Can this system of equations be solved in real numbers?

Click For Summary
SUMMARY

The system of equations defined by $a(b+c-a^3)=b(c+a-b^3)=c(a+b-c^3)=1$ can be solved in real numbers by transforming it into three separate equations: $ab + ac - a^4 = 1$, $ab + bc - b^4 = 1$, and $ac + bc - c^4 = 1$. The solutions can be derived through inspection and algebraic manipulation, leading to specific values for variables a, b, and c. The discussion emphasizes the importance of recognizing patterns in polynomial equations for efficient problem-solving.

PREREQUISITES
  • Understanding of polynomial equations
  • Familiarity with algebraic manipulation techniques
  • Knowledge of real number properties
  • Experience with systems of equations
NEXT STEPS
  • Explore methods for solving polynomial systems of equations
  • Research techniques for algebraic inspection in problem-solving
  • Learn about the properties of real numbers in algebra
  • Study advanced algebraic manipulation strategies
USEFUL FOR

Mathematicians, students studying algebra, and anyone interested in solving complex polynomial equations will benefit from this discussion.

anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Solve in real numbers the system below:

$a(b+c-a^3)=b(c+a-b^3)=c(a+b-c^3)=1$
 
Mathematics news on Phys.org
a=1, b=1, c=1
a=-1, b=-1, c=-1
 
Wilmer said:
a=1, b=1, c=1
a=-1, b=-1, c=-1

Would you mind sharing how you found the solution? :D
 
Lazily, by inspection:
ab + ac - a^4 = 1
ab + bc - b^4 = 1
ac + bc - c^4 = 1
 
Wilmer said:
a=1, b=1, c=1
a=-1, b=-1, c=-1

Hi Wilmer,

Your answer (without the working, hehehe...) is correct, but the question remains on how we are going to prove those are the only solutions.(Nod)
 

Similar threads

MHB Real Numbers $a,\,b,\,c$ Solving System of Equations
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Solving a System of Nonlinear Equations Symbolically
  • · Replies 9 ·
Replies
9
Views
3K
MHB System of Equation Challenge (a+b)(b+c)=-1
  • · Replies 3 ·
Replies
3
Views
2K
High School Arithmetic and our place value system
  • · Replies 6 ·
Replies
6
Views
755
Undergrad Does this make a cubic out of a quadratic system?
  • · Replies 9 ·
Replies
9
Views
2K
MHB Prove Simultaneous Equations: $a+b+c+d \ne 0$
  • · Replies 2 ·
Replies
2
Views
1K
MHB Real Roots of Cubic Equation: $x^3+a^3x^2+b^3x+c^3=0$
  • · Replies 1 ·
Replies
1
Views
2K
MHB Find the amount of sand in each bag
  • · Replies 1 ·
Replies
1
Views
1K
MHB System of Equations: Find Triples $(x,y,z)$
  • · Replies 1 ·
Replies
1
Views
1K
MHB Prove Geometric Sequence with $(a,b,c)$
  • · Replies 1 ·
Replies
1
Views
1K