Question

Solve this system of two equations with two unknowns using appropriate software: $$ \begin{aligned} &x^{3}-y^{2}=10.5 \\ &3 x y+y=4.6 \end{aligned} $$

Short answer

Answer: To solve a system of nonlinear equations using Mathematica, follow these steps: 1. Choose the appropriate software (i.e., Wolfram Mathematica). 2. Write the given equations in the Mathematica format using double equal signs (==). 3. Use the `NSolve` function to solve the system of equations. 4. Interpret the solutions given by Mathematica in a list format, finding the numerical values for x and y for each solution.

Step-by-step solution

Choose the appropriate software for solving the nonlinear equations

In this case, we will use the popular and powerful mathematical software Wolfram Mathematica to solve this nonlinear system of equations. Yet, there are other alternative software that can help you to solve similar problems like MATLAB or Python's Sympy library.

Write the equations in Mathematica format

First, we need to write the given equations in the format that Mathematica accepts: `eq1 = x^3 - y^2 == 10.5; eq2 = 3*x*y + y == 4.6;` Make sure to use double equal signs (==) for the equations.

Solve the system of equations using Mathematica command

To solve the system of nonlinear equations in Mathematica, use the following `NSolve` function: `solutions = NSolve[{eq1, eq2}, {x, y}]` The `NSolve` function will return the numerical solutions for the unknowns x and y.

Interpret the solutions given by Mathematica

After running the `NSolve` command, Mathematica will provide the numerical solutions for the system of equations (in most cases, multiple solutions). The solutions will be in a list format, and you may need to scroll through the list to find the numerical values for x and y for each solution. That's it! You have successfully solved the system of nonlinear equations using Mathematica. Follow these steps whenever you need to solve similar problems.