Problem 35 - College Algebra

Question

In solving a system of dependent equations in three variables, one student simply said that there are infinitely many solutions. A second student expressed the solution set as \(\\{(4 z+3,5 z-1, z)\\} .\) Which is the better form of expressing the solution set and why?

Step-by-Step Solution

Answer

The second student provided a more helpful and informative solution by expressing it in parametric form. This form not only confirms the presence of infinite solutions, but also represents them in a way that clearly displays the relationship between the variables. Thus, it is a better method of presenting the solution set for a system of dependent equations in three variables.

Step 1: Understanding the Concept

In the realm of linear algebra, when either a system of equations is dependent, it implies that the equations aren't really unique, but rather, they're multiple representations of the same equation. Hence, there are infinitely many solutions. That being said, expressing these infinite solutions may be done in various ways, one of which is a simple statement such as 'infinite solutions', and another is in a parametric form.

Step 2: Evaluating the Two Forms of the Solution

The first student's answer, 'infinite solutions', while true in its essence, lacks detail or context. It doesn't explain to what variables these solutions apply, nor does it give any insight into the relationship between the variables. On the other hand, the second student's answer, expressed in a set, \(\{(4 z+3,5 z-1, z)\}\), is more informative. Here, the dependencies of the variables are clear. We can see that the value of 'z' will determine the value of the other two variables, giving us the ability to generate the infinite solutions by changing the parameter 'z'.

Step 3: Identifying the Better Form

Although both forms of expressing the solution are correct, the parametric form used by the second student is more useful and informative because it provides a clear structure on how the variables are related and helps in generating the infinite solutions by simply replacing 'z' with different values. Therefore, it is considered a better method to represent the solutions to a system of dependent equations in three variables.