Problem 76 - College Algebra

Question

Find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether either line through the points rises, falls, is horizontal, or is vertical. \((a-b, c)\) and \((a, a+c)\)

Step-by-Step Solution

Answer

The slope of the line through the points \((a-b, c)\) and \((a, a+c)\) is 'b'. The line rises.

Step 1: Identify the coordinates of two points

The coordinates of the two points are \((a-b, c)\) and \((a, a+c)\). So, we have: \[ x1 = a-b, \quad y1 = c, \quad x2 = a, \quad y2 = a+c\]

Step 2: Calculate the slope

The slope (m) of a line passing through two points \((x1, y1)\) and \((x2, y2)\) is given as \((y2 - y1)/(x2 - x1)\). Substitute values from Step 1: \(m = (a+c - c)/(a - (a - b)) = b\). So, our slope 'm' is 'b'.

Step 3: Determine the line properties

Since 'b' represents a positive real number, the slope 'm' is also positive. Hence, the line through the points rises.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves studying geometric figures through the use of coordinates. It brings together algebra and geometry, allowing problems to be solved algebraically by using coordinate points.

In the context of the exercise, points on a graph are represented by their x (horizontal) and y (vertical) coordinates. In this case, the points are \( (a-b, c) \) and \( (a, a+c) \). Correct identification of coordinates is the first critical step in coordinate geometry, which lays the groundwork for further calculations like slope, distance, and equation of a line.

Slope Calculation

The slope of a line is a measure of its steepness and can tell us whether a line is rising, falling, horizontal, or vertical. To calculate the slope—denoted as 'm'—we use the formula: \[ m = \frac{y2 - y1}{x2 - x1} \] where \( (x1, y1) \) and \( (x2, y2) \) are the coordinates of two distinct points on the line.

The slope indicates the rate at which y increases or decreases as x increases. A positive slope means the line is rising, a negative slope implies it is falling, zero indicates a horizontal line, and an undefined slope signifies a vertical line.

In our exercise, after plugging in the coordinates, we find the slope to be a positive real number 'b', which confirms that the line is indeed rising as it moves from left to right.

Positive Real Numbers

In mathematics, positive real numbers are all the numbers greater than zero. These numbers can be whole numbers, decimals, or fractions as long as they are not negative.

They play an essential role in determining the characteristics of the slope of a line in coordinate geometry. If a slope is a positive real number, it indicates a line rising from left to right. As in our exercise, since 'b' is given to be a positive real number, this assures us that the line's direction is upward, which helps analysts and students alike to visualize the line's behavior on the graph.