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 $20$ 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.

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?

Lucía tries to solve the following non-linear system, which describes the relationship between two economic variables:

On her exam, she writes the following steps:

• Step 1: I isolate $y$ from the first equation: $y = 2x + 1$.

• Step 2: I substitute into the second equation: $x^2 + (2x + 1)^2 = 13$.

• Step 3: I calculate the square: $x^2 + 4x^2 + 1 = 13 \Rightarrow 5x^2 + 1 = 13$.

• Step 4: I isolate $x$: $5x^2 = 12 \Rightarrow x = \pm\sqrt{2,4}$.

1. a) Identify in which step Lucía's error is located and explain with mathematical precision which identity or algebraic property she has applied incorrectly.
2. b) Correct the procedure starting from step 2 and solve the system correctly, indicating all pairs of solutions $(x, y)$.

The following system of non-linear equations is proposed to be solved:

To solve it by substitution, three students suggest different paths:

• Irene: Isolate $y$ in the first equation and substitute its expression into the second.

• David: Isolate $x$ in the first equation and substitute its expression into the second.

• Clara: Isolate $y^2$ in the second equation ($y^2 = 2x^2 - 7$), take the square root $y = \pm\sqrt{2x^2 - 7}$, and substitute into the first.

1. a) Reasonably analyze the three proposals: explain which one is the most convenient and what inconveniences or unnecessary algebraic risks the other two introduce.
2. b) Solve the system following the optimal strategy you have chosen and express all solutions as coordinate pairs $(x, y)$.

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):

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

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 euros) 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 $2 \text{ metres}$ and its height is decreased by $3 \text{ metres}$, 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 fair trade cooperative in Granada wants to fit out a rectangular showroom. The architectural project establishes two constraints: the total perimeter of the premises must be $34\text{ m}$ and the usable area must be exactly $60\text{ m}^2$. Furthermore, accessibility regulations require the minimum width of the shorter side to be at least $4\text{ m}$.

1. a) Before operating algebraically, make a reasoned estimate: if the premises were perfectly square with that perimeter, what area would it have? From this upper bound, explain why the shorter side cannot be neither very close to $0$ nor greater than $8,5\text{ m}$, and estimate a plausible pair of integers for the dimensions.
2. b) Set up the corresponding non-linear system of equations, solve it by substitution, and find the exact dimensions of the premises ($x$ and $y$).
3. c) Check whether the solutions obtained confirm your initial estimate and determine, with justification, whether the premises will comply with accessibility regulations.

A cooperative oil mill in Jaén analyzes the economic viability of its new line of organic extra virgin olive oil. The average packaging cost per box, $y$ (in euros), depends on the number of daily boxes produced, $x$, according to the rational function $y = \frac{800 + 10x}{x}$, where $800\text{ \euro}$ corresponds to daily fixed costs and $10\text{ \euro}$ to the unit variable cost. On the other hand, market studies determine that the unit selling price, $y$ (in euros per box), follows the linear demand law $y + x = 70$.

1. a) Set up the non-linear system of equations that allows determining the daily production levels $x$ at which the unit selling price exactly equals the average packaging cost (break-even points).
2. b) Solve the system and state the solution pairs $(x, y)$ obtained.
3. c) Interpret the business meaning of the results: between what values of daily box production will the cooperative make a profit (that is, the selling price will exceed the average cost)? Justify your conclusion by analyzing what would happen if they produced $30\text{ boxes}$ per day versus if they produced only $10\text{ boxes}$.

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 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.

Calculate the dimensions of a rectangular garden knowing that its diagonal measures exactly $\sqrt{113}$ metres and that, if we increased its base by 2 metres and decreased its height by 1 metre, 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).

In an urban planning department, they need to define on a cartographic map the layout of two infrastructures: a straight tram line and a parabolic perimeter road. To allow passenger transfers, both lines must intersect at exactly two stations whose known coordinates are $A(1, 4)$ and $B(3, 8)$ (with distances expressed in kilometers), having no other point in common.

1. a) Deduce the linear equation of the tram's trajectory in the form $y = mx + n$ from the coordinates of both stations.
2. b) Knowing that the parabolic road corresponds to a quadratic function symmetric with respect to the vertical axis of the form $y = a \cdot x^2 + c$, calculate with justification the values of the parameters $a$ and $c$ so that it passes through both stations.
3. c) Set up the system of equations with the expressions obtained and algebraically prove why there are no other intersection points between both infrastructures besides points $A$ and $B$.
4. d) Technicians are planning a second tram line parallel to the first one (with the same slope $m$). Calculate what value its y-intercept $n$ must take to be tangent to the parabolic road, touching it at a single interchange station.

A digital marketing agency in Barcelona manages user acquisition for a financial services platform. The daily advertising spend $y$ (in hundreds of euros) and the number of registered clients $x$ (in tens of people) are linked by the company's budget according to the linear relationship $y - 3x = -4$. On the other hand, due to audience saturation algorithms, the campaign's performance corresponds to the non-linear model with a radical:

1. a) Isolate the most suitable linear variable and substitute it into the second equation to transform it into a quadratic polynomial equation in a single unknown.
2. b) Solve the resulting equation to obtain the candidate values of $x$ and determine their corresponding values of $y$.
3. c) Rigorously check each of the pairs $(x, y)$ obtained in the two original equations. Identify which of them is an "extraneous solution" (false root), explain with algebraic rigor why it appeared during the solving procedure, and determine the only real operating point $(x, y)$ of the advertising campaign, interpreting its values in tens of clients and hundreds of euros.