Problem 115 - College Algebra Essentials

Question

Prove that the equation of a line passing through \((a, 0)\) and \((0, b)(a \neq 0, b \neq 0)\) can be written in the form \(\frac{x}{a}+\frac{y}{b}=1\). Why is this called the intercept form of a line?

Step-by-Step Solution

Answer

The equation of a line passing through the points \((a, 0)\) and \((0, b)\) can be indeed written as \(\frac{x}{a} + \frac{y}{b} = 1\). This is called the intercept form because 'a' and 'b' are the x-intercept and y-intercept of the line respectively.

Step 1: Understanding Line Equation

The equation of a line passing through two distinct points \((x_1, y_1)\) and \((x_2, y_2)\) can be given as \((y - y_1) = \frac{{y_2 - y_1}}{{x_2 - x_1}} \cdot (x - x_1)\). Substituting the given points \((a, 0)\) and \((0, b)\), we get the equation for our line.

Step 2: Substitute the Given Points

The points to be substituted are \((a, 0)\) and \((0, b)\). Let's consider \((x_1, y_1) = (a, 0)\) and \((x_2, y_2) = (0, b)\) and substitute in our equation.

Step 3: Calculate the Slope of the Line

Calculate the slope of the line using the formula \(\frac{{y_2 - y_1}}{{x_2 - x_1}}\), we get \(\frac{{b - 0}}{{0 - a}} = -\frac{b}{a}\). Adding this to our equation.

Step 4: Plug in the Slope and Points into Line Equation

Substitute the slope \(-\frac{b}{a}\) and points \((a, 0)\) into the line equation. This gives us \(y - 0 = -\frac{b}{a} \cdot (x - a)\). Simplifying it to get \(\frac{1}{a} \cdot x + \frac{1}{b} \cdot y = 1\).

Key Concepts

Line Equation

In coordinate geometry, the equation of a line plays a fundamental role in describing how a line behaves on a graph. There are several forms of line equations, but one of the most straightforward ways to express a line is by using two points it passes through. The general formula is called the point-slope form:

• \((y - y_1) = m(x - x_1)\), where \(m\) is the slope.

In the exercise, the goal was to find the equation of a line passing through points \((a, 0)\) and \((0, b)\). By substituting these points into the point-slope formula, you can derive the line's equation in the intercept form as:

• \(\frac{x}{a} + \frac{y}{b} = 1\).

This form is very useful for quickly identifying the x-intercept (\(a\)) and y-intercept (\(b\)) of a line. Remember, the intercept form tells you the points where the line crosses the axes.

Slope

The slope of a line is a measure of its steepness and direction. It is calculated by the change in y-values divided by the change in x-values between two points on the line:

• Slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

In our case, the points are \((a, 0)\) and \((0, b)\), leading to a slope calculation of:

• \(m = \frac{b - 0}{0 - a} = -\frac{b}{a}\).

The negative slope means the line is descending as you move from left to right. This concept helps us understand whether the line rises or falls as it moves across the coordinate plane. The slope directly influences the equation of the line and its graph.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This field connects algebra and geometry, allowing us to describe geometric shapes and their properties algebraically.

Key concepts in coordinate geometry include:

• Points, defined by pairs of coordinates (e.g., \((x, y)\)).
• Lines, which can be described using equations like the point-slope form or intercept form.
• Distance and midpoints between points, calculated using specific formulas.

In the specific problem of finding a line's equation, coordinate geometry helps us transition from two given points \((a, 0)\) and \((0, b)\) to an algebraic expression \(\frac{x}{a} + \frac{y}{b} = 1\). This equation not only provides a way to accurately graph the line but also offers insights into the relationship between its algebraic and geometric properties.