Problem 63 - Algebra and Trigonometry

Question

(a) Show that if the \(x\) . and \(y\) intercepts of a line are nonzero numbers \(a\) and \(b\) , then the equation of the line can be written in the form $$ \frac{x}{a}+\frac{y}{b}=1 $$ This is called the twe-intercept form of the equation of a line. (b) Use part (a) to find an equation of the line whose \(x\) -intercept is 6 and whose \(y\) -intercept is \(-8\) .

Step-by-Step Solution

Answer

Equation: \(\frac{x}{6} + \frac{y}{-8} = 1\)

Step 1: Understand the intercepts

A line's intercepts are points where the line crosses the axes. The \(x\)-intercept \(a\) is the x-coordinate of the point where the line crosses the x-axis \((a, 0)\), and the \(y\)-intercept \(b\) is the y-coordinate of the point where the line crosses the y-axis \((0, b)\).

Step 2: Equation of a line using intercepts

The general equation of a line is \(y = mx + c\). To convert this line into the intercept form \(\frac{x}{a} + \frac{y}{b} = 1\), we utilize the points where the line would intersect the axes, specifically \((a, 0)\) and \((0, b)\).

Step 3: Derive the equation from intercepts

The point \((a, 0)\) means when \(y = 0\), \(x = a\). Substituting in \(\frac{x}{a} + \frac{y}{b} = 1\), we have: \(\frac{a}{a} + \frac{0}{b} = 1\), which simplifies to 1. The point \((0, b)\) means when \(x = 0\), \(y = b\). Substituting in, we have: \(\frac{0}{a} + \frac{b}{b} = 1\), which again simplifies to 1. Thus, the line satisfies the equation at both intercepts.

Step 4: Solve part (b) using intercept form

Given \(x\)-intercept \(a = 6\) and \(y\)-intercept \(b = -8\). Substituting these into \(\frac{x}{a} + \frac{y}{b} = 1\), we have \(\frac{x}{6} + \frac{y}{-8} = 1\). This is the line's equation using the intercept form.

Key Concepts

x-intercept

The x-intercept of a line is a fundamental concept that refers to the point where the line crosses the x-axis. At this intersection, the y-coordinate is always zero. This means the x-intercept is the x-coordinate of the point on the graph where the line meets the x-axis.
For instance, if a line crosses the x-axis at the point (3, 0), then the x-intercept is 3. This simply tells us that when y is zero, x equals the x-intercept value.
Finding the x-intercept can be quite useful, especially when given in the context of the two-intercept form of a line's equation. In this form, you can easily identify the x-intercept by looking at the denominator in the fraction containing the x variable. If the equation of the line in two-intercept form is \( \frac{x}{a} + \frac{y}{b} = 1 \), then the x-intercept is denoted by \( a \).
Understanding and identifying the x-intercept is crucial when working with different formats of linear equations, as it provides significant insight into the line’s interaction with the horizontal axis.

y-intercept

The y-intercept is just as essential as the x-intercept in understanding the graph of a line. It is the point where the line intersects the y-axis. At this particular point, the x-coordinate is always zero, so the coordinates of the y-intercept are typically written as (0, b), where b is the actual y-intercept value.
Imagine a line crossing the y-axis at the point (0, -4). In this scenario, the y-intercept is -4. It tells us that when x equals zero, the y-value is at the intercept, -4.
The y-intercept plays a significant role in various mathematical problems and real-world applications, as it often represents some initial value or starting point before any changes (denoted as x) occur.
In the two-intercept form \( \frac{x}{a} + \frac{y}{b} = 1 \), the y-intercept \( b \) can be directly identified from the equation as the denominator in the fraction containing the y variable.

• It gives information on how a line behaves concerning the y-axis.
• It helps in understanding the line's starting point when x is zero.

equation of a line

The equation of a line is a mathematical way to represent the linear relationship between x and y variables. There are different forms of line equations, such as the slope-intercept form \( y = mx + c \) and the standard form \( Ax + By = C \). Each offers unique insights into a line's properties.
The two-intercept form \( \frac{x}{a} + \frac{y}{b} = 1 \) is especially useful when you know the x-intercept (a) and y-intercept (b) of the line. This form provides a straightforward way to build the equation using simply the points where the line crosses the axes.
To understand how the equation reveals the behavior of the line:

• The coefficients \( a \) and \( b \) represent the direct intercept distances from the origin along their respective axes.
• By rearranging or solving this form, you can derive other line forms to study the slope or other characteristics of the line.

This format is intuitive for visualizing how far a line travels across the x and y axes before intersecting each axis, creating a geometric perspective on algebraic concepts.