Problem 69 - Introductory Algebra for College Students

Question

A formula in the form \(y=m x+b\) models the cost, \(y,\) of a four-year college \(x\) years after \(2010 .\) Would you expect \(m\) to be positive, negative, or zero? Explain your answer.

Step-by-Step Solution

Answer

The slope 'm' is expected to be positive, as generally, the cost of education has been increasing over the years, which translates to 'for every additional year after 2010, the cost of the four-year college would increase'.

Step 1: Understand the Linear Equation

The given is a linear equation \(y=m x+b\) which represents the cost, \(y\), of a four-year college \(x\) years after 2010. In this form of equation, \(m\) represents the slope or the rate of change. Interpretation is, for each increase by one unit of \(x\), \(y\) will increase by \(m\) unit. Here, we need to predict if \(m\) is positive, negative or zero.

Step 2: Contextual Understanding

Predicting the trend of \(m\) would depend on what we understand about the cost of education over the years. Generally, the cost of college increases each year due to inflation and other factors. This would suggest that every additional year after 2010, the cost of college would go up.

Step 3: Estimation of Slope 'm'

Since we established that the cost of college increases each year, the rate of change \(m\) which represents this increase can be expected to be positive. Hence, the slope 'm' is likely to be positive.