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. \((1,1)\) and \((2,2)\)

Step-by-Step Solution

Answer

The slope of the line is 1.

Step 1: Calculate the Change in Y

To find the slope of the line passing through the points \((1,1)\) and \((2,2)\), start by calculating the difference in the y-values of the two points. This is done by subtracting the y-coordinate of the first point from the y-coordinate of the second point, resulting in the change in y: \( \Delta y = y_2 - y_1 = 2 - 1 = 1 \).

Step 2: Calculate the Change in X

Next, calculate the difference in the x-values of the two points. Subtract the x-coordinate of the first point from the x-coordinate of the second point to find the change in x: \( \Delta x = x_2 - x_1 = 2 - 1 = 1 \).

Step 3: Divide the Changes to Find the Slope

The slope of a line is given by the formula \( m = \frac{\Delta y}{\Delta x} \). Substitute the changes you calculated in the previous steps: \( m = \frac{1}{1} = 1 \). This means the slope of the line is 1.

Key Concepts

Change in Y

When you want to find the slope of a line, one critical step is calculating the 'change in y'. This refers to the difference between the y-coordinates of two points on the line. Let's break it down:

1. Identify the y-coordinates of the two given points. In our example, the points are (1, 1) and (2, 2). The y-coordinates are 1 and 2.
2. Subtract the y-coordinate of the first point from the y-coordinate of the second point. This calculation gives you the change in y.
3. Formula: \[ \Delta y = y_2 - y_1 \]
In our case: \( 2 - 1 = 1 \).

By understanding this simple subtraction, you can determine how much the y-values rise or fall as you move from one point to another along the line.

Change in X

Once you have calculated the 'change in y', the next step is to find the 'change in x'. This is done similarly by calculating the difference in the x-coordinates of the two points.

1. Identify the x-coordinates of the two given points, which in our example are 1 and 2.
2. Subtract the x-coordinate of the first point from the x-coordinate of the second point.
3. Formula:\[\Delta x = x_2 - x_1\]
In our case: \( 2 - 1 = 1 \).

This process tells us how much the x-values move horizontally as we move from one point to another on the line.

Linear Equation

In coordinate geometry, a 'linear equation' represents a straight line. The slope, a key aspect of this equation, indicates the steepness of the line.

The general form of a linear equation is:\[y = mx + b\]
Here:

• \( m \) is the slope, showing how much y changes with a one-unit change in x.
• \( b \) is the y-intercept, where the line crosses the y-axis.

For our problem, we found that the slope \( (m) = 1 \). This shows the line rises one unit vertically for every one unit it moves horizontally. Understanding the linear equation helps in visualizing and graphing linear relationships.

Coordinate Geometry

Coordinate geometry involves studying geometric figures using a coordinate system. It's the bridge connecting algebraic equations and geometric figures. Points are plotted using coordinates in the form \((x, y)\).

Key components include:

• Coordinates: Show exact position on the plane.
• Slope: Measures the angle or steepness of a line.
• Distance: The length between two points.

In our exercise, we focused on calculating the slope using coordinates. By doing so, we understand the direction and tilt of the line that connects two points. This knowledge is fundamental in further exploring graphs and equations in coordinate geometry.