Problem 15 - College Algebra with Modeling and Visualization

Question

If possible, find the slope of the line passing through each pair of points. $$ (-0.5,9.2),(-0.3,7.6) $$

Step-by-Step Solution

Answer

The slope of the line is -8.

Step 1 - Understand the Slope Formula

The slope (abla) of a line passing through two points (x_1, y_1) and (x_2, y_2) is calculated using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 2 - Identify Values

Identify and assign the given points to (x_1, y_1) and (x_2, y_2). Here, (x_1, y_1) = (-0.5, 9.2) and (x_2, y_2) = (-0.3, 7.6).

Step 3 - Substitute Values into Slope Formula

Substitute the values for (x_1, y_1) and (x_2, y_2) into the slope formula:\[ m = \frac{7.6 - 9.2}{-0.3 + 0.5} \]

Step 4 - Calculate the Differences

Calculate the differences: - For the numerator: 7.6 - 9.2 = -1.6 - For the denominator: -0.3 + 0.5 = 0.2

Step 5 - Compute the Slope

Now, calculate the slope by dividing the differences:\[ m = \frac{-1.6}{0.2} = -8 \]

Step 6 - Conclusion about the Slope

The slope of the line passing through the points (-0.5, 9.2) and (-0.3, 7.6) is −8.

Key Concepts

Slope Formula

The slope formula is a fundamental concept used in mathematics to determine how steep a line is. Understanding this is crucial for graphing lines and interpreting data in geometry and algebra. When we talk about the slope of a line, we're essentially asking how much the line goes up or down as we move from one point to another along it.

The formula for calculating slope (represented by the letter \( m \)) is given by:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula helps us find the 'rise' over the 'run' — where 'rise' is the change in the vertical direction (\( y \)-coordinates) and 'run' is the change in the horizontal direction (\( x \)-coordinates).

This straightforward formula tells us how much the line "rises" or "falls" for each step it takes "horizontally" from one point to another.

Point Coordinates

Points on a graph are represented by coordinates that tell us their exact location. Coordinates are typically in the form of \( (x, y) \), where \( x \) specifies the horizontal position, and \( y \) specifies the vertical position.

For our example, we have two points: \( (-0.5, 9.2) \) and \( (-0.3, 7.6) \). These both are pairs of numbers that we use in the real world for plotting and calculating slopes.

To effectively utilize this in the slope formula, we assign these coordinates as follows:

• First Point: \( (x_1, y_1) = (-0.5, 9.2) \)
• Second Point: \( (x_2, y_2) = (-0.3, 7.6) \)

With these assignments, the slope formula can now be easily applied to find the slope of the line that connects these two points.

Calculating Slope

Once we have the coordinates identified and arranged for the slope formula, calculating the slope becomes a simple substitution process. Using our points, \( (-0.5, 9.2) \) for \( (x_1, y_1) \) and \( (-0.3, 7.6) \) for \( (x_2, y_2) \), we substitute these values into the slope formula:

\[ m = \frac{7.6 - 9.2}{-0.3 - (-0.5)} \]
Taking each part of the formula step by step:

• Calculate the 'rise': \( 7.6 - 9.2 = -1.6 \)
• Calculate the 'run': \( -0.3 + 0.5 = 0.2 \) (noticing treating negative accordingly)

Finally, we divide the rise by the run to get the slope:

\[ m = \frac{-1.6}{0.2} = -8 \]
Thus, the slope of the line through these points is \( -8 \). This negative result indicates that the line slopes downward as it moves from left to right.