Problem 88 - Brief 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 \(a x+b y=c\) (for \(a\) and \(b\) not both zero) includes vertical and nonvertical lines.]

Step-by-Step Solution

Answer

The equation \( a x + b y = c \) can be in slope-intercept form if \( b \neq 0 \) or a vertical line if \( b = 0 \).

Step 1: Start with the given equation for part (i)

For part (i), we start with the general linear equation: \[ a x + b y = c \] where \( b eq 0 \).

Step 2: Solve for y for part (i)

To express \( y \) in terms of \( x \), first isolate the term involving \( y \):\[ b y = -a x + c \] Now, divide every term by \( b \) to solve for \( y \):\[ y = -\frac{a}{b} x + \frac{c}{b} \] This is the slope-intercept form of a line: \( y = m x + b \), where \( m = -\frac{a}{b} \) is the slope and \( \frac{c}{b} \) is the y-intercept.

Step 3: Start with the given equation for part (ii)

For part (ii), start with the same initial equation:\[ a x + b y = c \] with \( b = 0 \) and \( a eq 0 \).

Step 4: Simplify equation for part (ii)

With \( b = 0 \), the equation reduces to:\[ a x = c \] Solve for \( x \) by dividing both sides by \( a \):\[ x = \frac{c}{a} \] This represents a vertical line at \( x = \frac{c}{a} \), independent of \( y \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a special way of writing the equation of a line so that you can easily see the slope and the y-intercept. This form is written as \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) stands for the y-intercept. Knowing how to express a line in this form makes it simpler to graph or analyze.

To transform a general linear equation \( ax + by = c \) into the slope-intercept form, especially when \( b eq 0 \), follow these steps:

• Isolate the \( y \) term by manipulating the equation to: \( by = -ax + c \).
• Then divide every term by \( b \) to solve for \( y \). This gives: \( y = -\frac{a}{b}x + \frac{c}{b} \).

In this final equation, \( -\frac{a}{b} \) is the slope, showing how steep the line is, and \( \frac{c}{b} \) indicates where the line crosses the y-axis. It's a versatile format used in many situations, from graphing to understanding line relationships.

Vertical Line Equation

A vertical line behaves differently in terms of its equation. It is represented by an equation where \( b = 0 \) in the general form \( ax + by = c \). This simplifies to \( ax = c \) given that \( a eq 0 \). When you solve for \( x \), the equation becomes \( x = \frac{c}{a} \). This represents the fact that every point on the line has the same x-coordinate.

Features of a vertical line include:

• It runs parallel to the y-axis.
• It does not have a y-intercept because it does not touch the y-axis at any point.
• Its slope is undefined, since the rise is along the y-axis without any horizontal change.

Understanding vertical lines is important, especially because they serve as boundaries or reference lines in various applications. They signify a constant x value over the range of y values.

General Linear Equation

The general linear equation, \( ax + by = c \), is a foundational concept in understanding lines in algebra. It encompasses all types of straight lines, whether they slope upwards, slope downwards, are perfectly horizontal, or are vertical.

For the general equation, the values of \( a \), \( b \), and \( c \) determine the line's orientation and position:

• If \( b eq 0 \), the line is neither vertical nor horizontal, and the equation can be transformed into the slope-intercept form.
• If \( b = 0 \) and \( a eq 0 \), the line is vertical, as represented by \( x = \frac{c}{a} \).
• If \( a = 0 \) and \( b eq 0 \), the line is horizontal, following the form \( y = \frac{c}{b} \).

This equation is versatile and can describe any linear relationship between two variables, making it a crucial part of linear algebra and mathematical graphing.