Problem 4 - 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.. $$(-1,3) \text { and }(2,4)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (-1,3) and (2,4) is 1/3, indicating a rising line.

Step 1: Find the difference in 'y' coordinates

In the given points (-1,3) and (2,4) the 'y' coordinates are 3 and 4 respectively. Hence, \( y2 - y1 = 4 - 3 = 1 \)

Step 2: Find the difference in 'x' coordinates

In the given points (-1,3) and (2,4) the 'x' coordinates are -1 and 2 respectively. Hence, \( x2 - x1 = 2 - (-1) = 3 \)

Step 3: Calculate the slope

Using the formula for finding the slope between two points \( m = (y2 - y1)/(x2 - x1) \), we substitute values, we get \( m = 1/3 \)

Step 4: Determine the line's direction

From the slope obtained in step 3, we can infer the direction of the line. As the slope is positive, the line is rising.

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial in interpreting the relationship between two points on a graph. The slope is a measure of how steep a line is, and it is calculated by finding the ratio of the vertical change to the horizontal change between two points.

In mathematics, the slope is typically represented by the letter 'm' and is obtained using the formula: \[ m = \frac{\Delta y}{\Delta x} \] where \( \Delta y \) is the change in the y-coordinates (vertical change) and \( \Delta x \) is the change in the x-coordinates (horizontal change).

For instance, consider the coordinates \((-1,3)\) and \((2,4)\). The slope can be calculated as follows:

• Change in y (\( \Delta y \)): \( 4 - 3 = 1 \)
• Change in x (\( \Delta x \)): \( 2 - (-1) = 3 \)
• Slope (m): \( m = \frac{1}{3} \)

This indicates that for every three units of horizontal movement to the right, there is a one unit of vertical movement upwards. Positive slope, such as \( \frac{1}{3} \), tells us that the line is rising as it moves from left to right.

Algebraic Equations

Algebraic equations form the backbone of finding the slopes of lines. These equations contain variables and constants, and they can be simple or complex depending on the number of terms they have. In the context of finding the slope of a line, the most important algebraic equation is the one for a straight line, also known as the linear equation, which is usually expressed in the form: \[ y = mx + b \]

The 'm' in this equation represents the slope of the line, and 'b' is the y-intercept, the point where the line crosses the y-axis. When we have two points, like \((-1,3)\) and \((2,4)\), we can plug these into the slope formula to create an algebraic equation representing the slope of the line connecting these points. By mastering the use of algebraic equations, students can easily solve for the slope and graph linear relationships.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This usually refers to the Cartesian coordinate system, which is a two-dimensional plane divided into four quadrants by the x-axis (horizontal) and y-axis (vertical).

When working in coordinate geometry, every point on the plane can be represented by a pair of numbers \((x, y)\), which corresponds to its horizontal and vertical positions respectively. The slope of a line in this system is a fundamental concept as it demonstrates how lines relate to one another within the space. For example, the points \((-1,3)\) and \((2,4)\) are situated in the plane and once plotted, the slope calculated tells us how the line between these points will look in this system, thus connecting algebra with geometric visualization.

Undefined Slope

In some cases, a line will have what is known as an 'undefined slope'. This happens when a line is vertical, meaning there is no horizontal change between the two points, no matter how much vertical change there is. Mathematically, it would mean that we're trying to divide by zero when calculating the slope, which is not possible.

A vertical line has an equation in the form of \( x = a \), where 'a' is the x-coordinate that the line crosses. Since the x-value remains constant, the line does not 'rise' or 'fall' as it does with lines that have a defined slope. When students encounter points like \((3,2)\) and \((3,5)\), they should recognize that since the x-coordinates are the same and the y-coordinates differ, the slope is undefined, representing a vertical line.