Problem 26 - Calculus Single Variable

Question

Write the intercept form of the equation of the line determined by the given data. \(x\) -intercept \(5, y\) -intercept \(1 / 3\)

Step-by-Step Solution

Answer

The intercept form of the line is \( \frac{x}{5} + 3y = 1 \).

Step 1: Understand the Intercept Form

The intercept form of the equation of a line is given by \( \frac{x}{a} + \frac{y}{b} = 1 \), where \(a\) is the x-intercept and \(b\) is the y-intercept.

Step 2: Identify the Intercepts

From the problem, we have the x-intercept \(a = 5\) and the y-intercept \(b = \frac{1}{3}\). These values will be used to write the equation.

Step 3: Substitute the Intercepts into the Intercept Form

Substitute \(a = 5\) and \(b = \frac{1}{3}\) into the equation \( \frac{x}{a} + \frac{y}{b} = 1 \). This gives us \( \frac{x}{5} + \frac{y}{\frac{1}{3}} = 1 \).

Step 4: Simplify the Equation

To simplify \( \frac{x}{5} + \frac{y}{\frac{1}{3}} = 1 \), rewrite \( \frac{y}{\frac{1}{3}} \) as \( 3y \), because dividing by a fraction is equivalent to multiplying by its reciprocal. Thus, the equation becomes \( \frac{x}{5} + 3y = 1 \).

Key Concepts

Equation of a Line

When discussing the equation of a line, it's helpful to understand the different forms that such equations can take. A line in a plane can be represented in multiple forms like the slope-intercept form, point-slope form, and the intercept form. Each one serves a different purpose and is suitable for specific kinds of information about the line.

For instance, the intercept form of a line’s equation is very handy when you know the points where the line crosses the x-axis and y-axis. It is given by the equation:

• \( \frac{x}{a} + \frac{y}{b} = 1 \)

In this context, \( a \) and \( b \) are the x-intercept and y-intercept, respectively. This format makes it easy to see the intercepts directly from the equation. So, if you can identify these intercepts, you can jump right to writing this form of the equation.

This method is simple, and once you understand it, you will find it practical for solving a wide range of problems involving linear equations.

X-Intercept

The x-intercept of a line is the point where the line crosses the x-axis. To understand this better, imagine a horizontal axis where a line slants and intersects. That specific point where it lands on the x-axis is known as the x-intercept.

In mathematical terms, this is the value of \( x \) when \( y = 0 \). In the context of the intercept form of a line's equation, represented as \( \frac{x}{a} + \frac{y}{b} = 1 \), the value \( a \) represents the x-intercept. It's important because knowing \( a \) allows you to know one key characteristic about the line without needing further data.

In our example problem, if the x-intercept \( a \) is given as 5, then the point where the line crosses the x-axis is (5, 0). This concise point tells us a lot about the placement and slope of the line in relation to the horizontal axis.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. At this point, the value of \( x \) is zero. By seeing where a line hits the y-axis, you're observing the y-intercept.

This intercept provides a fundamental piece of information for setting the stage of our linear equation in intercept form. In the equation \( \frac{x}{a} + \frac{y}{b} = 1 \), \( b \) is used to indicate the y-intercept. This allows us to understand and pinpoint where the line aligns vertically.

For example, in our given problem, the y-intercept \( b \) is \( \frac{1}{3} \). As a result, the line passes through (0, \( \frac{1}{3} \)) on the y-axis. Seeing the line’s equation written in intercept form makes deduction easier. Knowing the y-intercept is particularly useful for graphing or visualizing the line's interaction with the axis, even in a purely algebraic context.