Problem 79 - Thinking Mathematically

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. Almost every line in the rectangular coordinate system can be expressed in slope-intercept form, except for vertical lines.

Step 1: Understand the statement

The statement says that every line in the rectangular coordinate system can be expressed in slope-intercept form. This means that any line should be expressed as \(y = mx + c\).

Step 2: Consider the case of vertical lines

For a vertical line, the slope is undefined because the vertical line goes straight up and down, and does not 'rise over run'. Recall that the slope \(m\) is defined as the ratio of the vertical change (rise) to the horizontal change (run). As there is no horizontal change for vertical lines, calculating the slope will result in division by zero, which is undefined.

Step 3: Correction of the false statement

The incorrect part of the statement is 'every', the correct statement should allow for the exception of vertical lines. So, the corrected statement should be 'Almost every line in the rectangular coordinate system has an equation that can be expressed in slope-intercept form, except for vertical lines.'

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line is a popular way to represent linear equations. It is written as \( y = mx + c \) where:

• \( y \) is the dependent variable.
• \( m \) represents the slope of the line, which describes how steep the line is. The slope is found by dividing the change in the vertical direction (rise) by the change in the horizontal direction (run).
• \( x \) is the independent variable.
• \( c \) is the y-intercept, which is the point where the line crosses the y-axis.

This form is especially useful because it directly shows the slope and y-intercept, making it easier to graph the line.
For example, in the equation \( y = 2x + 3 \), the slope \( m \) is 2, and the y-intercept \( c \) is 3. This means that as you move 1 unit along the x-axis, \( y \) will increase by 2 units, and the line crosses the y-axis at \( y = 3 \).
This form works well for almost any line in the rectangular coordinate system except vertical lines.

Vertical Lines

Vertical lines pose a unique situation in coordinate geometry. In a vertical line, all the points have the same x-coordinate, which means that the line does not rise over run. Rather, it just goes straight up and down.
The equation of a vertical line can be expressed as \( x = a \), where \( a \) is the constant x-coordinate of all points on the line.
Since a vertical line has no horizontal change, its slope is considered undefined. This is because you cannot divide by zero (which is the horizontal change or 'run'), as division by zero is not defined in mathematics.

• Since the slope is undefined, vertical lines cannot be expressed in slope-intercept form.
• They never intersect the y-axis, meaning there is no y-intercept.

Understanding vertical lines is crucial because it highlights the limitations of using the slope-intercept form for certain equations.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane with two axes: the horizontal x-axis and the vertical y-axis. This system is foundational for graphing various types of equations.
Each point in this system is denoted by an ordered pair \((x, y)\), where the x-value measures horizontal distance and the y-value measures vertical distance from the origin \((0, 0)\).
In this system:

• Vertical lines run parallel to the y-axis and have equations of the form \( x = a \).
• Horizontal lines run parallel to the x-axis and have equations of the form \( y = b \).
• Other lines can usually be expressed in the slope-intercept form \( y = mx + c \).

The rectangular coordinate system is essential for visualizing the relationships between variables, plotting graphs, and understanding geometric concepts involving lines and curves.