Exercises: Systems of equations

In a role-playing video game, the experience points of two skills, Strength ($x$) and Magic ($y$), must satisfy the conditions of the following system of equations:

Check, by carrying out the necessary operations, which of these pairs of values is the correct solution to the system for the character to level up:

1. a) $x = 10, y = 5$
2. b) $x = 8, y = 7$

Translate into algebraic language and write down the corresponding system of equations for the following situation:

"In a café, we were charged $5$ € for two soft drinks ($x$) and a bag of crisps ($y$). If we buy one soft drink and one bag of crisps, the price is $3$ €."

(It is not necessary to solve it, just set up the system).

Solve the following system by the substitution method. Notice that in the second equation the unknown $y$ is already isolated, which makes the process easier:

Solve the following system using the equating method. To do this, equate the two expressions representing the value of $x$:

Solve the following system using the elimination method. Notice that the unknown $y$ has opposite coefficients ($+1$ and $-1$), so you can add the two equations directly to eliminate it:

Aoife has bought $2$ pens and $3$ rubbers at the stationer's for a total of $4$ €. Seán has bought $2$ pens and $1$ rubber for $2$ €.

If $x$ is the price of a pen and $y$ that of a rubber, the system representing the situation is:

Find the price of each item using the elimination method (subtracting both equations).

On a small farm there is a total of $12$ animals, consisting of chickens ($x$) and rabbits ($y$). If we count all their legs, there is a total of $34$ legs. The system of equations describing this situation is:

Solve the system using whichever method you prefer to find out how many chickens and how many rabbits there are on the farm.

Look closely at the coefficients of the unknowns in the following system of linear equations:

1. a) If you had to choose between the substitution, elimination, and equating methods, which one would you consider most direct and convenient to solve this system? Justify your answer by looking at the characteristics of the coefficients.
2. b) Solve the system using the method you defended in the previous section and formally check that the pair $(x, y)$ obtained satisfies both equations at the same time.

We want to design a system of two linear equations with two unknowns whose unique solution is exactly the pair $(x = 4, \; y = -2)$.

1. a) Write a first equation of the form $ax + by = c$ (with $a$ and $b$ non-zero integers) that holds for these values. Show the numerical verification of how you obtain the constant term $c$.
2. b) Come up with a second linear equation that also admits this solution, but making sure that it is not equivalent (proportional) to the first one to prevent the lines from being coincident.
3. c) Solve your resulting system to prove that, indeed, the only shared solution is $(4, -2)$.

In a strategy video game, to unlock an upgrade you need to collect blue gems and red gems. We know that three times the number of blue gems plus twice the number of red gems gives a total of 34 energy. Furthermore, the difference between the number of blue and red gems is only 3.

1. a) Set up a system of equations to find out how many gems of each type you need.
2. b) Check whether the pair of values $x = 8, y = 5$ (where $x$ are the blue gems and $y$ are the red gems) is the solution to the system.

Solve the following system of equations using the substitution method. Remember that it is advisable to rearrange for the unknown with the simplest coefficient:

Look at the following figure made up of two rectangles with different properties. Rectangle A has a perimetre of $22\text{ cm}$. Rectangle B is such that its base is twice that of rectangle A and its height is three times that of rectangle A, having a total perimetre of $52\text{ cm}$.

Calculate the dimensions $x$ (base) and $y$ (height) of rectangle A by setting up and solving a system using the equating method.

In the school canteen, a group of friends paid $\text{€}\ 15$ for 4 soft drinks and 2 sandwiches. The next day, another group paid $\text{€}\ 17.50$ for 3 soft drinks and 4 sandwiches. Use the elimination method to find the price of a soft drink and that of a sandwich.

Find the solution to the following system of equations using the method you consider most appropriate. Before solving it, check whether it is necessary to simplify the expressions by expanding brackets:

A mother's age is currently three times that of her son. However, in 12 years' time, the mother's age will be only twice that of her son. How old are they both now? Use a table to organise the age data before setting up the system.

To organise a charity festival, two types of tickets were sold: children's tickets at $\text{€}\ 4$ and adult tickets at $\text{€}\ 7$. At the end of the event, a total of 240 people attended and the total raised was $\text{€}\ 1350$. How many children and how many adults attended the festival?

Two cyclists, Aoife and Seán, are a distance of $45\text{ km}$ apart. Aoife sets off from point A at a constant speed of $12\text{ km/h}$ and Seán sets off at the same time from point B to meet Aoife at $18\text{ km/h}$.

If $x$ is the time they take to meet and $y$ is the distance covered by Ana:

1. a) Set up a system of equations relating the time and distance for both.
2. b) How long will it take them to cross paths and at what distance from point A will it happen?

