How can it be possible for a set of inhomogeneous simultaneous equations to have no unique solution?
It is indeed possible for a set of inhomogeneous simultaneous equations to have no unique solution. This occurs when the equations are dependent or inconsistent.
To understand why this happens, we need to determine the difference between dependent and inconsistent equations:
1. Dependent Equations: In this case, the equations are not truly independent, and one equation can be written as a linear combination of the others. This means that one or more equations are redundant, and they do not provide any new or additional information. Consequently, there are infinitely many solutions that satisfy the given set of equations.
2. Inconsistent Equations: When the equations contradict each other and cannot be satisfied simultaneously, we say they are inconsistent. In other words, there is no solution that simultaneously satisfies all the equations.
To determine whether a set of inhomogeneous simultaneous equations has no unique solution, we can use a method called Gaussian elimination or matrix algebra.
Here's how to approach it using Gaussian elimination:
1. Organize the equations in a matrix form, by writing the coefficients of the variables and the constant terms in the system.
2. Perform row operations to transform the matrix into reduced row-echelon form. These operations include multiplying a row by a constant, adding or subtracting rows, and exchanging rows.
3. If the matrix is transformed into reduced row-echelon form, examine the resulting matrix:
a. If there are rows of the form [0 0 0 ... a] (where a is not zero) followed by a row of the form [0 0 0 ... 0], the system has no solution and is inconsistent.
b. If there are any free variables (variables that do not have a leading 1 in their column) in the rows corresponding to the variables, the system has infinitely many solutions and is dependent.
c. If none of the above conditions are met, the system has a unique solution.
By following these steps, you can determine whether a set of inhomogeneous simultaneous equations has no unique solution and understand the reasons behind it.