The elimination method is a useful tool for solving equations with multiple variables. To use this method, you need to first rearrange your equation so that all the terms with the variable you are solving for (in this case, v0) are on one side of the equation, and all the other terms are on the other side. Then, you can add or subtract the equations together to eliminate one of the variables, leaving you with an equation that only contains the variable you are solving for. From there, you can solve for v0 using basic algebraic techniques.
While both the substitution and elimination methods are used to solve equations with multiple variables, they differ in their approach. The substitution method involves replacing one variable with an equivalent expression and then solving for the remaining variable. The elimination method, on the other hand, involves manipulating the equations to eliminate one of the variables entirely, leaving an equation with only one variable to solve for.
The elimination method can be used for linear equations with two or more variables. However, it may not be the most efficient method for solving certain equations. For example, if the equations have coefficients that are difficult to cancel out, the substitution method may be a better option.
If you end up with a contradiction (an equation that has no solutions) or an identity (an equation that is true for all values of the variable), this means that there is no unique solution for the system of equations. This can happen if the equations are parallel or if they represent the same line. In this case, there may be an error in your calculations or the equations may be inconsistent.
One common mistake when using the elimination method is forgetting to perform the same operation on both sides of the equation (e.g. adding or subtracting the same number from both sides). This can lead to incorrect solutions. It is also important to be careful when combining like terms and to double check your final solution by plugging it back into the original equations to ensure it satisfies all the equations simultaneously.