Problem 8 - Algebra A Combined Function

Question

Find the slope of the line that passes through the given points. See Examples 1 and 2. $$ (-2,-3) \text { and }(-2,5) $$

Step-by-Step Solution

Answer

The slope is undefined.

Step 1: Identify the Points

The problem provides two points which are (-2, -3) and (-2, 5). These represent two coordinates in the form \((x_1, y_1)\) and \((x_2, y_2)\). Let's label them as:\((x_1, y_1) = (-2, -3)\) and \((x_2, y_2) = (-2, 5)\).

Step 2: Recall the Slope Formula

The slope \(m\) 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 3: Calculate the Change in Y

Find the difference between the y-coordinates:\(y_2 - y_1 = 5 - (-3) = 5 + 3 = 8\).

Step 4: Calculate the Change in X

Find the difference between the x-coordinates:\(x_2 - x_1 = -2 - (-2) = -2 + 2 = 0\).

Step 5: Determine if the Slope is Defined

Substitute the changes in \(y\) and \(x\) into the slope formula:\[m = \frac{8}{0}\]Since division by zero is undefined, the slope of this line is not a number.

Key Concepts

Coordinates

In mathematics, "coordinates" are used to identify the position of a point in a space using ordered pairs. When dealing with a two-dimensional plane, as we are here, coordinates are expressed as \(x, y\). The first number is the \(x\)-coordinate, representing how far along the horizontal axis a point is. The second number is the \(y\)-coordinate, indicating the vertical position.For example, in the exercise, the coordinates given are \((-2, -3)\) and \((-2, 5)\). These two points tell us:

• For \((-2, -3)\): \(x = -2\) and \(y = -3\)
• For \((-2, 5)\): \(x = -2\) and \(y = 5\)

Understanding coordinates is crucial because they help us visualize where a line or point is located. Coordinates form the basic building block for calculating slopes and other geometric properties.

Slope Formula

The slope of a line is essentially its steepness, which tells us how for every unit move in the \(x\) direction, the \(y\) value changes. The slope formula provides a standardized method to calculate this change between two points.The formula to determine the slope \(m\) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This fraction represents the "rise" over the "run," where:

• "Rise" is the change in \(y\)-coordinates, \( y_2 - y_1 \).
• "Run" is the change in \(x\)-coordinates, \( x_2 - x_1 \).

Applying this to our points \((-2, -3)\) and \((-2, 5)\), we calculated the changes:

• Change in \(y\): \(5 - (-3) = 8\)
• Change in \(x\): \(-2 - (-2) = 0\)

The slope formula gives a clear mathematical tool to compute the line's slope, essential for understanding line behavior on graphs.

Division By Zero

Division by zero occurs when a number is divided by zero, a situation that falls outside standard arithmetic rules. In mathematical terms, division by zero is undefined; it doesn't yield a result within the ordinary set of numbers.In the context of the slope formula, if after applying the formula, the denominator (which is the change in \(x\) coordinate or \(x_2 - x_1\)) is zero, we encounter this undefined scenario. Why is this significant? Because dividing by zero tells us:

• The vertical distance (rise) cannot be matched by a horizontal distance (run). This indicates a vertical line.
• It suggests a fundamental difference in direction orientation compared to lines with calculable slopes.

Therefore, encountering division by zero in this context usually means the line is vertical, and its slope cannot be expressed as a number.

Undefined Slope

An undefined slope occurs when a line is perfectly vertical. In our exercise, after we computed the slope using the points, the zero denominator indicated division by zero.A vertical line means:

• The x-coordinates of the points are identical, hence \(x_1 = x_2\).
• There is no horizontal movement, leading to an undefined run.

Let's understand why the slope is undefined. In the slope formula, when you have zero in the denominator (because there's no change in \(x\)), the fraction essentially breaks the rules of division, leading to no definitive number.Recognizing that a slope is undefined is pivotal as it characterizes specific types of lines. In these cases, instead of a numerical slope value, we refer to the line simply as vertical.