Problem 62 - Calculus Volume 1

Question

For each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. (6,4) and (4,-3)

Step-by-Step Solution

Answer

The slope is 7/2, and the line is increasing.

Step 1: Determine the change in y-values

For the points \((6, 4)\) and \((4, -3)\), find the change in the y-values by subtracting the second y-coordinate from the first y-coordinate: \(4 - (-3) = 4 + 3 = 7\). So, the change in y-values (\(\Delta y\)) is 7.

Step 2: Determine the change in x-values

Next, calculate the change in x-values by subtracting the second x-coordinate from the first x-coordinate: \(6 - 4 = 2\). Thus, the change in x-values (\(\Delta x\)) is 2.

Step 3: Calculate the slope of the line

The slope \(m\) of the line is given by the formula: \(m = \frac{\Delta y}{\Delta x}\). Using the values from the previous steps, \(m = \frac{7}{2}\). This means the slope of the line is 7/2.

Step 4: Determine the type of line

Since the slope \(7/2\) is positive, the line passing through points \((6,4)\) and \((4,-3)\) is increasing. A line has a positive slope when it rises as it moves from left to right.

Key Concepts

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. These equations represent straight lines when plotted on a graph. The standard form of a linear equation in two variables is expressed as:

• General form: \(Ax + By = C\)
• Slope-intercept form: \(y = mx + b\)
• Point-slope form: \(y - y_1 = m(x - x_1)\)

Here, \(m\) represents the slope, \((x_1, y_1)\) is a point on the line, and \(b\) is the y-intercept where the line crosses the y-axis. Understanding these forms will allow you to construct and manipulate linear equations effectively, which is key to solving problems related to the slope of a line.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. Points in this system are defined by ordered pairs or triples (like \((x, y)\) or \((x, y, z)\)) that correspond to positions on a plane or in space. This method is crucial for understanding the positional relationships between geometric figures.

• The Cartesian coordinate system divides a plane into four quadrants using an x-axis (horizontal) and a y-axis (vertical).
• Each point on the plane is represented by a coordinate, which is essentially its address on the graph.

Coordinate geometry bridges the gap between algebraic equations and geometric figures by allowing equations to represent shapes. This connection makes it possible to derive important information, such as the slope of a line or distances between points, with precision.
Consider the points \((6, 4)\) and \((4, -3)\) in the problem above. Using their coordinates, we can calculate changes in \(x\) and \(y\) values, and thus determine the slope of the line through these points.

Line Characteristics

The characteristics of a line are primarily defined by its slope and direction. Lines can be classified based on the value of their slopes:

• Positive slope: The line rises as it moves from left to right, indicating an increasing relationship.
• Negative slope: The line descends as you move from left to right, showing a decreasing trend.
• Zero slope: If a line is horizontal, it has a slope of zero, indicating no vertical change as \(x\) increases.
• Undefined slope: A vertical line has an undefined slope because the change in \(x\) is zero, causing division by zero in the slope formula.

In the case of the points \((6, 4)\) and \((4, -3)\), the slope is calculated to be \(\frac{7}{2}\), which is positive. This means the line is increasing. Recognizing these characteristics helps in interpreting data and understanding the relationship between variables.