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.

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.

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 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.

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.

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?

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.

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.

Aoife 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 Aoife'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:

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

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

• Niamh: 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 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.

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 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.

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.

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

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?

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.

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.

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.