Exercises: Non-linear systems

Solve the following system of non-linear equations:

$ \begin{cases} x - y = 2 \\ x^2 + y^2 = 34 \end{cases} $

Solve the system:

$ \begin{cases} x + y = 7 \\ x \cdot y = 12 \end{cases} $

Determine the solutions to the system:

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

Calculate two positive integers knowing that their sum is $15$ and their product is $56$.

The perimeter of a rectangle is $28 \text{ cm}$ and its area is $48 \text{ cm}^2$. Determine the lengths of its sides by setting up a system of equations.

Solve the following system with radicals:

$ \begin{cases} \sqrt{x} + y = 5 \\ x - y = 1 \end{cases} $

Solve the system with algebraic fractions:

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

The difference between two natural numbers is $3$ and the difference between their squares is $33$. What are these numbers?

Marcos is solving this non-linear system to find two unknown numbers:

$ \begin{cases} x + y = 9 \\ x^2 + y^2 = 41 \end{cases} $

In his notebook, he has written out the following procedure:

"I isolate $y$ in the first equation: $y = 9 - x$. I substitute it into the second:

$x^2 + (9 - x)^2 = 41 \implies x^2 + 81 - x^2 = 41 \implies 81 = 41$.

Since I get an absurd equality ($81 = 41$), I conclude that the system has no real solution."

1. a) Find and explain with reasoning the algebraic error that Marcos made when expanding the power.
2. b) Correct his approach, solve the resulting quadratic equation, and find the pairs of solutions $(x, y)$.
3. c) Check whether the solutions obtained satisfy the two conditions of the original system.

The City Council of Sevilla has launched an urban gardens plan and has assigned a neighbourhood association a rectangular plot with an area of $60\text{ m}^2$. To protect the crops from rabbits, the association has a closed roll of $32\text{ metres}$ of wire mesh to fence exactly the entire outer perimeter.

1. a) Let $x$ be the length and $y$ be the width of the plot, and set up a non-linear system of equations that reflects the perimeter and area information.
2. b) Solve the system using the substitution method by isolating an unknown in the linear equation.
3. c) Interpret the result: you will get two pairs of mathematical solutions $(x, y)$, but how many different rectangular plots do they represent on actual ground? Justify your answer.

Solve the following system of exponential equations:

$ \begin{cases} 2^x + 3^y = 17 \\ 2^{x+1} - 3^{y-1} = 13 \end{cases} $

Find the solution to the logarithmic system:

$ \begin{cases} \log(x) + \log(y) = 2 \\ x - y = 15 \end{cases} $

The diagonals of a rhombus differ by $4 \text{ cm}$ and its area is $96 \text{ cm}^2$. Calculate the length of both diagonals.

A group of friends decides to buy a birthday present together costing €$120$. At the last minute, $2$ more friends join the group, meaning each of the initial friends has to pay €$3$ less. Set up a system of equations and find how many friends there were initially.

Solve the system of radical equations:

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

Solve the following non-linear system:

$ \begin{cases} x^2 + xy = 24 \\ y^2 + xy = 12 \end{cases} $

A van travels a distance of $300 \text{ km}$. If it increased its average speed by $10 \text{ km/h}$, it would take one hour less to complete the journey. Determine the original speed and the time taken.

If the base of a rectangle increases by $2 \text{ cm}$ and its height increases by $1 \text{ cm}$, the area increases by $22 \text{ cm}^2$. Knowing that the original base is twice the height, find the initial dimensions of the rectangle.

An agricultural cooperative from El Ejido (Almería) is going to build a greenhouse shed with a rectangular floor plan. The technical project specifies that the outer perimeter of the foundation must be exactly $90\text{ metres}$ and the usable interior surface area must reach $450\text{ m}^2$.

1. a) Estimate: Before solving any equation, suggest at least two pairs of integer dimensions (length and width) whose sum is $45\text{ metres}$ (the semi-perimeter). Calculate mentally or approximately the area of each pair and indicate which of your proposals is closest to $450\text{ m}^2$.
2. b) Set up the non-linear system of equations corresponding to the perimeter and the area, and solve it algebraically by substitution.
3. c) Check: Compare the exact solutions with your estimates from section a) and reason whether the resulting dimensions are feasible to allow an agricultural tractor of $1,80\text{ metres}$ wide to pass and manoeuvre inside.

