Problem 51 - Introductory Algebra for College Students

Question

Excited about the success of celebrity stamps, post office officials were rumored to have put forth a plan to institute two new types of thermometers. On these new scales, \(^{\circ} E\) represents degrees Elvis and \(^{\circ} M\) represents degrees Madonna. If it is known that \(40^{\circ} E=25^{\circ} M, 280^{\circ} E=125^{\circ} M\) and degrees Elvis is linearly related to degrees Madonna, write an cquation expressing \(E\) in terms of \(M.\)

Step-by-Step Solution

Answer

The equation expressing E in terms of M is \(E = 2.4M - 20\).

Step 1: Formulate Two Equations

From the problem, we get two equations: (1) \(E_1 = 40\) when \(M_1 = 25\) and, (2) \(E_2 = 280\) when \(M_2 = 125\).

Step 2: Calculate the Slope (m)

The slope of a line in a linear relationship is the change in y over the change in x. In our case, the slope `m` can be calculated with the formula \((E_2 - E_1) / (M_2 - M_1)\). Substituting the numerical values, we get \((280-40) / (125-25) = 240 / 100 = 2.4\). So, the slope \(m = 2.4\).

Step 3: Calculate the y-intercept (b)

To calculate the y-intercept \(b\), we rearrange the equation to the form \(b = y - mx\). Substituting for \(y\) and \(x\) from one of our initial equations and \(m\) from Step 2, we get \(b = E_1 - m * M_1 = 40 - 2.4 * 25 = -20\). So, \(b = -20\).

Step 4: Write the Linear Equation

With the values of \(m\) and \(b\) calculated in Steps 2 and 3, we now express `E` in terms of `M`. The linear equation is: \(E = 2.4M - 20\).

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It allows us to solve problems and find unknown values by creating relationships between variables. In the problem at hand, we're using algebra to find the equation that relates two new temperature scales: degrees Elvis (\(^{\circ} E\)) and degrees Madonna (\(^{\circ} M\)). By interpreting the given conditions, we form two algebraic equations based on known values.

• (1) \(E_1 = 40\) when \(M_1 = 25\)
• (2) \(E_2 = 280\) when \(M_2 = 125\)

By utilizing these equations, we can find the slope and intercept of the line that connects the two temperature scales. Algebra simplifies the understanding of how these two temperatures relate in a linear fashion, allowing us to express one variable in terms of another.

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations. It is expressed as \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. This form makes it easy to understand the direction and position of the line at a glance.**Finding the Slope**The slope \(m\) is calculated as the change in \(y\) over the change in \(x\). In our problem, this becomes \[m = \frac{E_2 - E_1}{M_2 - M_1} = \frac{280 - 40}{125 - 25} = \frac{240}{100} = 2.4\]The slope tells us that for every degree increase in Madonna, there is a 2.4-degree increase in Elvis.**Determining the Y-intercept**The y-intercept \(b\) is where the line crosses the y-axis. Calculated as:\[b = E_1 - m \cdot M_1 = 40 - 2.4 \times 25 = -20\]This means that when \(M = 0\), \(E\) would be \(-20\).This form is particularly useful in graphing and solving real-world problems, as it translates to easily understandable data.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to represent real-world scenarios. It helps in predicting and understanding complex systems by simplifying them into equations.In this exercise, creating a linear model between degrees Elvis and degrees Madonna gives a clear depiction of how the values change relative to each other. By converting a verbal problem into a mathematical equation, we provide a concise description of their relationship:\[E = 2.4M - 20\]This model shows how degrees Elvis can be calculated for any given value of degrees Madonna.**Benefits of Mathematical Modeling**

• It simplifies complex systems into understandable equations.
• Allows predictions of future behavior within the system.
• Gives a clear visual representation when graphed.

This example highlights how mathematical modeling, through the use of algebra and slope-intercept form, provides a practical tool for solving real-world problems efficiently.