Non-linear systems of equations

The perimeter of a rectangular area designed for an escape room is $20$ metres and its total area is $24$ square metres.

Calculate the dimensions (length and width) of the room by setting up and solving a system of equations.

Find the solutions to the following system of equations:

$\begin{cases} x^2 + y^2 = 25 \\ x - y = 1 \end{cases}$

Which geometric shape does the first equation correspond to? What about the second?

A surveillance drone follows a straight-line security path described by the equation $x - y = 1$. At the same time, an intruder moves along the perimeter of a restricted area, following a path given by the parabolic function $y = x^2 - 5x + 6$. Determine the exact points of the plane where both paths cross so that the drone can intercept the intruder.

The figure shows both paths to help you get oriented; you must calculate the coordinates of the crossing points yourself.

Find two natural numbers knowing that their difference is equal to $4$ and that the sum of their squares is $106$. Set up a system of non-linear equations and solve it using the substitution method.

A biological study analyses the interaction between two populations of microorganisms in a laboratory. The number of individuals in each population, $x$ and $y$, must satisfy the following system of exponential equations:

$\begin{cases} 2^x + 2^y = 10 \\ 2^{x-y} = 4 \end{cases}$

Determine the values of $x$ and $y$, applying a change of variable if necessary.

Solve the following system of nonlinear equations using the substitution method:

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

The figure shows the parabola and the line corresponding to the two equations.

1. a) Solve the system by substitution and find the coordinates of all solutions $(x, y)$.

1. b) Briefly explain what the solutions you obtained represent geometrically (intersection points between a parabola and a line) and check that your results agree with the figure.

In the design of a new logo for a clothing brand, a circle and a straight line are used. The circle is centred at the origin and its equation is $x^2 + y^2 = 25$. The cutting line passes through the design following the equation $2x + y = 1$. Find the points of intersection between the circle and the line that define the boundaries of the logo.

Determine two natural numbers knowing that their sum is $10$ and that the sum of their squares is $58$. To do so, translate the problem into a system of equations and solve it by substitution.

Solve the following system of equations using whichever method you consider most appropriate (substitution, equating, or elimination):

$\begin{cases} x \cdot y = 12 \\ (x + 2)(y - 1) = 12 \end{cases}$

An online shop selling custom T-shirts has determined that the daily net profit $B$ (in hundreds of euros) and the number of T-shirts sold $x$ are related by two different optimisation models. The first model suggests that $B + x^2 = 10x + 11$, while the second model, based on variable costs, states that $B - 2x = -5$. Calculate the sales values $x$ for which both models yield the same net profit.