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:

$$\begin{cases} x + y = 15 \\ 2x - y = 9 \end{cases}$$

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:

$$\begin{cases} 3x + y = 10 \\ y = 2x \end{cases}$$

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

$$\begin{cases} x = y + 5 \\ x = 2y + 2 \end{cases}$$

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:

$$\begin{cases} 4x + y = 13 \\ x - y = 2 \end{cases}$$

Lucía has bought $2$ pens and $3$ rubbers at the stationer's for a total of $4$ €. Pedro 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:

$$\begin{cases} 2x + 3y = 4 \\ 2x + y = 2 \end{cases}$$

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

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.

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.