Problem 118 - College Algebra

Question

A farmer finds 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 relationship is \(y = -0.5n + 45\).

Step 1: Understand the Problem

We need to find a linear equation of the form \(y = mn + b\), where \(n\) is the number of stalks and \(y\) is the yield per plant. We are given two data points: (30, 30) and (34, 28).

Step 2: Calculate the Slope (m)

The slope \(m\) is calculated as the change in yield per plant divided by the change in number of stalks. Using the points (30, 30) and (34, 28), the slope \(m\) is calculated as:\[ m = \frac{28 - 30}{34 - 30} = \frac{-2}{4} = -0.5 \]

Step 3: Write the Linear Equation Using Point-Slope Form

Using the slope \(m = -0.5\) and one of the points (30, 30), we use the point-slope form \(y - y_1 = m(n - n_1)\) to write the equation. Substituting the values, we get:\[ y - 30 = -0.5(n - 30) \]

Step 4: Simplify to Slope-Intercept Form

Distribute the slope and simplify to get the equation in the form \(y = mn + b\). Starting with:\[ y - 30 = -0.5n + 15 \]Add 30 to both sides to get:\[ y = -0.5n + 45 \]

Step 5: Verify the Equation with the Given Points

Substitute both points into the equation \(y = -0.5n + 45\) to verify it matches:For (30, 30), \(30 = -0.5(30) + 45 = 30\).For (34, 28), \(28= -0.5(34) + 45= 28\). The equation is verified.

Key Concepts

Slope Calculation

The concept of slope is central to understanding linear equations. It represents how much one variable changes with respect to another. In the farmer's problem, it tells us how the yield per plant changes as the number of plants changes.
The slope is often referred to as "rise over run." In mathematical terms, it's the change in the vertical axis (yield) divided by the change in the horizontal axis (number of stalks).

• The formula for slope is: \( m = \frac{\Delta y}{\Delta n} \), where \( \Delta y \) is the change in yield and \( \Delta n \) is the change in number of stalks.
• For the given points (30, 30) and (34, 28), the calculation is: \( m = \frac{28 - 30}{34 - 30} = \frac{-2}{4} = -0.5 \).

This tells us that for every additional stalk planted, the yield per plant decreases by 0.5 ounces.

Point-Slope Form

Point-slope form is a particularly useful way to write the equation of a line when you know one point and the slope of the line. It's especially handy in scenarios where you're working with real-world data, just like our farmer's situation.
The form is given by the equation:

• \( y - y_1 = m(n - n_1) \)
• The variables \( y_1 \) and \( n_1 \) represent the coordinates of a known point on the line.
• The slope \( m \) is calculated as shown earlier (\(-0.5\)).

Using the point (30, 30) and the calculated slope \( -0.5 \), we plug these values into the point-slope form: \[ y - 30 = -0.5(n - 30) \].This equation describes the relationship between the number of bean stalks and the yield for any given number of stalks.

Slope-Intercept Form

To make equations more user-friendly for prediction and interpretation, we often convert them to slope-intercept form.
This form makes it clear what the starting value (intercept) of our dependent variable is when the independent variable is zero. It has the structure:

• \( y = mn + b \)
• where \( m \) is the slope and \( b \) is the y-intercept.

Starting with the point-slope equation \( y - 30 = -0.5(n - 30) \), we distribute and simplify: \[ y = -0.5n + 45 \].Now, you can easily see that the y-intercept is 45. This means if no stalks were planted, the theoretical yield would be 45. This illustrates a baseline production level for this model.

Verifying Solutions

Verification is an essential step in problem-solving, especially for ensuring that your derived equation accurately models the data points.
To verify, you substitute the original data points into your final equation to check for consistency. Here’s how it works:

• For the equation \( y = -0.5n + 45 \):
• Substitute the point (30, 30), resulting in \( 30 = -0.5(30) + 45 \). Simplifying yields 30, confirming it fits.
• Substitute the point (34, 28), resulting in \( 28 = -0.5(34) + 45 \). Again, simplifying yields 28.

Both calculations confirm that our linear model correctly represents the original data.

Algebraic Problem-Solving

Algebraic problem-solving involves using mathematics to analyze and solve equations based on given data. For linear equations, it's about finding relationships and making predictions. Here are ways to approach these problems:

• Understand the Problem: Break down what you know and what you need to find. Identify data points and the type of relationship (linear, in this case).
• Calculate and Formulate: Use algebraic techniques like calculating the slope and transforming equations to usable forms.
• Solution Verification: Always test your solution against known values to ensure accuracy.
• Application: Use your final equation to make predictions or understand trends. Here, you could predict yields for different amounts of stalks planted.

By following these steps, you streamline problem-solving and enhance your understanding of algebraic concepts.