Problem 92 - Applied Calculus

Question

i. Show that the general linear equation \(a x+b y=c\) with \(b \neq 0\) can be written as \(y=-\frac{a}{b} x+\frac{c}{b}\) which is the equation of a line in slope-intercept form. ii. Show that the general linear equation \(a x+b y=c\) with \(b=0\) but \(a \neq 0\) can be written as \(x=\frac{c}{a},\) which is the equation of a vertical line. [Note: Since these steps are reversible, parts (i) and (ii) together show that the general linear equation \(a x+b y=c \quad\) (for \(a\) and \(b\) not both zero) includes vertical and nonvertical lines.]

Step-by-Step Solution

Answer

i) Rearrange to slope-intercept form: \( y = -\frac{a}{b}x + \frac{c}{b} \). ii) For vertical lines: \( x = \frac{c}{a} \).

Step 1: Rearrange the Linear Equation for Non-Vertical Lines

Start with the general linear equation: \[ a x + b y = c \]Assuming \( b eq 0 \), rearrange this equation to solve for \( y \) in terms of \( x \). First, subtract \( a x \) from both sides:\[ b y = -a x + c \]Then, divide every term by \( b \) to isolate \( y \):\[ y = -\frac{a}{b}x + \frac{c}{b} \]This is the slope-intercept form of a line, where the slope \( m = -\frac{a}{b} \) and the y-intercept \( b = \frac{c}{b} \).

Step 2: Determine the Equation for Vertical Lines

Consider the general linear equation:\[ a x + b y = c \]For a vertical line, \( b = 0 \) and \( a eq 0 \). Substitute \( b = 0 \) into the equation:\[ a x + 0 \cdot y = c \]This simplifies to:\[ a x = c \]Then, solve for \( x \) by dividing both sides by \( a \):\[ x = \frac{c}{a} \]This defines a vertical line, where \( x \) is constant.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a useful way to represent the equation of a line, particularly when you need to quickly identify the slope and y-intercept. The standard form is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept, the point where the line crosses the y-axis.
To derive this form from the general linear equation \( ax + by = c \) when \( b eq 0 \), we follow a simple rearrangement process. First, isolate the term \( by \) by moving \( ax \) to the other side: \( by = -ax + c \).
Next, divide everything by \( b \) to solve for \( y \):

• \( y = -\frac{a}{b}x + \frac{c}{b} \)

This equation is now in slope-intercept form, with the slope \( m = -\frac{a}{b} \) and the intercept \( b = \frac{c}{b} \). Understanding this form makes it easy to graph lines and compare slopes of different lines to determine if they are parallel (equal slopes) or perpendicular (slopes are negative reciprocals).

Vertical Lines

Vertical lines are a unique type of linear graph characterized by having no slope or an undefined slope. In the general form \( ax + by = c \), when \( b = 0 \) and \( a eq 0 \), the line becomes vertical.
Substitute \( b = 0 \) into the equation: \( ax = c \). Solve for \( x \) by dividing both sides by \( a \):

• \( x = \frac{c}{a} \)

This results in a vertical line, meaning regardless of the value of \( y \), \( x \) is constant. These lines run parallel to the y-axis and do not cross it. Since they lack a y-intercept, you can't express them in slope-intercept form.
Vertical lines are significant in geometry and analysis as they indicate regions without horizontal movement. They also divide the plane into two parts, creating distinct visual separations in graphical representations.

General Form of a Line

The general form of a line equation, \( ax + by = c \), is a versatile way to express any line on a Cartesian plane. It includes both vertical and non-vertical lines, provided at least one of \( a \) or \( b \) is not zero.
If \( b eq 0 \), the line is non-vertical and can be converted into the familiar slope-intercept form \( y = mx + b \). This conversion involves isolating \( y \) on one side of the equation.
If \( b = 0 \), we have a vertical line expressed as \( x = \frac{c}{a} \). These transformations illustrate the adaptability of the general form to model any line.
The general form is particularly useful in algebraic operations, such as adding and subtracting equations when solving systems of equations. It is also frequently used in analytical geometry to find intersections and analyze linear multiplicity.
Understanding both the strengths and limitations of each form enhances our ability to work flexibly with lines, whether graphically or algebraically.