Nonlinear 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:

$xy = 12$

$x - y = 4$

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.

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 $7\,200$ pesos, splitting the cost equally. At the last minute, two more friends join the group, which reduces each person's share by $180$ pesos. 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.

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 $660\,000$ pesos. At the end of the second year, the amount rises to $726\,000$ pesos. Set up a non-linear system to calculate the initial principal $C$ and the interest rate $r$, and solve it.

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.

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.

Solve the following system containing algebraic fractions:

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

$xy = 6$

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

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

$x^2 + y^2 = 13$

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 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:

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

$x^2 - y^2 = 99$

Solve the following mixed system combining exponential and logarithmic equations:

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

$\log_2 x + \log_2 y = 3$

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. Make sure to check whether the solution obtained is valid in the original equation:

$\sqrt{x} + y = 5$

$x - y = 1$

Find the solution to the following radical system:

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

$x - y = 4$

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

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

$x + y = 11$

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

$x^3 + y^3 = 35$

$x + y = 5$

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):

$x^2 + y^2 = 10$

$x + y = k$