the width. Set up a system of equations and determine the dimensions of the plot.

A math exam consists of $30$ questions. For each correct answer, $5$ points are added, and for each incorrect or blank answer, $2$ points are subtracted. A student has obtained $94$ points in total. How many questions have they answered correctly?

In a winery, two types of wine are mixed, one at $\$4$/gallon and another at $\$5,50$/gallon, to obtain a barrel of $300$ gallons of mixture that will be sold at $\$4,60$/gallon. How many gallons of each type of wine have been used to make the mixture?

A sports shop sells soccer balls, basketballs, and volleyballs. The price of a soccer ball 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 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 have been purchased.

A family allocates a part of their monthly income to pay the mortgage and another part to food. Last month, the sum of these two expenses was $\$1200$. This month, the mortgage payment has increased by $5\%$ and the food expense has decreased by $10\%$, with the total expense for these two items being $\$1170$. 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 $6000$ sq ft and the length of its diagonal is $130$ ft. 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 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 out what the number is.

Two faucets fill a tank in $6$ hours if they are opened at the same time. If the first faucet takes $5$ hours less than the second to fill the tank separately, set up a system of equations (which will include algebraic fractions) to determine how long each faucet would take to fill it individually.

In a chemistry laboratory, there are three solutions of the same acid with concentrations of $10\%$, $20\%$, and $40\%$. We 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 must be mixed?

A cyclist makes a round trip between two towns separated by a distance $d$. On the outward journey, with a tailwind, they travel at a constant speed of $30$ mph. On the return journey, with a headwind, they travel at $20$ mph. If the total time taken for the complete round trip is $2,5$ hours, set up a system of equations and find the distance between the two towns and the time taken for each journey.

A company distributes its annual profits among its three partners A, B, and C. Partner A receives double that of 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 company manufactures three models of computers: basic, medium, and advanced. The total daily production is $400$ units. The manufacturing cost is $\$300$ for the basic, $\$500$ for the medium, and $\$800$ for the advanced, with a total daily cost of $\$195000$. If the company decides to increase the production of the basic model by $20\%$ and reduce that of the advanced model by $10\%$, the 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 triangle knowing that its perimeter is $60$ inches and that the height drawn to the hypotenuse measures $12$ inches. 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 out how many medals of each type the team has won.

Two trains depart simultaneously from two stations A and B separated by $450$ miles, traveling on parallel tracks and in opposite directions. They pass each other after $3$ hours. If the train leaving A had departed $1$ hour before the train from B, they would have passed each other $306$ miles from station A. Set up a system of equations to calculate the constant speeds of both trains.

A box without a lid is to be made from a rectangular sheet of cardboard by cutting squares of side $5$ inches from each corner and folding up the edges. The length of the original sheet is double the width. If the volume of the resulting box must be $1500$ cubic inches, set up the necessary equation or system and determine the dimensions of the original cardboard sheet.

A system of linear equations dependent 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 means physically for the system to be inconsistent.

A farmer wants to fence a rectangular field that borders a straight river, so no fence is needed on the river side. They have exactly $400$ feet of fencing. Knowing that the enclosed area must be $20000$ sq ft, 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 $120$ ounces. It is known that the weight of the gold and silver together exceeds the weight of the copper by $40$ ounces. When the piece is submerged in water, it loses $10$ ounces of its apparent weight due to hydrostatic upthrust. If it is known 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 ounces of each metal the piece contains.