Solving Linear Systems in Row Echelon Form

In this article, we explore the process of solving a linear system given in row echelon form. Specifically, we will analyze and solve the given system:

Given System

The given linear system in row echelon form is:

a x y - z 2w 4

w 5

Analysis of the Given System

To solve a linear system, it’s crucial to understand the number of equations and the number of unknowns. In the given system, we have two equations with five unknowns: x, y, z, w, a.

Equation 1:

a x y - z 2w 4

Equation 2:

w 5

Solving the Given System

Let's start by substituting the value of w from the second equation into the first equation:

w 5

a x y - z 2(5) 4

a x y - z 10 4

a x y - z -6

Thus, the simplified system of equations is:

w 5

a x y - z -6

Conclusion

From the analysis, it's evident that the given system cannot be fully solved because it contains fewer equations than unknowns. In such cases, we can express some variables in terms of others, leading to a particular solution, a general solution, or a parametric form depending on the number of free variables.

Note: In the given system, since there are four unknowns and only two equations, the system is underdetermined. Therefore, there are infinitely many solutions. We can solve for two variables in terms of the other two.

Additional Information and Context

When dealing with linear systems in row echelon form or reduced row echelon form, the key is to ensure that the system has a unique solution, a particular solution, or no solution at all. Here are a few additional points to consider:

Underdetermined Systems: Systems with fewer equations than unknowns have infinite solutions. These systems can be solved to express the variables in terms of free variables. Overdetermined Systems: Systems with more equations than unknowns often have no solution or a unique solution if the equations are consistent. Consistency: A system is consistent if it has at least one solution. If a system is inconsistent, then it has no solution.

Conclusion

While the given system cannot be fully solved due to having fewer equations than unknowns, understanding the solution process is essential for tackling similar problems. By expressing one or more variables in terms of the others, we can gain valuable insights into the nature of the system.

Keywords

Linear system, row echelon form, solving equations