- #1
anemone
Gold Member
MHB
POTW Director
- 3,883
- 115
Find all real $x$ and $y$ that satisfy the system $x^3+y^3=7$ and $x^2+y^2+x+y+xy=4$.
The system of equations $x^3+y^3=7$ and $x^2+y^2+x+y+xy=4$ is a set of two equations with two variables, $x$ and $y$, that must be solved simultaneously. This means that the values of $x$ and $y$ must satisfy both equations at the same time.
The degree of an equation is the highest power of the variable present. In this system, the first equation has a degree of 3 and the second equation has a degree of 2.
Yes, this system can be solved algebraically by using techniques such as substitution or elimination. However, the solutions may involve complex numbers.
Yes, this system can also be solved graphically by plotting the two equations on the same coordinate plane and finding the points of intersection. Additionally, numerical methods such as Newton's method or the bisection method can be used to approximate the solutions.
This system has an infinite number of solutions, as there are infinitely many combinations of $x$ and $y$ that satisfy both equations. However, there may be a finite number of real solutions depending on the values of the coefficients in the equations.