Problem 16 - Thinking Mathematically

Question

Calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, is horizontal, or is vertical. \((3,-4)\) and \((3,5)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points \((3,-4)\) and \((3,5)\) is undefined because the calculation involves division by zero. This means that the line is vertical.

Step 1: Identify the given points

The given points are \((3,-4)\) and \((3,5)\). Assume \((x_1, y_1)\) equal to \((3,-4)\) and \((x_2, y_2)\) equal to \((3,5)\)

Step 2: Substitute the points into the slope formula

When we substitute these values into the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), we get \(m = \frac{5 - (-4)}{3 - 3}\)

Step 3: Simplify the equation

Simplifying the equation in the numerator we have \(5 - (-4) = 9\). But in the denominator we have \(3 - 3 = 0\)

Step 4: Calculate the slope

As division by zero is undefined in mathematics, the slope \(m\) of this equation is undefined

Step 5: Determine the direction of the line

Since the slope is undefined, which means the denominator of our slope equation was zero (i.e., \(x_2 - x_1\) was zero), this means that the line is vertical

Key Concepts

Slope Formula

Understanding the slope of a line is essential in many fields of math and science. The slope is a measure of how steep a line is and can be determined using the slope formula. This formula is simply a ratio of the vertical change (\textdelta y) to the horizontal change (\textdelta x) between two points on a line.

Mathematically, we express this as: \( m = \frac{y_2 - y_1}{x_2 - x_1} \),where \( (x_1, y_1) \) and \( (x_2, y_2) \) are coordinate points on the line, and \( m \) is the slope. Positive values of \( m \) indicate the line rises from left to right, while negative values indicate it falls. A zero slope means the line is horizontal, hence, no vertical change as you move along the line.

Undefined Slope

In certain circumstances, the slope formula produces a situation where we cannot determine a numerical value—this is known as an undefined slope. It occurs whenever the denominator in the slope formula, \( x_2 - x_1 \), equals zero. Since division by zero is not permitted in mathematics, the slope doesn't exist in this scenario. A line with an undefined slope is one that runs perfectly vertical, hence it doesn't have any run (horizontal change), only rise or fall (vertical change).

Vertical Line

A vertical line is one where all points on the line have the same x-coordinate. Because of this constant x-value, the horizontal change (\textdelta x) will always be zero, leading to an undefined slope. Contrary to horizontal lines that have a slope of zero, vertical lines are described by equations in the form of \( x = \text{constant} \). They serve as a powerful visual aid to indicate infinite steepness or an immeasurable rate of change.

Coordinate Points

Coordinate points are the foundation of plotting graphs on the Cartesian plane. Each point is defined by an x (horizontal) and y (vertical) coordinate, written as \( (x, y) \). By knowing the coordinates of two points, you can calculate the slope of the line that passes through them using the slope formula. Additionally, these points help dictate the shape and position of geometric figures on a graph and are crucial in understanding the underlying relationships between algebraic equations and their graphical representations.