Problem 5 - Introductory Algebra for College Students

Question

In Exercises \(1-10,\) find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. $$(4,-2) \text { and }(3,-2)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (4,-2) and (3,-2) is 0. This means the line is horizontal.

Step 1 - Identify the points

Identify the two points that the line passes through. They are \((4,-2)\) and \((3,-2)\). The x-coordinate of the first point (x1) is 4 and the y-coordinate of the first point (y1) is -2. Similarly, the x-coordinate of the second point (x2) is 3 and the y-coordinate of the second point (y2) is -2.

Step 2 - Apply the Slope Formula

Apply the slope formula \(m=(y2-y1)/(x2-x1)\). Substituting the given points into the formula gives \(m=(-2--2)/(3-4)\).

Step 3 - Calculate the Slope

Carrying out the operations in the equation will yield the slope of the line. In this case, \(m=0/(-1)=0\).

Step 4 - Interpret the Slope

Since the slope is equal to zero, this indicates that the line is horizontal. Hence, the line neither rises nor falls, and it is not vertical.

Key Concepts

Slope Formula

Understanding how to find the slope of a line is vital in mathematics, especially when dealing with linear equations. The slope, represented by the letter m, describes how steep a line is.

The slope formula is \(m = \frac{{y2 - y1}}{{x2 - x1}}\), where \((x1, y1)\) and \((x2, y2)\) are the coordinates of two distinct points on a line. You subtract the y-coordinates of the points and divide that by the subtraction of the x-coordinates of the same points. This ratio represents the rate at which the line rises or falls as you move from left to right on the coordinate plane.

Horizontal Line

A horizontal line is a straight line that moves from left to right and has a constant y-value for all points along its length.

When using the slope formula for a horizontal line, we notice that the change in the y-coordinates is zero (since y1 equals y2), making the slope zero. This indicates that the line is perfectly flat, and there is no rise or fall as you move along the line. In the context of a graph, if you were to walk along a horizontal line, you wouldn't be climbing uphill or descending downhill—the elevation remains the same.

Undefined Slope

In contrast to horizontal or sloped lines, vertical lines feature a unique characteristic: an undefined slope.

The reason behind this is quite intuitive. In the slope formula, when a line is vertical, the x-coordinates of any two points on the line will be the same (because x1 equals x2). This results in a division by zero when calculating the slope, which is mathematically undefined. Remember, any time you're asked for the slope of a vertical line, the correct answer is 'undefined' rather than zero. It's like trying to climb an infinitely high wall with no horizontal movement — there's no slope to measure.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is the two-dimensional grid where we plot points, lines, and curves. It consists of two number lines: the horizontal x-axis and the vertical y-axis.

When we discuss slopes, we're essentially discussing the geometry on this plane. Being able to navigate the coordinate plane is essential in understanding how the slope formula works because it requires precise identification of points and their x and y values. Every point on this plane is defined by a pair of coordinates, which are used to determine the slope, direction, and position of the lines drawn on the plane.