Problem 75 - Introductory Algebra for College Students

Question

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. A more accurate statement would be:' All non-vertical lines in the rectangular coordinate system have an equation that can be expressed in slope-intercept form.'

Step 1: Understand the concept of slope-intercept form

Firstly, it is important to understand the concept of the slope-intercept form of a line. The slope-intercept form is expressed as \(y = mx + c\), where \(m\) is the slope, \(x\) is the x-coordinate, \(c\) is the y-intercept, and \(y\) is the y-coordinate.

Step 2: Evaluate the case for non-vertical lines

For every non-vertical line in the rectangular coordinate system, we are able to calculate a specific slope and y-intercept. Therefore, any non-vertical line can be expressed in the slope-intercept form.

Step 3: Evaluate the case for vertical lines

A vertical line, on the other hand, does not have a defined slope, since it goes straight up and down, and there is no delta of \(x\) for a delta of \(y\). Thus, the slope (which could be considered as infinite) doesn't fit into the real numbers. So, a vertical line's equation is usually written as \(x = a\), where a is the x-intercept, which doesn't conform to the slope-intercept form.

Step 4: Develop the final statement

Having evaluated all possible cases, we determine that the original statement is false. A correct statement would be 'All non-vertical lines in the rectangular coordinate system have 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 x-axis and the vertical y-axis. Together, they form a grid where every point can be specified by a pair of numerical coordinates \( (x, y) \) representing its horizontal and vertical positions.

This system is essential for graphing equations and analyzing geometric shapes. It allows us to easily illustrate the relationship between numbers and provides a visual representation of equations. When plotting a line on this grid, we use the equation of the line to determine which points lie on it, thus bringing us to the concept of the equation of a line in its various forms.

Equation of a Line

The equation of a line is a way to express the relationship between the x and y coordinates of all points that lie on the line. One of the most common forms of this equation is the slope-intercept form \( y = mx + c \). Here \( m \) represents the slope, which indicates the steepness and direction of the line, while \( c \) is the y-intercept, the point where the line crosses the y-axis.

By using this formulation, we can identify key characteristics of the line, such as its slope and how it behaves in relation to the axes. This form is particularly user-friendly because it allows you to quickly graph a line by identifying its y-intercept and using the slope to find other points on the line. It's important to note, however, that not every line can be represented this way, especially vertical lines, which we will explore further.

Vertical Line Equation

Vertical lines are unique in the coordinate system because they run parallel to the y-axis and thus have an undefined or infinite slope. Unlike other lines, the vertical line's equation cannot be expressed in the slope-intercept form since there is no variation in y for any change in x; the value of x remains constant for all points on the vertical line.

Therefore, the equation for a vertical line is given as \( x = a \), where \( a \) is the x-coordinate at which the line intersects the x-axis. This simplicity allows us to understand that for every point on this line, no matter how far up or down it extends, the x-coordinate will always be the same. Recognizing the limitations of the slope-intercept form in relation to vertical lines helps in grasping the full scope of line equations within the coordinate system.