Can You Solve These Unique System of Equations?

  • MHB
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In summary, a system of equations having positive real solutions means that the equations can be solved for positive values that satisfy all the equations. To determine if a system has positive real solutions, you can use algebraic methods to solve the equations and check if the resulting solutions are real numbers greater than zero. It is possible for a system to have both positive and negative real solutions, and this can be applied in various fields such as economics, physics, and engineering. Other types of solutions for systems of equations include negative real solutions, complex solutions, and no solutions.
  • #1
anemone
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MHB
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Here is this week's POTW:

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Find all positive real solutions to the following system of solution:

$x^3+y^3+z^3=x+y+z$

$x^2+y^2+z^2=xyz$

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Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!
 
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  • #2
I apologize as I just realized the reply to this POTW #413 has gone amiss.

I just checked and no one answered to this POTW. (Sadface) Nevertheless, you can refer to the suggested solution by other below:

We have $xyz=x^2+y^2+z^2>Y^2+z^2\ge 2yz$. Hence $x>2$ and $x^3>x$. Similarly, $y^3>y$ and $z^3>z$. Adding them up gives $x^3+y^3+z^3>x+y+z$ and this contradicts to what is given and hence, there are no solutions to the system.
 

FAQ: Can You Solve These Unique System of Equations?

What is a system of equations?

A system of equations is a set of two or more equations that contain two or more variables. The solution to a system of equations is a set of values that satisfy all of the equations in the system.

What are positive real solutions?

Positive real solutions are solutions to a system of equations that are positive numbers. In other words, they are values that are greater than zero and can be represented on the number line.

How do you find positive real solutions to a system of equations?

To find positive real solutions to a system of equations, you can use a variety of methods such as substitution, elimination, or graphing. These methods involve manipulating the equations to isolate the variables and solve for their values.

Why are positive real solutions important?

Positive real solutions are important because they represent real-world solutions to problems. In many scientific and mathematical applications, it is necessary to find solutions that are positive in order to make sense in the context of the problem.

Can a system of equations have more than one positive real solution?

Yes, a system of equations can have more than one positive real solution. This occurs when the equations intersect at multiple points on the graph or when there are multiple sets of values that satisfy all of the equations in the system.

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