Problem 38 - Algebra for College Students

Question

Find the slope of the line that contains each of the following pairs of points. $$(-2.7,19.3),(5.46,-3.28)$$

Step-by-Step Solution

Answer

The slope of the line is \(-2.77\).

Step 1 - Identify the points

Begin by identifying and labeling the given points. Let the first point be \((x_1, y_1) = (-2.7, 19.3)\) and the second point be \((x_2, y_2) = (5.46, -3.28)\).

Step 2 - Recall the slope formula

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

Step 3 - Substitute the coordinates into the formula

Substitute the values of the coordinates into the slope formula: \[ m = \frac{-3.28 - 19.3}{5.46 - (-2.7)} \]

Step 4 - Simplify the numerator

Calculate the difference in the y-coordinates: \[ -3.28 - 19.3 = -22.58 \]

Step 5 - Simplify the denominator

Calculate the difference in the x-coordinates: \[ 5.46 - (-2.7) = 5.46 + 2.7 = 8.16 \]

Step 6 - Divide the numerator by the denominator

Divide the result from Step 4 by the result from Step 5 to find the slope: \[ m = \frac{-22.58}{8.16} \]

Step 7 - Simplify the fraction

Perform the division to simplify the fraction: \[ m = -2.77 \] (rounded to two decimal places)

Key Concepts

Finding Slope

Finding the slope of a line is an essential skill in coordinate geometry. The slope represents the steepness or incline of the line. To find the slope between any two points, you need to understand the concept of the 'rise' and 'run'. The rise is the change in the y-values, and the run is the change in the x-values. In simple terms:

• Rise = vertical change (difference in y-coordinates)
• Run = horizontal change (difference in x-coordinates)

Once you have these two values, calculating the slope becomes straightforward. Knowing the slope helps you understand how one quantity changes in relation to another.

Slope Formula

The slope formula is a mathematical tool that helps calculate the slope of a line passing through two specific points. It is expressed as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This formula requires two coordinates \((x_1, y_1)\) and \((x_2, y_2)\). By substituting these coordinates into the formula, you can easily find the slope:

• Identify the coordinates of the first point and label them \(x_1, y_1\)
• Identify the coordinates of the second point and label them \(x_2, y_2\)
• Subtract the y-coordinates to find the rise: \(y_2 - y_1\)
• Subtract the x-coordinates to find the run: \(x_2 - x_1\)
• Divide the rise by the run: \( m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \)

In our example, substituting the given points \((x_1, y_1) = (-2.7, 19.3)\) and \((x_2, y_2) = (5.46, -3.28)\), the slope calculation simplifies to:
\[ \frac{-3.28 - 19.3}{5.46 - (-2.7)} = \frac{-22.58}{8.16} = -2.77 \] (rounded to two decimal places).

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses algebraic formulas to describe and analyze geometrical shapes. It is based on a coordinate system where each point has a specific location identified by coordinates \(x, y\). Understanding coordinate geometry allows us to:

• Identify and plot points on a coordinate plane
• Find distances between points
• Determine the slope of lines
• Analyze linear equations

In our exercise, coordinate geometry helps us find the slope of a line through two specified points. This slope tells us how the line moves through the plane, indicating whether it rises, falls, or remains constant as we travel from one point to another. By applying the principles of coordinate geometry, students can solve problems in geometry and algebra, gaining a deeper understanding of mathematical relationships and structures.