Exercises: Non-linear systems of equations

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.

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

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

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.

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?

The difference between two numbers is $2$ and their product is $15$. Calculate these numbers by setting up a system of equations.

Solve the following system using the elimination method, taking advantage of the fact that both unknowns are squared:

$\begin{cases} x^2 + y^2 = 13 \\ x^2 - y^2 = 5 \end{cases}$

State all possible pairs of solutions $(x, y)$.

A young graphic designer is creating a logo that consists of a circle centered at the origin and a line that crosses it. The design software gives him the following equations for the strokes:

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

Calculate the points of intersection where the line and the circle cross.

In a right-angled triangle, the hypotenuse is $5$ cm long. If one leg is $1$ cm longer than the other, how long are the two legs?

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.

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.

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

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

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.

Two investors have deposited their savings in different cryptocurrencies. The relationship between their monthly profits ($x$ and $y$, expressed in thousands of pesos) is given by a logarithmic system:

$\begin{cases} \log(x) + \log(y) = \log(6) \\ x - 2y = 1 \end{cases}$

Calculate how much each investor has made this month, knowing that profits must be positive values.

The area of a rectangular garden is $180 \text{ m}^2$. If its base is increased by $2 \text{ meters}$ and its height is decreased by $3 \text{ meters}$, the resulting area remains $180 \text{ m}^2$. Calculate the original dimensions (base and height) of the garden.

Solve the system of equations with algebraic fractions, carrying out the necessary transformations to obtain a polynomial equation:

$\begin{cases} \dfrac{1}{x} + \dfrac{1}{y} = \dfrac{5}{6} \\ x + y = 5 \end{cases}$

A graphic design studio is creating a new circular logo crossed by a straight line. The circle of the logo is defined by the equation $x^2 + y^2 - 6x - 4y + 4 = 0$, while the cutting line follows the line with equation $y - x = 2$. Using a nonlinear system of equations, determine the points of intersection where the line cuts the circle, in order to place some anchor nodes.

An online store selling custom T-shirts has determined that the daily net profit $B$ (in hundreds of pesos) and the number of T-shirts sold $x$ are related by two different optimization 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.

Calculate the dimensions of a rectangular garden knowing that its diagonal measures exactly $\sqrt{113}$ meters and that, if we increased its base by 2 meters and decreased its height by 1 meter, the total area would not change. Set up and solve the corresponding system of equations.

In a biotechnology laboratory, the growth of two bacterial populations follows different patterns. Population $A$ is governed by the expression $2^x - 3^y = 7$, while the relationship between their densities at a critical moment in the experiment satisfies $2^{x+1} + 3^{y+1} = 59$. Find the values of $x$ and $y$ by solving the system using a suitable change of variable.

Find two integers such that the difference of their squares is 56 and the sum of their reciprocals is equal to $\frac{1}{45}$ multiplied by the sum of the original numbers.

A projectile is launched following a parabolic trajectory given by the equation $y = -x^2 + 8x - 7$. At the same time, a tracking laser beam is activated from the origin of coordinates following the linear trajectory $y = 2x$. Determine the points in space (coordinates $x, y$) where the laser beam intercepts the projectile.

Solve the following non-linear system of equations involving rational and quadratic terms:

$\begin{cases} \frac{x+y}{x-y} + \frac{x-y}{x+y} = \frac{10}{3} \\ x^2 + y^2 = 20 \end{cases}$

Note: Make a change of variable in the first equation to simplify the process.

It is desired to build a wooden frame for a rectangular mirror such that the total area (mirror plus frame) is $4000 \text{ cm}^2$. The mirror has an area of $2800 \text{ cm}^2$. If the frame has a constant width of $5 \text{ cm}$ on all sides, calculate the outer dimensions of the frame (total length and width).