Problem 44 - College Algebra and Calculus: An Applied Approach

Question

Find the slope and \(y\) -intercept (if possible) of the line specified by the equation. Then sketch the line. \(2 x+5=0\)

Step-by-Step Solution

Answer

The given line \(2x + 5 = 0\) has an undefined slope, and does not have a y-intercept. The line can be sketched as a vertical line passing through the point -5/2 on the x-axis.

Step 1: Identify the Type of Line

This equation \(2x + 5 = 0\) represents a vertical line x = -5/2. Vertical lines have an undefined slope as they do not 'rise over run'.

Step 2: Slope of the Line

A vertical line like the one represented by the given equation does not have a slope, as the concept of rise over run doesn't apply to vertical lines. Hence, the slope is undefined.

Step 3: y-Intercept of the Line

A vertical line does not intersect the y-axis except when the line is the y-axis itself, so for this equation, there is no y-intercept.

Step 4: Sketching the Line

To sketch this line, draw a straight, vertical line that passes through the point -5/2 on the x-axis. The line doesn't intersect with the y-axis as expected.

Key Concepts

Understanding Slope

When we talk about the slope of a line in mathematics, we refer to its steepness or inclination. The slope is calculated as the 'rise' (change in the y-coordinates) divided by the 'run' (change in the x-coordinates). This can be represented by the formula: \[ m = \frac{\Delta y}{\Delta x} \] where \( m \) is the slope. However, it's important to note that vertical lines, such as the one in our original exercise equation \( 2x + 5 = 0 \), have an undefined slope. Why? Because vertical lines do not 'run'; they don't extend horizontally.

• For horizontal lines, the slope is 0.
• For vertical lines, the slope is undefined because they stretch infinitely upwards or downwards without any horizontal movement.

Understanding the concept of slope is crucial, especially when distinguishing between different types of lines such as horizontal and vertical.

Grasping the Y-Intercept

The y-intercept of a line is the specific point where it crosses the y-axis. This is often denoted as \( (0, b) \) in the linear equation \( y = mx + b \). At this intersection, the value of \( x \) is always zero, which makes it easy to spot on a graph. However, vertical lines such as the one derived from the equation \( 2x + 5 = 0 \) do not have a y-intercept. This is because they aren't crossing the y-axis—rather, they run parallel (or coincidentally with) it. Of course, there's an exception: when the line itself *is* the y-axis, in which case the whole line could be considered as crossing every point on the y-axis. Knowing if a line has a y-intercept is very helpful for graphing and understanding its behavior in the Cartesian coordinate system.

Equation of a Line Basics

The equation of a line, usually written as \( y = mx + b \), is a fundamental aspect of algebra. Here, \( m \) represents the slope, and \( b \) the y-intercept. This form is known as the slope-intercept form and is especially useful for quickly graphing a line. For vertical lines—like the line described by the equation \( 2x + 5 = 0 \)—we use a different approach. This equation can simply be rearranged to \( x = -\frac{5}{2} \), indicating that every point on that line has \( x = -\frac{5}{2} \). This is known as the standard form for a vertical line, and it helps to highlight that such lines do not follow the familiar \( y = mx + b \) format.

• Horizontal lines can be written as \( y = c \).
• Vertical lines are represented as \( x = d \), showing that their nature diverges from the standard slope-intercept form.

Understanding these differences in line equations helps in identifying and graphing them correctly.