Problem 16 - Elementary Algebra

Question

Find the slope of the line determined by each pair of points. $$(-4,-5),(-4,9)$$

Step-by-Step Solution

Answer

The slope is undefined (vertical line).

Step 1: Identify Coordinates

First, identify the coordinates of the given points. The points are \((-4, -5)\) and \((-4, 9)\). Here, Point 1 is \((-4, -5)\) and Point 2 is \((-4, 9)\).

Step 2: Apply the Slope Formula

The formula to find the slope \(m\) of a line given 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 Coordinates into Slope Formula

Substitute the coordinates of the given points into the slope formula: For Point 1 \((x_1, y_1) = (-4, -5)\) and for Point 2 \((x_2, y_2) = (-4, 9)\), substitute into the formula: \[ m = \frac{9 - (-5)}{-4 - (-4)} \].

Step 4: Perform the Calculations

First, simplify the numerator: \(9 - (-5) = 9 + 5 = 14\). Then the denominator becomes: \(-4 - (-4) = -4 + 4 = 0 \). So, the slope is \( \frac{14}{0} \).

Step 5: Determine the Nature of Slope

Since the denominator of the slope calculation is 0, the slope is undefined. This means the line is vertical.

Key Concepts

Vertical Line

A vertical line is a straight line that runs up and down, parallel to the y-axis on a coordinate plane. All the points on a vertical line have the same x-coordinate, which means the line does not slant left or right, just straight through the top and bottom of the plane.

• Example: If you have two points, \((-4, -5)\) and \((-4, 9)\), they are both on the vertical line \(x = -4\) since their x-coordinates are identical.
• Visual Aspect: Imagine drawing a line through these points - it would look like a straight line that never touches the horizontal (x-axis) except at one fixed point.

Vertical lines are significant in geometry because they help define shapes and directions clearly and are a basis for understanding more complex curve structures.Due to their unique property of having the same x-coordinate for all points, vertical lines have a special slope called an 'undefined slope', which we will discuss next.

Undefined Slope

The undefined slope is a concept you encounter with vertical lines. Slope generally refers to the steepness or tilt of a line and is calculated by the change in y over the change in x between two points.

• Formula: The slope \(m\) is described as \( m = \frac{y_2 - y_1}{x_2 - x_1}\).
• For a vertical line, the change in x, or \(x_2 - x_1\), is zero.

Since you cannot divide by zero mathematically, the slope is 'undefined.' This is different from a horizontal line where the change in y, \(y_2 - y_1\), is zero, resulting in a slope of zero. Here, for the vertical line connecting \((-4, -5)\) and \((-4, 9)\), the denominator becomes zero,making the slope undefined.

Coordinates of Points

When dealing with the slope of a line, the coordinates of points serve as a crucial starting point. They tell us exactly where each point lies on the graph.

• Definition: Each point on a two-dimensional plane is denoted by \((x, y)\), where \(x\) is the horizontal position, and \(y\) is the vertical position.
• Purpose: Knowing the coordinates allows us to find the slope or to understand the line's direction and length.
• Example: In our original problem, we have points \((-4, -5)\) and \((-4, 9)\). Their x-coordinates are identical (\(-4\)), indicating the line connecting them is indeed vertical.

By interpreting the coordinates of the points, we get crucial information needed for determining how a line behaves and are thereby essential in slope calculations.

Slope Formula

The slope formula is essential for determining how a line slants on a graph. It essentially quantifies how a line moves up or down as it moves from left to right across the graph.

• Formula: The formula for the slope \(m\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
• Process: Plug the respective \(x\) and \(y\) coordinates into the formula to find the slope.
• Application: For the points \((-4, -5)\) and \((-4, 9)\), you substitute directly into the slope formula, yielding \( m = \frac{9 - (-5)}{-4 - (-4)} = \frac{14}{0} \).

Here, you see another example of a vertical line since the denominator is zero, resulting in an undefined slope.Understanding the slope formula is essential for graphing, comparing slopes, and solving geometric problems that involve lines on the coordinate plane.