Problem 22 - Precalculus with Limits

Question

In Exercises 17-28, find the slope and \(y\)-intercept (if possible) of the equation of the line. Sketch the line. \( 3y + 5 = 0 \)

Step-by-Step Solution

Answer

The slope of the line is 0 and the y-intercept is -5/3. The line is a horizontal line which crosses the y-axis at -5/3.

Step 1: Rearrange to Slope-Intercept Form

Rearrange the equation to get it into slope-intercept form (y = mx + b). Here, subtract 5 from both sides of the equation to isolate 'y'. The equation becomes: 3y = -5. Then, divide each side by 3 to solve for 'y'. This gives: y = -5/3.

Step 2: Identify the Slope and Y-Intercept

Next, identify the slope (m) and the y-intercept (b). Since the equation is now in the form y = mx + b, and there's no 'x' in the given equation, m = 0 (as there's no x-term present) and the y-intercept b = -5/3.

Step 3: Sketch the Line

Using the slope and y-intercept, sketch the line. Since the slope is 0, the line is horizontal. The y-intercept is -5/3, which is where the line crosses the y-axis. So, this is a horizontal line that crosses the y-axis at -5/3.

Key Concepts

Finding Slope and Y-Intercept

Understanding the slope and y-intercept of a linear equation is crucial for analyzing and sketching the line it represents. The slope of a line, denoted as 'm', indicates the steepness and the direction of the line. A positive slope means the line rises as it moves from left to right, while a negative slope indicates that it falls. A slope of zero means the line is horizontal, and an undefined slope means the line is vertical.

The y-intercept is the point where the line crosses the y-axis. It's given by the value 'b' in the slope-intercept form of a line, which is written as \( y = mx + b \). To find the slope and y-intercept for any given line, one must rearrange the equation into this form. For instance, in the equation \( 3y + 5 = 0 \), rearranging gives us \( y = -\frac{5}{3} \), indicating a slope (m) of 0, and a y-intercept (b) of \( -\frac{5}{3} \).

The slope being 0 reveals that the line is horizontal, and the fact that the x-term is missing from the equation further supports this characteristic. On a graph, this y-intercept means our line will pass through the y-axis at \( -\frac{5}{3} \) without rising or falling.

Sketching Linear Equations

Sketching linear equations is a visual representation of the relationship outlined by the equation. To sketch the line represented by an equation in slope-intercept form, begin by plotting the y-intercept on the y-axis. Since the y-intercept is a starting point, that's where you'll draw a dot on the graph. From there, use the slope to determine the direction and steepness of the line.

For example, in a horizontal line with a slope of 0, such as the equation \( y = -\frac{5}{3} \), only the y-intercept needs to be plotted since the slope indicates that the line will be flat, paralleling the x-axis. You won't need to 'rise over run' as you might with a non-zero slope since the line won't rise or fall as it runs. The y-intercept, in this case, \( -\frac{5}{3} \) is where the line touches the y-axis. Simply draw a straight line through this point that extends infinitely in both directions along the y-value of \( -\frac{5}{3} \) to complete your sketch.

Horizontal Line Equation

The equation of a horizontal line is a special type of linear equation that can be quickly identified by its standard form. For any horizontal line, the slope (m) will always be 0 because there is no vertical change as the line moves horizontally. This leads to the general form of the equation \( y = b \), where 'b' represents the y-intercept.

In the given exercise, after adjusting the given equation \( 3y + 5 = 0 \) to slope-intercept form, we found that \( y = -\frac{5}{3} \). Here, since there is no 'x' term present, it tells us immediately that we have a horizontal line. Such lines run parallel to the x-axis and cross the y-axis at the given y-intercept, in this case, \( -\frac{5}{3} \). Keep in mind that the x-coordinate for any point on a horizontal line can be any real number, while the y-coordinate will always be the constant value of the y-intercept.