Problem 14 - Brief Applied Calculus

Question

Find the slope (if it is defined) of the line determined by each pair of points. \((6,-4)\) and \((6,-3)\)

Step-by-Step Solution

Answer

The slope of the line is undefined; it's a vertical line.

Step 1: Identify the given points

We are given two points: \((6, -4)\) and \((6, -3)\). These are the coordinates \((x_1, y_1) = (6, -4)\) and \((x_2, y_2) = (6, -3)\).

Step 2: Determine the formula for the slope

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \[m = \frac{y_2 - y_1}{x_2 - x_1}\].

Step 3: Substitute the values into the slope formula

Using the coordinates \((x_1, y_1) = (6, -4)\) and \((x_2, y_2) = (6, -3)\), substitute into the formula: \[m = \frac{-3 - (-4)}{6 - 6}\].

Step 4: Simplify the expression

Calculate \(-3 - (-4) = -3 + 4 = 1\) and \(6 - 6 = 0\). Thus, the slope \(m = \frac{1}{0}\).

Step 5: Interpret the result

Since division by zero is undefined, the slope \(m\) is not defined. This means the line is a vertical line.

Key Concepts

Understanding Coordinates

Coordinates are a pair of numbers used to locate a point on a plane. The most common system is the Cartesian coordinate system. Here, each point is defined by an

• x-coordinate, which represents the horizontal position, and a
• y-coordinate, which indicates the vertical position.

For instance, in the point

• (6, -4), 6 is the x-coordinate, and -4 is the y-coordinate.
• Similarly, for the point (6, -3), 6 is the x-coordinate, and -3 is the y-coordinate.

Coordinates are essential for plotting and identifying the position of points on a graph. They help in establishing the relationship between different points, such as finding the slope of a line that connects two points. In the case of finding the slope of a line, these coordinates are vital information required to plug into the slope formula.

The Concept of a Vertical Line

A vertical line is a line that moves up and down on a graph rather than side to side. It is distinct because all the x-coordinates of the points on this line are the same. For example, if we consider points with coordinates (6, -4) and (6, -3), all of these points lie on a line where x is always 6.

This unique characteristic leads to the fact that a vertical line does not tilt even slightly to the left or right. It simply goes straight up and down, making it exceptional among lines. Vertical lines are special because they don't have a well-defined slope, which makes them stand out in equations and graphs.

Exploring Undefined Slope

Slope is a measure of steepness of a line, calculated as the ratio of change in y-coordinates to the change in x-coordinates. The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). An undefined slope occurs when the change in x-coordinates (\(x_2 - x_1\)) is zero. This happens with vertical lines because all x-values are identical.

When you try to compute the slope for points like

• (6, -4) and (6, -3), it results in division by zero: \(\frac{1}{0}\), which makes the slope undefined.

This is because the difference in x-coordinates is zero, indicating no horizontal movement between the points. Thus, in mathematical terms, the idea of "undefined" arises because you cannot divide by zero.