Exercises: Non-linear systems

Solve the following non-linear system, which represents the intersection between a parabola and a line:

$y = x^2 - 4x + 3$

$y = x - 1$

Find the solutions to the following system formed by the equation of a circle and a line:

$x^2 + y^2 = 25$

$x + y = 7$

Find the values of $x$ and $y$ that satisfy the following system:

$xy = 12$

$x - y = 4$

Solve the following system of exponential equations by expressing both powers with the same base:

$2^{x+y} = 32$

$2^{x-y} = 2$

Determine the solution to the logarithmic system by applying the laws of logarithms:

$\log x + \log y = 1$

$x - y = 3$

Solve the following system containing algebraic fractions:

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

$xy = 6$

Solve the following radical system. Make sure to check whether the solution obtained is valid in the original equation:

$\sqrt{x} + y = 5$

$x - y = 1$

The sum of two positive numbers is $10$ and the sum of their squares is $58$. Set up a non-linear system of equations and solve it to find these numbers.

Marcos has solved the following non-linear system with radicals on the blackboard:

His steps were as follows:

1. Isolates $y$ in the first equation: $y = x + 2$.

2. Substitutes into the second equation: $\sqrt{x + 4} = x + 2$.

3. Squares both sides: $x + 4 = (x + 2)^2 \implies x + 4 = x^2 + 4x + 4$.

4. Collects like terms and factors: $x^2 + 3x = 0 \implies x(x + 3) = 0$, obtaining $x_1 = 0$ and $x_2 = -3$.

5. Calculates the associated ordinates: $y_1 = 0 + 2 = 2$ and $y_2 = -3 + 2 = -1$.

Marcos claims that the system has two solutions: $(0, 2)$ and $(-3, -1)$. Find the conceptual error made by Marcos, explain why the squaring operation can introduce extraneous solutions, and provide the correct solution set, justifying your answer.

To solve the following system of exponential equations:

two students propose different strategies:

• Alba proposes isolating $y$ by applying logarithms in the first equation, writing $y = \log_3(11 - 2^x)$, and then substituting this expression into the second equation.

• Bruno proposes applying the properties of powers first to the second equation ($2^{x+1} = 2 \cdot 2^x$ and $3^{y+1} = 3 \cdot 3^y$) and carrying out the change of variables $u = 2^x$ and $v = 3^y$.

Compare both methods and justify which one is more suitable and why. Next, solve the system using the optimal path and explain which mathematical property allows you to finally isolate the exact values of $x$ and $y$ from $u$ and $v$.

A rectangular plot has a perimeter of $40$ m and an area of $96$ m$^2$. Set up a non-linear system of equations to calculate the dimensions of the plot and solve it.

Solve the following system of exponential equations by applying a suitable change of variable:

$3^x + 2^y = 11$

$3^{x+1} - 2^{y-1} = 26$

Solve the logarithmic and quadratic system, checking the validity of the solution obtained:

$\log(x) - \log(y) = 1$

$x^2 - y^2 = 99$

Find the solution to the following radical system:

$\sqrt{x+y} + \sqrt{x-y} = 4$

$x - y = 4$

A cyclist completes a $60$ km journey at a constant speed. If they had travelled $2$ km/h faster, it would have taken them $1$ hour less to complete the journey. Set up and solve a non-linear system to find the average speed $v$ and the time $t$ taken.

A group of friends decides to buy a birthday present costing $120$ €, splitting the cost equally. At the last minute, two more friends join the group, which reduces each person's share by $3$ €. Set up and solve a system to find how many friends there were initially and how much each person paid.

In a right-angled triangle, the hypotenuse measures $13$ cm and the total perimeter is $30$ cm. Set up a system of equations to determine the lengths of the two legs.

Solve the following system formed by a rational equation and a quadratic equation:

$\frac{x+y}{x-y} = 5$

$x^2 + y^2 = 13$

An irrigation community in the Huerta de Valencia is going to prepare a rectangular basin for drip irrigation protected by a perimeter fence of $36\text{ m}$. Due to hydraulic requirements of the pumping system, the installation company specifies that the difference between the base area and the square of the shorter side must be exactly equal to $36\text{ m}^2$.

1. a) Model the situation using a non-linear system of equations, denoting by $x$ the length of the longer side and by $y$ that of the shorter side (with $x \ge y > 0$).
2. b) Solve the system algebraically by substitution.
3. c) Interpret the solutions obtained within the real-world context: how many different dimensional configurations are technically possible to build the basin while meeting all requirements? Justify the dimensions of each permissible option.

The boundary coverage of a rural telecommunications station is described by the circle with equation $x^2 + y^2 = 25$ (where $x$ and $y$ are measured in kilometres relative to the antenna located at the origin). A straight forest track follows the path $y = 2x - 5$.

