Exercises: Simultaneous 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 performing the necessary operations, which of these pairs of values is the correct solution to the system so that the character can level up:

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

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

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

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

Solve the following system using 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 that represent the value of $x$:

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

Lucy bought $2$ pens and $3$ erasers at the stationery store for a total of $\$4$. Michael bought $2$ pens and $1$ eraser for $\$2$.

If $x$ is the price of a pen and $y$ the price of an eraser, the system that represents 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, including chickens ($x$) and rabbits ($y$). If we count all their legs, there is a total of $34$ legs. The system of equations that describes this situation is:

Solve the system using the method of your choice 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 triple the blue gems plus double the red gems adds up to 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 if the pair of values $x = 8, y = 5$ (where $x$ are the blue ones and $y$ are the red ones) is the solution to the system.

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

Observe the following figure composed of two rectangles with different properties. Rectangle A has a perimeter of $22\text{ cm}$. Rectangle B is such that its base is double that of rectangle A and its height is triple that of rectangle A, adding up to a total perimeter 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 cafeteria, a group of friends paid $15\text{ \$}$ for 4 soft drinks and 2 sandwiches. The next day, another group paid $17,50\text{ \$}$ for 3 soft drinks and 4 sandwiches. Use the elimination method to find the price of a soft drink and a sandwich.

Find the solution to the following system of equations using the method you consider most appropriate. Before solving it, observe if it is necessary to simplify the expressions by removing parentheses:

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

To organize a charity festival, two types of tickets were sold: children's at $4\text{ \$}$ and adults' at $7\text{ \$}$. At the end of the event, it is known that a total of 240 people attended and that the total amount collected was $1350\text{ \$}$. How many children and how many adults attended the festival?

Two cyclists, Ashley and David, are at a distance of $45\text{ mi}$. Ashley leaves from point A at a constant speed of $12\text{ mph}$ and David leaves at the same time from point B to meet Ashley at $18\text{ mph}$.

If $x$ is the time they take to meet and $y$ is the distance traveled by Ashley:

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

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 store, a user buys 3 mechanical keyboards and 2 wireless mice for a total of $\$135$. The next day, taking advantage of an offer, they buy 2 keyboards and 4 mice for $\$111$. However, the offer consisted of a $10\%$ discount on the keyboards and a $20\%$ discount on the mice from their original price.

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

Find the dimensions of a rectangle knowing that its perimeter is $40$ in. If its base were tripled and its height halved, the perimeter would become $70$ in. Once the base ($x$) and height ($y$) are found, calculate the area of the original rectangle.

A chemical laboratory needs to prepare $10$ gallons of an acid solution at $25\%$. To do this, it has two containers: one with an acid concentration of $15\%$ and another with a concentration of $40\%$. How many gallons from each container must be mixed 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 $18$ greater than the original. Set up a system of equations to find this number.

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

Two cyclists, Olivia and Marcus, are at a distance of $36$ miles. If they travel toward each other, they take $45$ minutes to meet. If they travel in the same direction (Olivia chasing Marcus), Olivia takes $3$ hours to catch up with him.

Calculate the speed of each cyclist in mph by setting up a system of equations.

Solve the following system using the method you consider most appropriate, after removing parentheses and combining like terms:

Once solved, indicate whether the system is consistent independent, consistent dependent or inconsistent.