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 $100$ pesos for two soft drinks ($x$) and a bag of chips ($y$). If we buy one soft drink and one bag of chips, the price is $60$ pesos."

(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:

Lucía has bought $2$ pens and $3$ rubbers at the stationer's for a total of $80$ pesos. Pedro has bought $2$ pens and $1$ rubber for $40$ pesos.

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.

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 $\$300$ for 4 soft drinks and 2 sandwiches. The next day, another group paid $\$350$ 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 $\$80$ and adult tickets at $\$140$. At the end of the event, a total of 240 people attended and the total raised was $\$27\,000$. How many children and how many adults attended the festival?

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

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 $135$ pesos. The next day, taking advantage of a sale, they buy 2 keyboards and 4 mice for $111$ pesos. 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 $10$ liters 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 liters 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, Elena and Marco, are $36$ km apart. If they ride towards each other, they take $45$ minutes to meet. If they ride in the same direction (Elena chasing Marco), Elena 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 $85\text{ \$/m}^2$.