Problem 44 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the equation of the line with \(x\)-intercept \(a \neq 0\) and \(y\)-intercept \(b \neq 0\) can be written as $$ \frac{x}{a}+\frac{y}{b}=1 $$

Step-by-Step Solution

Answer

The line with intercepts \(a\) and \(b\) is \(\frac{x}{a} + \frac{y}{b} = 1\).

Step 1: Identify the Line in Intercept Form

A line that has an \( x \)-intercept of \( a \) and a \( y \)-intercept of \( b \) indicates that the line will pass through the points \((a, 0)\) and \((0, b)\). The equation of the line using these intercepts can be expressed in intercept form, which is \( \frac{x}{a} + \frac{y}{b} = 1 \).

Step 2: Understanding the Intercept Form Equation

In the equation \( \frac{x}{a} + \frac{y}{b} = 1 \), solving for \(x\) when \(y = 0\) gives the \(x\)-intercept as \( a \), and solving for \(y\) when \(x = 0\) gives the \(y\)-intercept as \( b \). Therefore, this equation satisfies the conditions provided in the problem.

Step 3: Concluding the Proof

Since the provided equation \( \frac{x}{a} + \frac{y}{b} = 1 \) successfully describes a line with the specified \(x\)- and \(y\)-intercepts, the proof is concluded. All points on the line satisfy this equation under the conditions of the given intercepts.

Key Concepts

Intercept Form

The intercept form of a linear equation is a way of writing the equation of a line using its intercepts on the x-axis and y-axis. This form highlights the intercepts directly and makes it easy to see where the line crosses these axes. The general formula for the intercept form of a line is \( \frac{x}{a} + \frac{y}{b} = 1 \), where:

• \( a \) is the x-intercept, the point where the line intersects the x-axis, meaning \( y = 0 \).
• \( b \) is the y-intercept, the point where the line intersects the y-axis, meaning \( x = 0 \).

This form is particularly useful because it directly shows how a line behaves across these influential points, simplifying both graphing and understanding the nature of the line.

x-intercept

The x-intercept of a line is the point where the line crosses the x-axis. At this point, the value of \( y \) is zero. To find the x-intercept from the intercept form equation \( \frac{x}{a} + \frac{y}{b} = 1 \), you set \( y = 0 \) and solve for \( x \). The calculation looks like this:

\[ \frac{x}{a} + \frac{0}{b} = 1 \]\[ \frac{x}{a} = 1 \]\[ x = a \]This tells us that the x-intercept is \( (a, 0) \), effectively establishing the value where the line meets the x-axis.

Understanding the x-intercept is crucial since it allows you to quickly identify one of the points on the line, aiding in both plotting the graph and confirming the slope's direction.

y-intercept

The y-intercept is where a line crosses the y-axis, which means at this point the value of \( x \) is zero. Using the intercept form equation \( \frac{x}{a} + \frac{y}{b} = 1 \), you can find the y-intercept by setting \( x = 0 \) and solving for \( y \). This process looks like this:

\[ \frac{0}{a} + \frac{y}{b} = 1 \]\[ \frac{y}{b} = 1 \]\[ y = b \]Thus, the y-intercept is \( (0, b) \).

Having a clear understanding of the y-intercept is valuable. It helps you visualize where the line starts on the vertical axis, which is essential for graphing and understanding the relationship represented by the equation.

Proof in Geometry

Proof in geometry involves demonstrating a statement's truth through logical reasoning and the use of established geometrical facts. In the context of the intercept form of a line, the proof shows how the relationship between intercepts translates into the equation \( \frac{x}{a} + \frac{y}{b} = 1 \).

• Firstly, identify the line fulfills the requirement of touching the x-axis at \( (a, 0) \) and the y-axis at \( (0, b) \).
• Secondly, substitute these intercepts into the equation to confirm its validity, showing both the x-intercept and y-intercept satisfy the line's equation.
• Lastly, establish that all other points on the line maintain this relationship, thereby proving the equation describes every point on the line under the conditions laid out.

This proof consolidates understanding by confirming that logically, the intercept form is consistent with what we know about the geometric properties of lines and their intercepts.