Problem 6 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

$$ \text { For Problems } \text { find the equation of the line with the given characteristics. } $$ $$ \text { Passing through points }(0, a) \text { and }(b, 0) $$

Step-by-Step Solution

Answer

The equation of the line is \(y = \frac {-a} {b}x + a\).

Step 1: Compute the Slope

The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula \(m = \frac {y2 - y1} {x2 - x1}\). Here, the points are (0, a) and (b, 0). So the slope \(m = \frac {0 - a} {b - 0} = \frac {-a} {b}\)

Step 2: Utilize the Point-Slope Form of a Line

This is an equation that allows us to create an equation utilising the slope and a given point on the line which is \(y - y1 = m(x - x1)\). The slope here is -a/b. We will use the point (0, a) hence \(y - a = \frac {-a} {b} *(x - 0)\) beacuse it's simpler, however, using the point (b, 0) would work as well.

Step 3: Simplify the Equation

To get the equation in standard form \(y = mx + b\), we simplify to get \(y = \frac {-a} {b}x + a\).

Key Concepts

Point-Slope Form

Understanding the point-slope form is crucial for writing the equation of a line when you know the slope and any point on the line. Imagine it as a personalized formula suited for the line's particular features.

The general equation is expressed as \( y - y_1 = m(x - x_1) \), where \( m \) is the slope, and \( (x_1, y_1) \) is the known point. To use this form correctly, plug in the slope for \( m \) and the coordinates of the point into \( x_1 \) and \( y_1 \) respectively. In the textbook exercise, with the slope being \( -\frac{a}{b} \) and the point \( (0, a) \) in use, the line's equation is quickly found by substituting these values into the point-slope formula.

Applying Point to the Formula

By inputting our known point and slope into the point-slope form, we get \( y - a = -\frac{a}{b}(x - 0) \). This equation now successfully captures the unique characteristics of the line, ready for further simplification if necessary.

Slope Calculation

The slope of a line is a measure of its steepness and can be positive, negative, zero, or undefined. To find the slope, denoted \( m \), you need two points on the line, say \( (x_1, y_1) \) and \( (x_2, y_2) \).

The slope between these two points is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This operation is much like finding the ratio of the vertical change to the horizontal change between the two points. In the context of our exercise, the slope \( m = \frac{-a}{b} \) emerges by putting the coordinates \( (0, a) \) and \( (b, 0) \) into this formula, signifying a line that falls from left to right.

Understanding Slope Sign

In this instance, the negative slope indicates that as we move along the line from left to right, the line descends. Should the slope have been positive, it would have suggested an ascending line instead. Thus, the slope is intrinsic to the line's direction and incline.

Linear Equations

Linear equations are the basis for understanding relationships between two variables that produce a straight-line graph. Such an equation in its simplest form is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept - the point where the line crosses the y-axis.

For a more comprehensive representation, linear equations can manifest in various forms, including point-slope, slope-intercept, and standard form \( Ax + By = C \). Each form has its uses in different situations and can be transformed from one to another.

From Point-Slope to Slope-Intercept

As illustrated by the exercise, converting a point-slope form into the slope-intercept form \( y = mx + b \) entails algebraic manipulation. Once settings from the original exercise like \( y - a = -\frac{a}{b}x \) are simplified—by distributing the slope and moving the \( a \) term to the other side—we achieve the friendlier slope-intercept rendition of \( y = -\frac{a}{b}x + a \). This final form is commonly preferred for graphing since it directly reveals the line's slope and y-intercept.