Problem 122 - College Algebra with Modeling and Visualization

Question

Exercises 121 and 122: The following data can be modeled by a linear function. Estimate the value of \(x\) when \(y=2.99\) $$ \begin{array}{ccccc} x & 1 & 2 & 3 & 4 \\ \hline y & -1.66 & 2.06 & 5.78 & 9.50 \end{array} $$

Step-by-Step Solution

Answer

The approximate value of \(x\) is 2.25.

Step 1: Understand the Problem

We are given a set of data points and need to find the linear function that describes the relationship between \(x\) and \(y\). We have to estimate the value of \(x\) when \(y = 2.99\).

Step 2: Define the Linear Model

A linear function is defined as \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. We need to calculate \(m\) and \(c\) using the provided data points.

Step 3: Calculate the Slope \(m\)

The formula for the slope \(m\) is calculated using two points \((x_1, y_1)\) and \((x_2, y_2)\): \(m = \frac{y_2 - y_1}{x_2 - x_1}\).Using points (1, -1.66) and (2, 2.06), \[m = \frac{2.06 - (-1.66)}{2 - 1} = 3.72\].

Step 4: Calculate the y-intercept \(c\)

Plug one of the points into the linear equation to solve for \(c\). Using \((1, -1.66)\) and \(m = 3.72\), \(-1.66 = 3.72 \times 1 + c\). Solve for \(c\), so \(c = -1.66 - 3.72 \times 1 = -5.38\). Thus, the equation is \(y = 3.72x - 5.38\).

Step 5: Substitute \(y=2.99\) to Find \(x\)

Using the equation \(y = 3.72x - 5.38\), substitute \(y=2.99\) and solve for \(x\). \[2.99 = 3.72x - 5.38\] \[2.99 + 5.38 = 3.72x\] \[8.37 = 3.72x\] \[x = \frac{8.37}{3.72} \approx 2.25\].

Key Concepts

Slope Calculation

The slope, often represented by the symbol \( m \), is a measure of how steep a line is. It tells us how much \( y \) changes for a given change in \( x \). To calculate the slope, we use two distinct points from given data. The formula is:

\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Let's break this down step by step:

• Choose two points such as \((x_1, y_1)\) and \((x_2, y_2)\). In our exercise, these points were \((1, -1.66)\) and \((2, 2.06)\).
• Subtract the first \( y \)-value from the second \( y \)-value: \(2.06 - (-1.66)\).
• Subtract the first \( x \)-value from the second \( x \)-value: \(2 - 1\).
• Divide the change in \( y \) by the change in \( x \) to get \( m = 3.72 \).

A positive slope means the line is rising, while a negative slope means it is falling. Here, with \( m = 3.72 \), the line is ascending steadily as \( x \) increases.

Y-Intercept

The y-intercept, denoted as \( c \), marks the point where the line crosses the y-axis. It represents the value of \( y \) when \( x \) is zero. To find \( c \) in the equation \( y = mx + c \), do the following:

• Insert the slope \( m \) and use one of the points like \((x, y) = (1, -1.66)\).
• Plug these into the linear equation: \(-1.66 = 3.72 \times 1 + c\).
• Solve for \( c \): Since \(3.72 \times 1 = 3.72\), re-arrange the equation as \(-1.66 = 3.72 + c\).
• This results in \( c = -5.38 \).

Thus, the y-intercept informs us that the line would cross the \( y \)-axis at -5.38 when extended backwards.

Estimation of Values

To find the estimated values for an unknown \( x \) or \( y \), we use the linear equation derived. Our equation is \( y = 3.72x - 5.38 \). By substituting a known quantity, we can estimate the corresponding unknown.

Here's how to solve for \( x \) when \( y = 2.99 \):

• Start by substituting \( y \) in the equation: \( 2.99 = 3.72x - 5.38 \).
• Balance the equation by adding 5.38 to both sides: \( 2.99 + 5.38 = 3.72x \).
• This simplifies to \( 8.37 = 3.72x \).
• Finally, divide both sides by 3.72 to isolate \( x \), resulting in \( x \approx 2.25 \).

This process shows how the linear equation helps predict unknown values, serving as a powerful tool for estimation in real-world applications.