Problem 39 - Calculus Early Transcendentals

Question

Find the slope of each line and a point on the line. Then graph the line. $$x=8+2 t, y=1$$

Step-by-Step Solution

Answer

Answer: The slope of the line is 0, and the line passes through the point (8, 1).

Step 1: 1. Rewrite the parametric equations as a single equation in terms of x and y

First, isolate the parameter t in one of the equations, and then substitute the expression for t into the other equation. In this case, isolating t in the x-equation:$$t = \frac{x-8}{2}$$Next, substitute this expression for t in the y-equation:$$y = 1$$

Step 2: 2. Determine the slope

The equation we derived in the previous step is $$y=1$$, which is the equation of a horizontal line. For horizontal lines, the slope is always 0, so the slope of this line is $$m=0$$.

Step 3: 3. Choose a point on the line

Since the equation is $$y=1$$, we can choose any point that lies on this line. One of the easiest points to choose is the point where x=8, which comes from the provided equation $$x=8+2t$$ when t=0. So, the point on the line is (8, 1).

Step 4: 4. Graph the line

To graph a horizontal line with a slope of 0 and passing through the point (8, 1), simply draw a straight line parallel to the x-axis and passing through the point (8, 1). The line will consist of all points with a y-coordinate of 1.

Key Concepts

Slope of a Line

The slope of a line is a fundamental concept in geometry and algebra, describing how steep a line is. It is represented by the letter \( m \) and calculated as the change in the y-coordinate divided by the change in the x-coordinate between two points on a line: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This equation is easy to remember and helps us determine whether a line is rising or falling.

A positive slope indicates a line rising from left to right, while a negative slope shows a line falling. A slope of 0 means the line is perfectly horizontal. The greater the absolute value of the slope, the steeper the line.

A vertical line has an undefined slope because the change in x is zero, causing division by zero. Understanding these variations of slope helps when dealing with different types of lines, making slope an essential part of graphing and analyzing linear equations.

Graphing Lines

Graphing lines involves plotting points and drawing straight lines through them. This visual process helps us better understand mathematical relationships and equations. We can use the slope-intercept form of a linear equation to graph lines quickly. This equation looks like \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, the point where the line crosses the y-axis.

Here is how to graph a line:

• Identify the y-intercept \( b \) and plot this point on the y-axis.
• Use the slope \( m \) as "rise over run" to find more points. For example, a slope of 2 means you rise 2 units for every 1 unit run to the right.
• Draw the line through these points, extending it in both directions.

This method conveniently applies to both positive and negative slopes. Remember to double-check your line includes the y-intercept and follows the correct steepness based on the slope.

Horizontal Lines

Horizontal lines are straightforward to understand and graph. They have a constant y-coordinate for every x-coordinate, resulting in an equation of the form \( y = b \). Since the y-coordinates don't change, the slope of a horizontal line is always 0. This makes them unique among linear equations.

To graph a horizontal line:

• Identify the y-coordinate; it remains constant across the line.
• Draw a straight line parallel to the x-axis at this y-coordinate level.
• Remember that the line extends infinitely in both horizontal directions.

Horizontal lines are commonly found in various scenarios, like the surface of a calm lake or the horizon in a landscape. They're defined only by their y-value, making them simple but important in understanding the full range of linear equations.