Non-linear systems
You already know the non-linear systems from 4th ESO: a line with a parabola, or a pair of numbers known by their sum and product. In Bachillerato they appear with exponentials, logarithms and radicals, but the strategy is still the same: rearrange the simplest equation and substitute.
What a non-linear system is
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 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: $xy = 12$ $x - y = 4$
Find the values of $x$ and $y$ that satisfy the following system:
$xy = 12$
$x - y = 4$
Exponentials and logarithms: change of variable
Solve the following system of exponential equations by expressing both powers with the same base: $2^{x+y} = 32$ $2^{x-y} = 2$
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: $\log x + \log y = 1$ $x - y = 3$
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 following system of exponential equations by applying a suitable change of variable:
$3^x + 2^y = 11$
$3^{x+1} - 2^{y-1} = 26$
Systems with radicals
Solve the following radical system. Make sure to check whether the solution obtained is valid in the original equation: $\sqrt{x} + y = 5$ $x - y = 1$
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: $\sqrt{x+y} + \sqrt{x-y} = 4$ $x - y = 4$
Find the solution to the following radical system:
$\sqrt{x+y} + \sqrt{x-y} = 4$
$x - y = 4$
Symmetric and higher-degree systems
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 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…
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 higher-degree system of equations. (Hint: use algebraic identities and factorisation to simplify the first equation): $x^3 + y^3 = 35$ $x + y = 5$
Solve the higher-degree system of equations. (Hint: use algebraic identities and factorisation to simplify the first equation):
$x^3 + y^3 = 35$
$x + y = 5$
Discussion with a parameter: tangency
A line can cut a curve at two points, at one point (if it is tangent) or at none. When the system has a parameter, discussing how many real solutions it has is equivalent to studying the discriminant of the resulting quadratic equation.
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…
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):
$x^2 + y^2 = 10$
$x + y = k$