Problem 46 - Precalculus

Question

Write down, and then simplify as much as possible, an expression for the slope of the line passing through the two points \((a, 1 / a)\) and \((x, 1 / x)\).

Step-by-Step Solution

Answer

The slope of the line is \(\frac{-1}{ax}\).

Step 1: Recall the Formula for the Slope of a Line

To find the slope of a line passing through two points, we use the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are the given points.

Step 2: Assign the Points to the Formula Variables

Identify \((x_1, y_1)\) as \((a, 1/a)\) and \((x_2, y_2)\) as \((x, 1/x)\). Substitute these values into the slope formula, yielding: \( m = \frac{\frac{1}{x} - \frac{1}{a}}{x - a} \).

Step 3: Simplify the Expression for the Numerator

Combine the fractions in the numerator: \( \frac{1}{x} - \frac{1}{a} = \frac{a - x}{ax} \). This can be achieved by getting a common denominator for the fractions.

Step 4: Substitute and Simplify the Full Expression

Substitute the simplified numerator into the slope expression: \( m = \frac{\frac{a - x}{ax}}{x - a} \). Simplify this complex fraction by flipping the denominator and multiply: \( m = \frac{a - x}{ax} \times \frac{1}{x - a} = \frac{a - x}{ax(x - a)} \).

Step 5: Simplify the Final Expression

Notice that \(a - x = -(x - a)\). Replace and simplify: \( m = \frac{-(x - a)}{ax(x - a)} = \frac{-1}{ax} \).

Key Concepts

Simplifying Expressions

When we talk about simplifying expressions, we aim to write them in the most concise form possible without changing their value. This process can involve eliminating unnecessary parts or combining like terms. Simplifying expressions allows us to see the essence of a mathematical relationship. In algebra, it usually means performing operations that reduce the expression to its core components.

For example, if we start with an expression like \( \frac{\frac{1}{x} - \frac{1}{a}}{x - a}\), we can simplify it step by step. The first move is to combine the fractions in the numerator by finding a common denominator, which results in \( \frac{a - x}{ax}\).



By tackling each part of the expression methodically, you transform a seemingly complex statement into a more manageable form. This process requires patience and attention to detail, as each step builds on the previous one.

Simplifying expressions is not only satisfying but also an essential skill in algebra and higher mathematics, as it lays the groundwork for solving more complex problems.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both, are algebraic expressions. They require special care when simplifying or performing operations due to their variable components. An understanding of algebraic fractions is crucial for solving problems involving ratios and proportions in algebra.

To work with algebraic fractions, key strategies include:

• Identifying a common denominator for addition or subtraction.
• Factoring numerators and denominators where possible.
• Simplifying by canceling common factors in both numerator and denominator.

In our original example, we had algebraic fractions in the form \( \frac{1}{x} - \frac{1}{a}\), and found the common denominator to be \( ax \). This allowed us to combine them into a single fraction: \( \frac{a - x}{ax} \). The idea is to make calculations easier by reducing the number of fractions you're working with.

The ability to handle algebraic fractions effectively opens the door to more complex algebraic operations and equations, laying a strong foundation as you progress in mathematics.

Linear Equations

Linear equations are equations of the first degree, meaning they involve only the first power of the variable. These equations take the general form \( y = mx + b\), where \( m \) is the slope and \( b \) is the y-intercept. Understanding linear equations is a cornerstone of algebra because they represent the simplest form of relationships between variables.

The concept of slope, \( m \), is crucial when working with linear equations, as it indicates the rate of change between the variables on the graph. For instance, calculating the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) involves the change in y-values over the change in x-values: \( m = \frac{y_2 - y_1}{x_2 - x_1}\). This slope dictates the steepness and direction of the line on a graph.

Linear equations facilitate solving a variety of practical problems, from calculating distances to predicting future values in data sets. Mastering linear equations opens up vast application opportunities, offering valuable insights into the relationships they describe.