Cian tries to solve the following system of linear equations using the elimination method:

To eliminate the unknown $y$, Cian decides to multiply the second equation by $3$ and writes the following in his notebook:

Next, he adds both rows term by term, obtaining $17x = 16$, and concludes that "the system gives a very strange fractional solution and there is surely a typo in the statement."

1. a) Locate and explain with reasoning the mathematical error that Cian made when transforming the second equation.
2. b) What fundamental property of equalities did Cian violate in his procedure?
3. c) Correct the complete working from the erroneous step onwards, solve the system, and find the exact solution $(x, y)$.

In a neighbourhood greengrocer's in Cork, a kilogram of oranges costs less than a kilogram of strawberries. A customer buys $3\text{ kg}$ of oranges and $2\text{ kg}$ of strawberries, paying a total of $€11$ at the checkout. Next, another customer buys $2\text{ kg}$ of oranges and $4\text{ kg}$ of strawberries for an amount of $€18$.

1. a) Before writing any equations, make a reasoned estimate: if a kilo of oranges cost $€4$, could the first customer's receipt make sense? Do you expect the price of a kilo of oranges to be greater or less than $€2$? Justify your initial estimates.
2. b) Set up the system of equations that models both purchases, solve it using the method of your choice, and find the exact price per kilo of each product.
3. c) Check whether the values obtained confirm your initial estimates and reflect on whether the resulting prices are consistent with standard shopping in a market.

Solve the following system of linear equations by first simplifying both expressions. To find the final solution, use the double elimination method (eliminating one unknown and then the other):

In a computer shop, a customer buys 3 mechanical keyboards and 2 wireless mice for a total of $135$ €. The next day, taking advantage of a sale, they buy 2 keyboards and 4 mice for $111$ €. However, the deal was that keyboards had a discount of $10\%$ and mice a discount of $20\%$ off their original price.

Calculate the original price of each item by setting up a system of equations.

Find the dimensions of a rectangle given that its perimetre is $40$ cm. If its base were tripled and its height halved, the perimetre would become $70$ cm. Once you have found the base ($x$) and the height ($y$), calculate the area of the original rectangle.

A chemistry laboratory needs to prepare $10$ litres of a $25\%$ acid solution. To do this, it has two containers: one with an acid concentration of $15\%$ and another with a concentration of $40\%$. How many litres must be mixed from each container to obtain the desired solution?

The sum of the two digits of a number is $12$. If we reverse the order of its digits, we obtain a new number that is $18$ units greater than the original. Set up a system of equations to find that number.

(Hint: A two-digit number is written as $10x + y$).

Two cyclists, Aoife and Cian, are $36$ km apart. If they ride towards each other, they take $45$ minutes to meet. If they ride in the same direction (Aoife chasing Cian), Aoife takes $3$ hours to catch up with him.

Calculate the speed of each cyclist in km/h by setting up a system of equations.

Solve the following system using the method you consider most appropriate, after expanding brackets and collecting like terms:

Once solved, state whether the system is consistent with a unique solution, consistent with infinitely many solutions, or inconsistent.

Calculate the cost of painting the lateral surface of a regular quadrangular frustum of a pyramid whose bases have side lengths of $4\text{ m}$ and $2\text{ m}$ and whose height is $1.5\text{ m}$, knowing that the price is $85\text{ €/m}^2$.

In an Irish city, the municipal electric bicycle rental service charges a fare made up of a fixed unlocking fee for the vehicle plus a fixed rate for each minute of the ride.

Seán took a $15\text{-minute}$ ride and the system charged him €1.90. Niamh used another bicycle of the same type for $25\text{ minutes}$ and paid a total of €2.70.

1. a) Model the situation using a system of two linear equations with two unknowns, explicitly defining what quantity each unknown represents and in what units each is measured.
2. b) Solve the system to determine how much it costs to unlock a bicycle and what the cost per minute of riding is.
3. c) A third user, Aoife, has an exact balance of €5 in her app wallet. What is the maximum integer number of minutes she will be able to pedal on a single ride without running out of balance? Justify your deduction by interpreting the result.

Consider the following system of linear equations, in which an unknown parameter $k$ appears:

1. a) Without solving the system by trial and error, deduce with reasoning what value $k$ must take so that the two lines representing the system geometrically are parallel and, therefore, the system has no solution (it is an inconsistent system).
2. b) A student claims: "If we replace the constant term $7$ with another suitable number, keeping the value of $k$ you just found, the system would then have infinitely many solutions." Determine whether this statement is true or false, explaining what geometric and algebraic condition the two equations must satisfy for this to happen, and what number should exactly replace $7$.
3. c) If the value $k = 1$ is assigned, how many solutions will the system have? Justify your answer looking exclusively at the proportionality ratio between the coefficients of the unknowns, without needing to calculate the explicit solution.