1. a) Before calculating the solutions algebraically, make a reasoned geometric estimate: if an off-road vehicle travels along the track and crosses the vertical axis at $x = 0$, is that point inside, on, or outside the coverage boundary? Estimate in which quadrants of the Cartesian plane the vehicle's entry and exit into the covered area will occur.
2. b) Set up and solve the system formed by both equations to determine the exact coordinates of the intersection points between the track and the coverage boundary.
3. c) Check whether the calculated points match your previous estimate and find the exact distance travelled by the off-road vehicle within the antenna's coverage radius.

The circular orbit of a satellite in a reference frame follows the equation $x^2 + y^2 = 25$, while the path of a comet describes the hyperbola $xy = 12$. Solve the system to find the exact coordinates of the intersection points of both paths.

Discuss for which values of the parameter $k$ the following system has exactly one solution (geometrically, this corresponds to the tangency condition between the circle and the line):

$x^2 + y^2 = 10$

$x + y = k$

Solve the following mixed system combining exponential and logarithmic equations:

$2^x \cdot 2^y = 64$

$\log_2 x + \log_2 y = 3$

Solve the higher-degree system of equations. (Hint: use algebraic identities and factorisation to simplify the first equation):

$x^3 + y^3 = 35$

$x + y = 5$

A sheet-metal cylindrical tank has an interior volume of $54\pi$ m³ and a curved surface area (excluding the bases) of $36\pi$ m². Set up a system to find the base radius $r$ and the height $h$ of the cylinder, and solve it.

A principal $C$ is invested at an annual compound interest rate $r$ (expressed as a decimal). At the end of the first year, the accumulated amount is $11000$ €. At the end of the second year, the amount rises to $12100$ €. Set up a non-linear system to calculate the initial principal $C$ and the interest rate $r$, and solve it.

Solve the following logarithmic system by applying change-of-base properties and substitution:

$\log_y x + \log_x y = \frac{5}{2}$

$xy = 27$

Solve the following radical system, making sure to check which solutions are valid by analysing the domain of the roots:

$\sqrt{x^2 - y^2} = \sqrt{33}$

$x + y = 11$

A team of engineers is designing the slip road merging a parabolic frontage road with equation $y = a \cdot x^2 + c$ into a straight road with equation $y = 4x - 5$. To ensure maximum safety during the merging manoeuvre, both roadways must join smoothly (that is, they must be tangent) exactly at the point with abscissa $x = 3$.

1. a) Formulate the inverse problem: determine the values of the real parameters $a$ and $c$ that define the parabola, knowing that the non-linear system formed by the line and the parabola must have the point with abscissa $x = 3$ as its unique solution. (Hint: analyse the required coefficients of the resolving quadratic equation for it to admit a double root at that value).
2. b) Using the discriminant $\Delta$ of the intersection equation, formally justify why the tangency condition requires $\Delta = 0$ and explain what it would geometrically represent for the layout of the roads if said discriminant were strictly positive or strictly negative.

Consider the following symmetric non-linear system depending on a non-zero real parameter $k$ ($k \ne 0$):

1. a) Instead of substituting $y = k - x$ directly into the second equation (which would require expanding $(k - x)^3$), justify why it is advantageous to treat it as a symmetric system and use the algebraic identity $x^3 + y^3 = (x + y)^3 - 3xy(x + y)$ to obtain the product $P = x \cdot y$ in terms of $k$.
2. b) Set up the quadratic equation $t^2 - S \cdot t + P = 0$ whose solutions provide the values of the unknowns $x$ and $y$.
3. c) Discuss with justification for which values of the parameter $k$ the system has two distinct real solutions, a unique double solution, or no real solutions. Furthermore, explain why if the pair $(x_0, y_0)$ is a solution with $x_0 \ne y_0$, the pair $(y_0, x_0)$ is necessarily also a solution, interpreting this property as a geometric symmetry with respect to the line $y = x$.

Topic Challenge. Design the emergency logistical support network for three municipalities in your province.

1. Territorial geometry: Locate three non-collinear municipalities ($A$, $B$, and $C$) on Google Maps or IGN, setting the origin $(0,0)$ at your high school. Calculate their relative distances in km, find the circumcentre by solving the system of perpendicular bisectors of the triangle, and determine the opening angle between two routes using the angle difference trigonometric ratios $\cos(\alpha - \beta)$.

2. Distribution with Gauss: Look up real population or electricity consumption data for those localities on INE or REE. Set up and solve a $3 \times 3$ linear system using Gaussian elimination to distribute a fund of $150\,000$ € among the three sites in a justified manner according to three demand conditions.

3. Coverage and feasibility: Demarcate the feasible distribution area in GeoGebra using a system of linear inequalities with two variables. Determine the intersection points of a logistics route with the $12,5\text{ km}$ coverage radius of an antenna by solving a non-linear system (line and circle), and analyze whether the logistics cost function $f(x)$ has symmetry (parity).

4. Final report: Write a two-page proposal with the GeoGebra screenshots, the solved systems, and your technical conclusion.

Assessment criteria:

• Algebraic rigour when applying Gaussian elimination and solving linear and non-linear systems.

• Precision in the geometric formulation (notable points, distances, and trigonometry).

• Correct modeling of the system of inequalities and consistent use of real data.