How do three simultaneous equations with three variables be solved?

Three simultaneous equations with three variables can be solved using various methods, such as substitution, elimination, or matrix methods. Here is a general outline of how to solve a system of three equations with three variables:

1. Write down the three equations in standard form:
- Equation 1: ax + by + cz = d
- Equation 2: ex + fy + gz = h
- Equation 3: ix + jy + kz = l

2. Choose a method to solve the system, such as substitution, elimination, or matrix methods.

3. Use the chosen method to solve for one variable in terms of the other two variables in one of the equations.

4. Substitute the expression for the variable found in step 3 into the other two equations to create two equations with two variables.

5. Solve the two equations with two variables using the same method as in step 3.

6. Once you have found the values of two variables, substitute them back into one of the original equations to solve for the third variable.

7. Check your solution by substituting the values of the variables back into all three original equations to ensure they satisfy all three equations.

By following these steps, you can solve a system of three equations with three variables and find the values of the variables that satisfy all three equations simultaneously.