Exploring the Number of Solutions for Equations with Three Variables

Understanding Equations with Three Variables

When dealing with equations that involve three variables, it's important to understand the number of solutions these equations can have. For a single equation in three variables, such as xyz 6 or ax2 - by cz 88, there can be an infinite number of solutions due to the fact that the number of distinct equations exceeds the number of unknown variables. In such cases, there are infinitely many solutions because the variables can take on a wide range of values.

Special Cases of Equations with Three Variables

Case 1: No Solutions

When the number of distinct equations equals the number of unknowns, there can be no solutions. For example, consider the following system:

Here, if we subtract the first equation from the third, we get x 2. However, if we subtract the first equation from the second, we get x 1. Similarly, subtracting the second from the third gives us x 3, all of which are contradictions. Therefore, this system has no solutions.

Case 2: Infinitely Many Solutions

On the other hand, when the equations are consistent and there are infinitely many solutions, it often occurs when one equation is a linear combination of the others. For instance, consider the system:

Here, the values work for any x 1 and any pair of y and z such that yz 0. In such cases, we call one of the variables a 'free variable' because its value can vary with the values of the other variables. For example, in this system, z -y for any value of y.

Case 3: A Single Solution

Finally, there are cases where the system has exactly one solution. For instance:

The single solution to this system is x 11-1, which means x 1, y 1, z 1.

Solving Linear Equations in Three Variables

For a more structured approach, let's consider a linear equation in three variables of the form Ax By Cz D. The solution to such a linear equation can be represented geometrically as a plane in three-dimensional space (R3). By setting any two variables to zero, we can find intercepts on the space, which help in identifying the plane.

Matrix Representation and Rank

To determine the number of solutions using matrices, we can form an augmented matrix (A|b) and a matrix A without the last column. If the rank of the augmented matrix (A|b) equals the rank of A and both are equal to the number of variables, then there is only one solution. If the ranks are equal to the number of variables, there are infinitely many solutions. If the rank of the augmented matrix is less than the rank of A, then there is no solution.

Conclusion

Understanding the number of solutions for equations with three variables is crucial in algebra and its applications. Whether it's an infinite number of solutions, no solutions, or a single solution, the concepts of linear equations in three variables and matrix manipulation provide a robust framework for solving such systems.