Problem 26 - Algebra 1

Question

Plot the points and find the slope of the line passing through the points. $$(-6,-1),(-6,4)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (-6,-1) and (-6,4) is undefined because it is a vertical line.

Step 1: Plot the points

Plot the points \((-6,-1)\) and \((-6,4)\) on a graph. Both of these points fall on the vertical line \(x = -6\).

Step 2: Calculate the change in y-coordinates and the change in x-coordinates

The formula for the slope of a line is \(m=\frac{y2-y1}{x2-x1}\), where \((x1,y1)\) and \((x2,y2)\) are the coordinates of two different points on the line. Before applying the formula, find the change in the y-coordinates, called delta y (\(\Delta y\)), and the change in the x-coordinates, called delta x (\(\Delta x\)). Since the given points are \((-6,-1)\) and \((-6,4)\), \(\Delta y = 4 - (-1) = 5\) and \(\Delta x = -6 - (-6) = 0\).

Step 3: Calculate the slope

Substitute the calculated values \(\Delta y\) and \(\Delta x\) into the formula for the slope. However, since \(\Delta x\) is 0, the slope is undefined. In other words, the slope of a vertical line is undefined, because vertical lines rise infinitely without any horizontal movement.

Key Concepts

Plotting Points

Understanding how to plot points is a fundamental aspect of coordinate geometry, which involves a pair of numerical coordinates used to determine the exact location of a point on a graph. The coordinate system is composed of two axes: the horizontal axis (x-axis) and the vertical axis (y-axis).

Each point is written as an ordered pair \( (x, y) \), where the first number represents the horizontal position, and the second number represents the vertical position relative to the origin \( (0, 0) \). To plot a point, start at the origin, move along the x-axis by the value of the x-coordinate, then move parallel to the y-axis by the value of the y-coordinate.

For instance, to plot the points \( (-6,-1) \) and \( (-6,4) \) from the exercise, you find the x-coordinate value of -6 on the x-axis, then move down to -1 and up to 4 on the y-axis, respectively. This will show both points lie on the same vertical line, indicating a special scenario in slope calculation.

Coordinate Geometry

Coordinate geometry is the study of geometric figures graphed on a coordinate plane. This field of mathematics is vital as it combines algebra with geometry and enables us to solve geometric problems using algebraic equations.

The coordinate plane is divided into four quadrants by the x-axis and y-axis, each with different signs for their coordinates. In the given exercise, the coordinates \( (-6,-1) \) and \( (-6,4) \) are both in the second quadrant, where x-values are negative and y-values are positive.

Coordinate geometry allows us to quantify concepts like distance, midpoint, slope, and more. The slope, or gradient, is a measure of how steep a line is and is crucial for understanding the line's direction and steepness.

Vertical Line Slope

When we talk about the slope of a vertical line in coordinate geometry, we delve into a unique situation. Unlike other lines, the slope of a vertical line is not a number, but rather is described as 'undefined'.

A vertical line has an infinite slope, because while it moves up or down, it does not move left or right at all—the change in the x-value \( (\Delta x) \) is zero. The mathematical formula used to calculate the slope \( m \) is \( m = \frac{\Delta y}{\Delta x} \).

However, in the case of a vertical line, where \( \Delta x = 0 \), you end up with a division by zero, which is undefined in mathematics. This is why the slope of a vertical line is always considered undefined, signifying it's completely vertical with no horizontal component.

Undefined Slope

An undefined slope occurs when we attempt to find the steepness of a vertical line. Since the slope equation involves dividing by the change in the x-values \( (\Delta x) \) of two points, and the x-values of a vertical line do not change, we face a mathematical impossibility—a division by zero.

The concept of an undefined slope in the given exercise is demonstrated with the points \( (-6,-1) \) and \( (-6,4) \) which provide \( \Delta x = 0 \). In everyday terms, if you think of slope as a ratio of rise over run, a vertical line rises without running, leading to an 'infinite' slope, and hence, it's mathematically expressed as undefined.

This distinction is crucial when understanding different types of lines and their slopes. It also helps in solving problems where you need to distinguish between lines that have actual slope values, including zero (horizontal lines), and those that are so steep they cannot be measured with a finite number (vertical lines with undefined slope).