Problem 32 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \((3.45,10.88)\) and \((3.45,-4.69)\)

Step-by-Step Solution

Answer

The slope is undefined because the line is vertical.

Step 1: Identify the coordinates

First, identify the coordinates of the two points. The coordinates are given as \( (3.45, 10.88) \) and \( (3.45, -4.69) \).

Step 2: Recall the slope formula

The formula to find the slope (m) between 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 coordinates \( (x_1, y_1) = (3.45, 10.88) \) and \( (x_2, y_2) = (3.45, -4.69) \) into the formula: \[ m = \frac{-4.69 - 10.88}{3.45 - 3.45} \]

Step 4: Simplify the equation

Calculate the differences in the numerator and the denominator: \[ m = \frac{-15.57}{0} \]

Step 5: Determine the nature of the slope

Since the denominator is 0, this indicates the line is vertical. The slope of a vertical line is undefined.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves studying geometric shapes using a coordinate system. By placing figures in a coordinate plane, we can easily analyze and solve problems related to distance, angle, slope, and other geometric properties. In our exercise, we use the Cartesian coordinate system, which includes an x-axis (horizontal) and a y-axis (vertical).
Each point on the plane is represented as an ordered pair (x, y). The first value, x, is the horizontal distance from the origin, and the second value, y, is the vertical distance.
By placing our points \( (3.45, 10.88) \) and \( (3.45, -4.69) \) on this plane, we visualize the line passing through them and use their coordinates for further calculations.

Slope Formula

Understanding the slope of a line is crucial in coordinate geometry. The slope measures the steepness and direction of a line. It is calculated using the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of two points on the line.
To find the slope, we subtract the y-coordinate of the first point from the y-coordinate of the second point and divide this difference by the difference of the x-coordinates.
In our exercise, substituting \( (3.45, 10.88) \) and \( (3.45, -4.69) \) into the formula, we end up with:
\[ m = \frac{-4.69 - 10.88}{3.45 - 3.45} \]
Simplifying, we get:
\[ m = \frac{-15.57}{0} \]
Since the denominator is zero, this calculation reveals an important concept.

Undefined Slope

When the denominator in the slope formula becomes zero, as in our exercise, the situation represents a vertical line. Vertical lines have an undefined slope because division by zero is not possible in mathematics.
For the points \( (3.45,10.88) \) and \( (3.45, -4.69) \), we see that the x-coordinates are the same, making the line vertical.
A useful characteristic to remember is:

• Horizontal lines have a slope of zero.
• Vertical lines have an undefined slope.

Understanding undefined slopes helps in graphing lines and equations correctly and recognizing specific line types in coordinate geometry.