Problem 83 - Brief Applied Calculus

Question

True or False: Every line can be expressed in the form \(a x+b y=c\).

Step-by-Step Solution

Answer

True: Every line can be expressed in the form \(ax+by=c\).

Step 1: Understanding Line Equation Forms

We start by recognizing that a line in a two-dimensional plane can be expressed in various forms. The standard form is typically written as \(ax+by=c\) where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables representing coordinates. This is known as the linear Diophantine equation and is applicable generally to most lines.

Step 2: Considering Vertical Lines

A vertical line has an undefined slope and is expressed by the equation \(x=k\), where \(k\) is a constant. Here, \(b=0\) in the general form \(ax+by=c\), which can still fit into the linear form with \(aeq0\) and \(c=k\). Example: For \(x=2\), the equation can be rearranged to \(1\cdot x+0\cdot y=2\), fitting the general form.

Step 3: Considering Horizontal Lines

A horizontal line has a zero slope and is expressed by the equation \(y=k\), where \(k\) is a constant. Here, \(a=0\) in the general form \(ax+by=c\), which again can fit into the linear form with \(beq0\) and \(c=k\). Example: For \(y=3\), it can be written as \(0\cdot x+1\cdot y=3\), which is indeed in the general form.

Step 4: Special Case Exclusion

There are no special cases of lines that cannot be expressed in this manner. Since both vertical and horizontal lines can be rewritten in the form \(ax+by=c\), along with lines having any other non-zero slopes, all lines fit this form. Thus, the statement that every line can be expressed in the form \(ax+by=c\) is indeed true.

Key Concepts

Standard Form

The standard form of a linear equation is expressed as \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables representing the coordinates on the Cartesian plane. This form is very versatile, allowing for easy rearrangement and manipulation of linear equations.

Here are some reasons why the standard form is useful:

• It provides a straightforward way to identify both the x and y intercepts of a line.
• The coefficients \( a \) and \( b \) provide insight into the line's orientation and steepness without needing further conversion.
• This form is particularly helpful when dealing with systems of linear equations, as it standardizes the equations for consistent comparison and operation.

Understanding the standard form helps make sense of any line's fundamental attributes and simplifies the process of graphing and solving linear equations.

Vertical Lines

Vertical lines on the Cartesian plane are unique because they have an undefined slope. This is due to the fact that all the \( y \)-coordinates are the same for every point on the line, implying vertically infinite rise, which does not work with the concept of slope.

A vertical line can be represented with the equation \( x = k \), where \( k \) is a constant. Transforming this into standard form, we have \( 1 \cdot x + 0 \cdot y = k \).

This fits the structure \( ax + by = c \) with \( a = 1 \), \( b = 0 \), and \( c = k \).

• Ex: For the vertical line \( x = 5 \), it translates to \( 1 \cdot x + 0 \cdot y = 5 \).
• It's important to note that vertical lines cannot be expressed in slope-intercept form since the slope does not exist.

By rearranging a vertical line's equation into standard form, even these unique slopes can be expressed structurally alongside other lines.

Horizontal Lines

Horizontal lines are characterized by a zero slope, meaning they do not rise as they move across the plane. The equation for a horizontal line is \( y = k \), where \( k \) is a constant. This line remains flat, parallel to the \( x \)-axis.

To express this in standard form, it becomes \( 0 \cdot x + 1 \cdot y = k \), aligning with the form \( ax + by = c \), where \( a = 0 \), \( b = 1 \), and \( c = k \).

• Ex: The equation for the horizontal line \( y = 3 \) is represented as \( 0 \cdot x + 1 \cdot y = 3 \).
• This indicates that the horizontal line's position depends purely on \( y \), while \( x \) can be any value.

Horizontal lines are easier to represent in standard form given their clear separation from the vertical axis, making them distinct from lines with non-zero slopes.

Slope

The concept of slope is fundamental in understanding linear equations. It represents how steep a line is on a graph, calculated as the change in \( y \)-coordinates divided by the change in \( x \)-coordinates between two distinct points on the line, often expressed as \( m = \frac{\Delta y}{\Delta x} \).

Slope provides critical information:

• A positive slope means the line rises from left to right.
• A negative slope implies the line falls from left to right.
• A zero slope corresponds to a horizontal line, and an undefined slope is a characteristic of vertical lines.

The slope is invaluable for deducing the behavior of lines, easily represented in non-standard forms like the slope-intercept form \( y = mx + b \), where \( m \) denotes the slope.

By converting between the standard and other forms, such as slope-intercept, one can seamlessly interpret, graph, and manipulate linear equations even when dealing with different kinds of slopes.