Problem 52 - Brief Applied Calculus

Question

Show that the linear equation \(\frac{x}{a}+\frac{y}{b}=1\) has \(x\) -intercept \((a, 0)\) and \(y\) -intercept \((0, b)\). (The \(x\) -intercept is the point where the line crosses the \(x\) -axis.)

Step-by-Step Solution

Answer

The x-intercept is \((a, 0)\) and the y-intercept is \((0, b)\).

Step 1: Identify the x-Intercept

To find the x-intercept, set \( y = 0 \) in the equation \( \frac{x}{a} + \frac{y}{b} = 1 \). This yields the equation \( \frac{x}{a} + 0 = 1 \) or simply \( \frac{x}{a} = 1 \).

Step 2: Solve for x-Intercept

Solve \( \frac{x}{a} = 1 \) for \( x \). Multiply both sides by \( a \) to get \( x = a \). Thus, the x-intercept is the point \((a, 0)\).

Step 3: Identify the y-Intercept

To find the y-intercept, set \( x = 0 \) in the equation \( \frac{x}{a} + \frac{y}{b} = 1 \). This yields the equation \( 0 + \frac{y}{b} = 1 \) or simply \( \frac{y}{b} = 1 \).

Step 4: Solve for y-Intercept

Solve \( \frac{y}{b} = 1 \) for \( y \). Multiply both sides by \( b \) to get \( y = b \). Thus, the y-intercept is the point \((0, b)\).

Step 5: Conclusion

Hence, the x-intercept of the line \( \frac{x}{a} + \frac{y}{b} = 1 \) is the point \((a, 0)\), and the y-intercept is the point \((0, b)\), as required.

Key Concepts

Understanding the x-intercept

The x-intercept of a linear equation is a key concept when understanding how a line behaves on a Cartesian coordinate plane. It is specifically the point where the line crosses the x-axis. At this intersection:

• The y-coordinate is always zero.
• You only need to solve for the x-coordinate.

For the equation \( \frac{x}{a} + \frac{y}{b} = 1 \), finding the x-intercept involves setting \( y = 0 \). This simplifies the equation to \( \frac{x}{a} = 1 \). Solving for \( x \), you multiply both sides by \( a \), resulting in \( x = a \). Thus, the x-intercept of the line is the point \((a, 0)\), showing where the line meets the x-axis.

Discovering the y-intercept

The y-intercept is equally important in analyzing a linear equation. This is where the line crosses the y-axis. Here, the specific details are:

• The x-coordinate is always zero.
• Focus on solving for the y-coordinate only.

With the equation \( \frac{x}{a} + \frac{y}{b} = 1 \), you find the y-intercept by setting \( x = 0 \). This reduces the original equation to \( \frac{y}{b} = 1 \). Solving for \( y \), you multiply both sides by \( b \), which gives \( y = b \). Consequently, the y-intercept is the point \((0, b)\). This helps illustrate where the line intersects with the y-axis.

Exploring Cartesian coordinates

The Cartesian coordinate system is a two-dimensional plane formed by two perpendicular lines, the x-axis and the y-axis. This system is fundamental for plotting and understanding the graph of any equation. A few crucial points about Cartesian coordinates are:

• The horizontal line is the x-axis, and the vertical line is the y-axis.
• Every point in this plane can be defined by a pair of numbers \((x, y)\).
• The point \((0,0)\) is known as the origin, where both axes intersect.

In the context of linear equations like \( \frac{x}{a} + \frac{y}{b} = 1 \), the Cartesian coordinate system allows you to effectively determine and plot locations such as the x-intercept \((a, 0)\) and the y-intercept \((0, b)\). Understanding this system is critical for visualizing how equations translate into lines on a graph.