Question

1. if the equations of a system are $3x + 2y = 6$ and $6x + 4y = 12$, the system is:

a. inconsistent
b. undefined
c. consistent and independent
d. consistent and dependent

1. how can understanding that a system of linear equations has infinitely many solutions benefit a data analyst working with big data?

a. it allows the analyst to understand that under certain conditions, data trends can follow a predictable pattern.
b. it indicates a need for more variables to be introduced into the model.
c. it is generally of no benefit as real data sets do not conform to theoretical models.
d. it suggests that the data is too corrupted to provide meaningful insights.

Explanation:

First Question

Step1: Simplify the second equation

Divide $6x + 4y = 12$ by 2:
$\frac{6x}{2} + \frac{4y}{2} = \frac{12}{2}$
$3x + 2y = 6$

Step2: Compare the two equations

The simplified second equation is identical to the first equation $3x + 2y = 6$, meaning they represent the same line, so there are infinitely many solutions. Systems with infinitely many solutions are consistent and dependent.

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Second Question

A linear system with infinitely many solutions means the equations are dependent, representing a consistent, predictable relationship. For a data analyst, this translates to recognizing that under specific conditions, data trends follow a stable, repeatable pattern, which aids in modeling and forecasting. Adding more variables (option b) is not indicated here, real data can align with theoretical patterns (so option c is wrong), and infinite solutions do not mean corrupted data (option d is wrong).

Answer:

d. Consistent and dependent