- #1

EM_Guy

- 217

- 49

a = xyz

b = xy+xz+yz

c = x + y + z

How do you solve x, y, and z?

b = xy+xz+yz

c = x + y + z

How do you solve x, y, and z?

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In summary, the conversation discusses solving for the variables x, y, and z in equations and finding the roots of a cubic function. The solution involves using a quadratic equation and solving for y (or z) to find a solution for z in terms of a, b, and c, which can then be used to find x and y. The cubic equation being discussed is considered to have a "nice" solution.

- #1

EM_Guy

- 217

- 49

a = xyz

b = xy+xz+yz

c = x + y + z

How do you solve x, y, and z?

b = xy+xz+yz

c = x + y + z

How do you solve x, y, and z?

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- #2

mfb

Mentor

- 37,259

- 14,095

b=(c-y-z)y + (c-y-z)z + yz = c(y+z)-y^2-zy-z^2

Solve this quadratic equation for y (or z), use both in a=xyz and hope that it has a nice solution?

- #3

EM_Guy

- 217

- 49

I have determined that z^3-cz^2+bz-a = 0. So, if we can find the roots of the cubic function, then we have z as a function of a, b, and c. Then, it should be straightforward to find x and y in terms of a, b, and c.

But I forget how to find the roots of a cubic function.

- #4

mfb

Mentor

- 37,259

- 14,095

Solutions of a cubic equation

z^3-cz^2+bz-a = 0 looks very nice in my opinion.

- #5

blue_raver22

- 2,250

- 0

To solve this system of 3 nonlinear equations, we can use a variety of methods such as substitution, elimination, or graphing. One approach is to use the substitution method, where we solve one equation for one variable and then substitute it into the other equations. In this case, we can solve for z in the first equation, which gives us z = a/(xy). We can then substitute this into the second equation to get b = xy + x(a/(xy)) + y(a/(xy)). Simplifying, we get b = 2xy + a. We can now solve for x in terms of y and a, which gives us x = (b-a)/(2y). We can then substitute this value for x into the third equation to solve for y, and then solve for z using the first equation. This approach requires some algebraic manipulation but can lead to exact solutions for x, y, and z. Another approach is to use numerical methods such as Newton's method or the bisection method to approximate the solutions. These methods involve iterative calculations and can be more time-consuming, but they can handle more complex nonlinear equations. Ultimately, the method chosen will depend on the specific equations and the desired level of accuracy.

A system of 3 nonlinear equations is a set of three equations where the unknown variables are raised to a power or contain a variable term multiplied by itself. These equations cannot be solved using basic algebraic methods and require more advanced techniques to find a solution.

To solve a system of 3 nonlinear equations, you can use methods such as substitution, elimination, or graphing. These methods involve manipulating the equations to eliminate variables and find a solution that satisfies all three equations simultaneously. Another option is to use numerical methods, such as Newton's method, to approximate a solution.

The main challenge of solving a system of 3 nonlinear equations is that there is no one set method that will work for all situations. Each system may require a different approach, and it may be difficult to determine which method will be the most effective. Additionally, these equations can have multiple solutions or no real solutions, making it challenging to find a solution that satisfies all three equations.

Yes, there are many software programs and calculators that have built-in functions for solving systems of nonlinear equations. These tools can be helpful in quickly finding a solution, but it is important to understand the underlying methods and assumptions used by the software to ensure the accuracy of the solution.

Systems of 3 nonlinear equations are commonly used in fields such as physics, engineering, and economics to model and solve complex systems. They can be used to analyze and predict the behavior of dynamic systems, such as chemical reactions, population growth, and financial markets. They are also used in optimization problems to find the most efficient solution given certain constraints.

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