Problem 85 - 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 the General Form of a Line

The equation given, \(ax + by = c\), is known as the general linear form or standard form of a line in a two-dimensional Cartesian coordinate system. \(a\), \(b\), and \(c\) are constants. This form can represent any line that is not vertical.

Step 2: Considering Vertical Lines

Vertical lines have undefined slopes and cannot be expressed in terms of \(y\). They are described by equations of the form \(x = k\), where \(k\) is a constant. In this case, the coefficient of \(y\) would be zero (\(b = 0\)), resulting in an equation \(ax = c\), which can also be written as \(ax + 0\cdot y = c\).

Step 3: Expressing Different Types of Lines

Lines with a slope of zero are horizontal and are described by equations of the form \(y = k\). This can be transformed into \(0x + y = k\), fitting the given form with \(a = 0\). This demonstrates that both horizontal and non-horizontal lines can be expressed as \(ax + by = c\).

Step 4: Conclusion

Since both non-vertical and vertical lines (\(by = 0\) or \(ax = c\)) fit the form \(ax + by = c\), we can conclude that any line in the plane can be expressed this way. Therefore, the statement is True.

Key Concepts

Vertical Lines

Vertical lines in a coordinate system have a unique characteristic: their slope is undefined. The equation that defines a vertical line takes the form \(x = k\), where \(k\) is a constant value. This tells us that for every point on the line, the x-coordinate remains the same. Even though vertical lines cannot be expressed traditionally with the slope-intercept form, \(y = mx + b\), they fit perfectly into the generalized equation structure. To fit the general form \(ax + by = c\), we set \(b = 0\). Therefore, the equation for a vertical line can be rearranged into \(ax = c\), simplifying to \(ax + 0y = c\). This shows that vertical lines are an exception only when considering slope, but they still conform to the general linear form equation.

Horizontal Lines

Horizontal lines are quite straightforward, as they have no vertical change, meaning their slope is zero. The defining equation for a horizontal line is \(y = k\), where \(k\) represents a constant. Every point along a horizontal line shares the same y-coordinate. These lines can easily be written in the form \(ax + by = c\) by recognizing that the slope \(m\) is zero. For horizontal lines, we set \(a = 0\) in the general equation. This converts the equation into \(0x + y = k\). So, by adjusting the parameters of the general form, horizontal lines are perfectly expressed without any complexities.

Linear Equations

Linear equations describe straight lines and can take many forms, including the general form \(ax + by = c\). In this form, \(a\), \(b\), and \(c\) are constants that determine the properties of the line, such as its slope and intercepts. This form is versatile and comprehensive, allowing it to represent all types of lines on a 2D plane, except those that are not perfectly straight, which do not exist in linear terms. The general form is especially useful for simplifying algebraic expressions and solving systems of equations.
One of the great advantages of the general form is that it can neatly encompass both horizontal and vertical lines by adjusting the coefficients \(a\) and \(b\). In simple terms, this form unifies the representation of lines, offering a standardized way to express any line, making problem-solving in coordinate geometry more systematic.