To find the intersection points between a parabolic trajectory and a straight ramp, the following system of equations is set up:

$ \begin{cases} y = x^2 - 4x + 1 \\ x + y = 5 \end{cases} $

Three members of a working team propose three different ways to solve it:

• Aitana proposes isolating $x$ in the second equation ($x = 5 - y$) and substituting it for both $x$'s in the first equation.

• Borja proposes isolating $y$ in the second equation ($y = 5 - x$) and equating both expressions for $y$.

• Carla suggests drawing the parabola and the line on graph paper to read the intersection points directly.

1. a) Compare Aitana's and Borja's procedures: justify which of the two algebraic methods is faster and carries a lower risk of making errors with signs and powers.
2. b) Solve the system following the more efficient strategy you chose and find the coordinates of the intersection points.
3. c) Evaluate Carla's graphical proposal: explain what visual advantage it offers and what its main limitation is compared to algebraic calculation if the intersection points do not have integer coordinates.

Find the dimensions of a right triangle whose hypotenuse measures $13 \text{ cm}$, knowing that if both legs increase their length by $2 \text{ cm}$, the area of the triangle increases by $19 \text{ cm}^2$.

Solve the following mixed system:

$ \begin{cases} \log_2(x) - \log_2(y) = 1 \\ 3^{x-y} = 81 \end{cases} $

Two hoses together fill a tank in $6$ hours. If the first hose takes $5$ hours less than the second to fill the tank on its own, how long does each hose need to fill it individually?

Solve the rational system:

$ \begin{cases} \frac{x+y}{x-y} = 5 \\ \frac{x^2}{y} = 9 \end{cases} $

A wire of length $40 \text{ cm}$ is cut into two pieces. Each piece is formed into a square. If the sum of the areas of both squares is $58 \text{ cm}^2$, what is the length of each piece of wire?

Solve the radical system by using a suitable change of variable or algebraic manipulation:

$ \begin{cases} \sqrt{x-1} + \sqrt{y+2} = 5 \\ x + y = 12 \end{cases} $

Find the points of intersection between the circle with equation $x^2 + y^2 - 4x - 6y + 11 = 0$ and the line $x - y = 1$. Represent the situation graphically.

You can do it on paper and send a photo with your answer.

A company manufactures two types of parts. The cost of producing $x$ type A parts and $y$ type B parts is given by the relation $C = x^2 + y^2$. We know that they manufacture a total of $14$ parts and that the total cost is €$100$. How many parts of each type do they manufacture if it is known that they produce more type A parts than type B?

In a video game's programming, a character's jump describes the parabola $y = -x^2 + 6x$, while a moving laser barrier moves along the line $y = 2x + k$, where $k$ is a parameter that adjusts the initial height of the beam.

1. a) Set up the non-linear system of equations formed by the parabola and the line, and obtain the quadratic equation that results after applying the substitution method (leaving the letter $k$ indicated).
2. b) Inverse problem: Determine what exact value $k$ must have for the laser beam to be tangent to the jump trajectory, that is, for the system to have a SINGLE real solution (the character grazes the laser at exactly one point without passing through it). Find the coordinates $(x, y)$ of that point of contact.
3. c) Analyse what values $k$ can take for the character to pass cleanly under the laser without ever touching it (the system has no real solution). Justify your answer by analysing the sign of the discriminant.

A community of owners in Madrid is going to install solar panels on the flat rooftop of their building, which is rectangular in shape with a total area of $180\text{ m}^2$. According to the Technical Building Code (CTE), it is mandatory to leave a $1\text{ metre}$ wide perimeter safety walkway clear around the entire outer contour for maintenance and evacuation tasks. Deducting this walkway, the central usable rectangular area reserved for the panels is reduced to $130\text{ m}^2$.

1. a) If we call the total length of the roof $x$ and its total width $y$, write the algebraic expression for the interior usable dimensions (subtracting the walkway on both sides) and set up the non-linear system of equations relating the total area to the usable area.
2. b) Expand the equation for the usable area and use the one for the total area to transform the system into an equivalent linear equation between $x$ and $y$.
3. c) Solve the system to calculate the exterior dimensions of the rooftop ($x$ and $y$) and the dimensions of the usable area for the panels.
4. d) Knowing that each standard photovoltaic panel measures $2\text{ m} \times 1\text{ m}$, calculate the theoretical maximum number of panels that fit by area in the usable zone and explain why, in practice, the actual arrangement of rows and columns might accommodate fewer panels.