Problem 118 - College Algebra

Question

A farmer fi ds there is a linear relationship between the number of bean stalks, \(n,\) she plants and the yield, \(y\), each plant produces. When she plants 30 stalks, each plant yields 30 oz of beans. When she plants 34 stalks, each plant produces 28 oz of beans. Find a linear relationship in the form \(y=m n+b\) that gives the yield when \(n\) stalks are planted.

Step-by-Step Solution

Answer

The linear equation is \( y = -0.5n + 45 \).

Step 1: Identify given points

The given points from the problem are (30, 30) and (34, 28). These points represent the relationship between the number of stalks, \( n \), and the yield per stalk, \( y \).

Step 2: Calculate the slope (m)

The slope \( m \) of the line can be calculated using the formula: \( m = \frac{y_2 - y_1}{n_2 - n_1} \). Here, \( y_1 = 30 \), \( y_2 = 28 \), \( n_1 = 30 \), and \( n_2 = 34 \). Thus, \( m = \frac{28 - 30}{34 - 30} = \frac{-2}{4} = -0.5 \).

Step 3: Use point-slope formula to find y-intercept (b)

We use the point-slope form \( y - y_1 = m(n - n_1) \) to find \( b \). Using the point (30, 30) and \( m = -0.5 \), \( y - 30 = -0.5(n - 30) \). Solving for \( y \): \( y = -0.5n + 15 + 30 \) simplifies to \( y = -0.5n + 45 \).

Step 4: Write the final linear equation

Using the slope \( m = -0.5 \) and intercept \( b = 45 \), the linear relationship is given by \( y = -0.5n + 45 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to write the equation of a straight line. It is one of the most common methods for expressing linear equations. In this form, the equation is written as \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is the point where the line intersects the y-axis.
To easily identify these components:

• Slope (\( m \)): It shows the steepness and direction of the line. A positive slope means the line rises as it moves to the right, while a negative slope means it falls.
• Y-intercept (\( b \)): This is the point where the line crosses the y-axis. It tells you the value of \( y \) when \( x \) is 0.

Using the slope-intercept form is beneficial because it allows for quick graphing of a line if the slope and intercept are known. Once the equation \( y = -0.5n + 45 \) was derived in the exercise, the farmer can easily predict the yield of beans for any number of stalks planted.

Point-Slope Formula

The point-slope formula is another powerful tool for dealing with linear equations. This formula is particularly useful when you know a point on the line and the slope of the line. The formula is expressed as \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1)\) is a known point on the line and \( m \) represents the slope.
For example, in the provided exercise:

• Point (\( x_1, y_1 \)): The farmer can use one of the given points, such as (30, 30).
• Slope (\( m \)): Calculated as -0.5.

Substituting these values in the formula helps find other aspects of the line, such as the y-intercept, which was crucial for finding the linear relationship. This method provides a bridge between knowing specific points and forming the full equation, making it easier to derive equations quickly.

Linear Relationships

Linear relationships describe situations where there is a constant rate of change between two variables. They can be represented graphically as straight lines.
Key aspects of linear relationships include:

• Constant Rate of Change: For every unit increase in one variable, there is a consistent change in the other variable. In this exercise, the slope \(-0.5\) indicates that for every additional stalk planted, each plant yields 0.5 oz less than the previous scenario.
• Graphical Representation: This consistent change appears as a straight line when graphed, making it easy to extrapolate and interpolate data.

In the farmer's case, the linear relationship \( y = -0.5n + 45 \) indicates a negative slope, showcasing that as more bean stalks are planted, the yield per stalk slightly decreases. Understanding such relationships helps in making informed decisions based on predictable outcomes, which is particularly valuable in fields like agriculture, economics, and data analysis.