Problem 84 - Applied Calculus

Question

True or False: Every line has a slope.

Step-by-Step Solution

Answer

False. Vertical lines do not have a defined slope.

Step 1: Understand the Definition of a Line

A line in a two-dimensional plane can be described by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This indicates that lines that pass through the x-axis and y-axis in some manner typically have a defined slope.

Step 2: Consider Vertical Lines

Vertical lines are represented by an equation of the form \(x = a\), where \(a\) is a constant. These lines are parallel to the y-axis. For vertical lines, the concept of slope does not apply in the traditional sense, as the slope would require division by zero.

Step 3: Identify Slopes in Other Lines

For all other lines, including horizontal lines and other sloped lines, the slope can be calculated as the change in y over the change in x, and it's not indeterminate. This confirms that these lines do indeed have a slope.

Step 4: Conclusion on Line Slopes

Since vertical lines do not have a defined slope, the statement "every line has a slope" is not true for vertical lines. Thus, the statement "Every line has a slope" is false because vertical lines are an exception.

Key Concepts

Vertical Lines

A vertical line is a unique type of line in geometry which runs straight up and down in a graph. This means it is perfectly parallel to the y-axis. You can represent vertical lines with an equation of the form \(x = a\), where \(a\) is a constant value. This indicates that for any point on the line, the x-coordinate remains consistent.

When it comes to slopes, vertical lines are special. Generally, slope is defined as the "rise over run," or the change in y over the change in x. However, for vertical lines, there is no horizontal change (since all points have the same x-coordinate), making the denominator zero in the slope calculation. This results in division by zero, which is undefined. Therefore, vertical lines are considered to have an undefined slope, making them an important exception in geometry and algebra.

Equation of a Line

The equation of a line is a fundamental tool in algebra and geometry used to describe a straight line in the coordinate plane. One common form of a line equation is the slope-intercept form, expressed as \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept.

• The slope \(m\) indicates how steep the line is and the direction it moves. A positive slope means the line ascends from left to right, while a negative slope means it descends.
• The y-intercept \(b\) is the point where the line crosses the y-axis. It shows the value of y when \(x\) is zero.

This form is particularly useful because it gives clear information about how to graph the line and understand its orientation. Other forms of line equations include the standard form \(Ax + By = C\), and point-slope form \(y - y_1 = m(x - x_1)\), where different information such as a specific point on the line may be more highlighted.

Horizontal Lines

Horizontal lines are a straightforward concept in the study of graphs and functions. These lines run parallel to the x-axis, extending left and right accordingly. The equation of a horizontal line has the form \(y = b\), where \(b\) is a constant.

For these lines, the y-coordinate remains the same for all points along the line, meaning there is no vertical change. As a result, when calculating the slope, the change in y over the change in x equals 0 over any number, which reliably results in a slope of 0. This defines horizontal lines uniquely as having a slope of zero, distinguishing them from vertical lines which have an undefined slope.