Problem 70 - College Algebra with Modeling and Visualization

Question

The intercept form of a line is \(\frac{x}{a}+\frac{y}{b}=1\) Determine the \(x\) -and y-intercepts on the graph of the equation. Draw a conclusion about what the constants a and b represent in this form. $$ \frac{x}{2}+\frac{y}{3}=1 $$

Step-by-Step Solution

Answer

x-intercept is 2 and y-intercept is 3; constants a and b are the intercepts.

Step 1: Understand the Intercept Form Equation

The intercept form of a line is given by \( \frac{x}{a} + \frac{y}{b} = 1 \). In this form, \( a \) and \( b \) are constants that help us to determine where the line crosses the x- and y-axes.

Step 2: Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( \frac{x}{2} + \frac{y}{3} = 1 \). This simplifies to \( \frac{x}{2} = 1 \). Multiplying both sides by 2 gives \( x = 2 \). So, the x-intercept is 2.

Step 3: Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( \frac{x}{2} + \frac{y}{3} = 1 \). This simplifies to \( \frac{y}{3} = 1 \). Multiplying both sides by 3 gives \( y = 3 \). So, the y-intercept is 3.

Step 4: Conclusion about Constants a and b

In the equation \( \frac{x}{a} + \frac{y}{b} = 1 \), the constant \( a \) represents the x-intercept and the constant \( b \) represents the y-intercept. Therefore, for the equation \( \frac{x}{2} + \frac{y}{3} = 1 \), \( a = 2 \), indicating the x-intercept at 2, and \( b = 3 \), indicating the y-intercept at 3.

Key Concepts

Understanding the x-intercept

The x-intercept is the point where a line crosses the x-axis on a graph. To find the x-intercept of a linear equation, you need to set the y value to zero, because the x-intercept represents the point where the y value is zero.
Let's look at the intercept form of a line: \( \frac{x}{a} + \frac{y}{b} = 1 \). For this equation, to find the x-intercept, you make the substitution \( y = 0 \). This results in \( \frac{x}{a} = 1 \), which simplifies to \( x = a \).
In the equation \( \frac{x}{2} + \frac{y}{3} = 1 \), set \( y \) to zero. This simplifies to \( \frac{x}{2} = 1 \), yielding \( x = 2 \). Hence, the x-intercept is 2. It shows where the graph cuts the x-axis, and illustrates that \( a \), in this equation, represents the x-intercept.

Understanding the y-intercept

The y-intercept is slightly different - it is the point where the line crosses the y-axis on the graph. At the y-intercept, the x value is zero because it is the location on the y-axis where the line meets it.
Using the intercept form \( \frac{x}{a} + \frac{y}{b} = 1 \), to find the y-intercept, set \( x = 0 \). This changes the equation to \( \frac{y}{b} = 1 \), which simplifies to \( y = b \).
For the specific equation \( \frac{x}{2} + \frac{y}{3} = 1 \), substitute \( x \) with zero. The equation simplifies to \( \frac{y}{3} = 1 \) which leads to \( y = 3 \). Therefore, the y-intercept is 3, showing that the graph crosses the y-axis at this point, and confirming \( b \) as the y-intercept.

Exploring Linear Equations

Linear equations represent straight lines and have a standard form, often expressed in different ways to serve various calculations. The intercept form \( \frac{x}{a} + \frac{y}{b} = 1 \) is particularly useful because it easily identifies where the line crosses the axes.
Linear equations can be plotted on a graph to visually demonstrate the relationship between x and y. The main feature of linear equations is that they graph to a straight line. This simplicity makes them a powerful tool for modeling basic relationships in algebra and geometry.
The intercept form provides unique insights by highlighting the x- and y-intercepts directly through the constants \( a \) and \( b \). In the case of \( \frac{x}{2} + \frac{y}{3} = 1 \), it quickly reveals that the line crosses the x-axis at 2 and the y-axis at 3, offering clarity in graphing and interpreting the equation's behavior.