Problem 39 - Algebra and Trigonometry

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

Which form is better largely depends on context. If a high level of detail is needed, the second student's parameter representation is more advantageous because it defines a method to iterate through the range of infinite possibilities, while the first student's assertion, although correct, lacks specificity.

Step 1: Understanding the Concept of Dependent Equations

In the context of linear algebra, a system of equations is considered dependent if all of the equations represent the same geometric entity, such as a line or a plane. That implies we have infinite solutions, as any point on that shared entity would satisfy all the included equations.

Step 2: Reviewing the Two Forms of Expressing the Solution Set

In reference to the above scenario, the first student proposed an accurate, but very general, observation that there are infinitely many solutions. The second student, on the other hand, provides a more explicit representation of those infinite solutions using a parameterized equation. By replacing \(z\) with any real number, he/she can uniquely generate different points on the solution set.

Step 3: Drawing the Conclusion

Regarding which form is better, it largely depends on the context. While the first student's assertion is correct and succinct, it lacks specific details. The second student's answer, however, offers a mechanism to generate specific solution points from the range of infinite possibilities. In situations where it's important to provide more detail or specificity, the second format might be more advantageous.