Problem 39 - 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's form of expressing the solution set, \(\{(4 z+3,5 z-1, z)\}\), is better. While both students correctly identified that there are infinitely many solutions, the second student's form effectively communicates the relationship between these solutions.

Step 1: Understanding Dependent System of Equations

Dependent system of equations refers to a system in which all equations are multiples of each other. These systems usually have infinite solutions.

Step 2: Looking at the first student's response

The first student stated that there are infinitely many solutions. Although this answer is technically correct, it does not provide any additional information about the nature or form of these solutions.

Step 3: Looking at the second student's response

The second student expressed the solution set as \(\{(4 z+3,5 z-1, z)\}\). This form not only shows that the solution set is infinite, but it also represents a specific pattern that the solutions follow.

Step 4: Conclusion

While both answers are correct in stating that the system of equations has infinite solutions, the second student's response is better. The reason is because it also communicates the relationship between the variables in the solution set.