Problem 75 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(74-77\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Every line in the rectangular coordinate system has an equation that can be expressed in slope-intercept form.

Step-by-Step Solution

Answer

The statement is false. It can be changed to 'Every line in the rectangular coordinate system, except vertical lines, has an equation that can be expressed in slope-intercept form' to become true.

Step 1: Understanding slope-intercept form

The equation of a line in slope-intercept form is \(y = mx + b\) where \(m\) is the slope of the line and \(b\) is the y-intercept of the line. It expresses how y changes as x changes.

Step 2: Understanding exceptions

However, not every line in a rectangular coordinate system can be expressed in slope-intercept form. The limitation comes with vertical lines, as they do not have a slope, and hence cannot be presented using the slope-intercept form.

Step 3: Correcting the statement

To make the statement true, it can be revised as: 'Every line in the rectangular coordinate system, except vertical lines, has an equation that can be expressed in slope-intercept form'.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a foundational element in algebra and geometry. It consists of two perpendicular axes: the horizontal axis (usually labeled as the x-axis) and the vertical axis (labeled as the y-axis). The point where these two axes intersect is called the origin, typically denoted as (0, 0).

Points on this plane are defined by ordered pairs of numbers \( (x, y) \) known as coordinates. The x-coordinate indicates the position along the horizontal axis, while the y-coordinate reveals the position along the vertical axis. By plotting these points, you can represent various mathematical objects, such as lines and curves.

Plotting Lines in the System

When it comes to lines, the rectangular coordinate system allows us to graph them and analyze their properties, such as direction, steepness, and position relative to the origin. The equation of a line in this system is useful for predicting values and solving real-world problems. It's crucial for students to master plotting points and understanding the system's layout, as these skills are widely applicable across different areas of mathematics and sciences.

Equation of a Line

The equation of a line encapsulates the relationship between the x and y coordinates of all the points that make up the line. One common form of a linear equation is the slope-intercept form, given by \( y = mx + b \), where \( m \) is the slope, representing how much y increases for a one-unit increase in x, and \( b \) is the y-intercept, showing where the line crosses the y-axis.

The slope \( m \) is a crucial element because it defines the steepness and direction of the line. A positive slope means the line is inclined upwards, while a negative slope indicates a downward incline. The y-intercept \( b \) provides an essential starting point for the line and is especially helpful when graphing.

Slope as a Rate of Change

Seeing the slope as a rate of change furthers the understanding of linear relationships in various contexts, such as physics (speed), economics (cost changes), and biology (growth rates). The beauty of the slope-intercept form lies in its directness and ease of use for graphing and interpreting the characteristics of lines.

Vertical Lines

Vertical lines in the rectangular coordinate system present an interesting exception to the general rules for equations of lines. These lines go straight up and down and have an undefined slope. Since the slope-intercept form \( y = mx + b \) requires a definable slope \( m \), vertical lines cannot be expressed in this form.

Instead, vertical lines have equations of the form \( x = k \), where \( k \) is the x-coordinate through which the line passes. This simplicity means that for any value of y, the x-coordinate remains the same, confirming the line's vertical nature. When discussing or teaching equations of lines, it's essential to note this distinction to avoid misconceptions that could lead to difficulties in more complex calculations.

Understanding Undefined Slope

Recognizing that the concept of an undefined slope results from the inability to divide by zero (as vertical lines would imply a change in y without any change in x) is a fundamental element of mathematics education. This emphasizes the importance of vertical lines as an educational topic, ensuring students don't apply slope-intercept form where it doesn't belong and develop a well-rounded understanding of different types of linear equations.