Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(23-28\), find the slope of the line containing the given two points. (1,1) \text { and }(2,2)

Step-by-Step Solution

Answer

The slope of the line is 1.

Step 1: Understand the formula for slope

The slope of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]where \(m\) represents the slope of the line.

Step 2: Identify the coordinates

Identify the two points given in the problem, \((1, 1)\) and \((2, 2)\). Here, \(x_1 = 1\), \(y_1 = 1\), \(x_2 = 2\), and \(y_2 = 2\).

Step 3: Substitute the coordinates into the formula

Substitute the coordinates into the formula:\[m = \frac{2 - 1}{2 - 1}\]Simplify the differences in the numerator and denominator.

Step 4: Simplify the expression

Simplify the expression to find the slope:\[m = \frac{1}{1} = 1\]Thus, the slope of the line is 1.

Key Concepts

Understanding the Two-Point Formula

The two-point formula is a fundamental concept in geometry that helps us determine the slope of a line. It's a simple method used when you have two points on a line:

• The first point is \((x_1, y_1)\),
• The second is \((x_2, y_2)\).

The formula to find the slope \(m\) is: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here’s how it works: You take the difference between the y-coordinates and divide it by the difference between the x-coordinates. This process effectively measures the "steepness" of the line. Remember, the slope tells us how much y increases or decreases as x increases. A positive slope means the line rises as it moves left to right, while a negative slope indicates it falls.

Exploring Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a powerful branch of mathematics that connects algebra and geometry. In coordinate geometry, every point is represented by an ordered pair \((x, y)\). These coordinates tell us where the point lies on a two-dimensional plane. Coordinate geometry allows us to:

• Plot points using their coordinates.
• Find distances between points.
• Understand the properties of shapes and lines.

By using coordinate geometry, you can represent geometric shapes algebraically and make calculations quickly and efficiently. In the context of this exercise involving slopes, coordinate geometry helps us easily determine the relationship between two points on a line.

Decoding Linear Equations

Linear equations are algebraic expressions that form straight lines when plotted on a graph. The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, or the point where the line crosses the y-axis. Each component of a linear equation serves a purpose:

• The slope \(m\) describes the line's direction and steepness.
• The y-intercept \(b\) tells you the starting value of y when \(x = 0\).

Linear equations are essential in understanding the behavior of lines in the coordinate plane. Knowing the slope through the two-point formula allows us to create these equations and predict how changes in one variable affect another. It's a crucial tool for modeling real-world scenarios in various fields, from physics to economics.