Problem 55 - Elementary Algebra

Question

Find the slope, if it exists, of the line through the given pairs of points. $$ \left(\frac{1}{3}, \frac{3}{4}\right),\left(\frac{2}{9},-\frac{5}{6}\right) $$

Step-by-Step Solution

Answer

Question: Find the slope of the line going through the points $\left(\frac{1}{3}, \frac{3}{4}\right)$ and $\left(\frac{2}{9},-\frac{5}{6}\right)$. Answer: The slope of the line is $\frac{171}{12}$.

Step 1: Identify the Coordinates

We are given these two points on the line: $$\left(\frac{1}{3}, \frac{3}{4}\right)\ \text{and}\ \left(\frac{2}{9},-\frac{5}{6}\right)$$ Let's label them as follows: $$x_1 = \frac{1}{3},\ y_1 = \frac{3}{4}\ \ \text{and}\ \ x_2 = \frac{2}{9},\ y_2 = -\frac{5}{6}$$

Step 2: Apply the Slope Formula

Now, plug the values into the slope formula: $$m = \frac{y_2 - y_1}{x_2 - x_1} =\frac{-\frac{5}{6} - \frac{3}{4}}{\frac{2}{9} - \frac{1}{3}}$$

Step 3: Find a Common Denominator and Simplify

In order to subtract the fractions, we need to find a common denominator for both the numerator's and denominator's fractions. The Least Common Multiple (LCM) for 6 and 4 is 12, while the LCM for 9 and 3 is 9. Rewrite the fractions: $$m = \frac{\frac{-10}{12} - \frac{9}{12}}{\frac{2}{9} - \frac{3}{9}}$$ Now, subtract the fractions: $$m = \frac{-19}{12} \cdot \frac{1}{-1/9}$$

Step 4: Divide by the Reciprocal

Let's rewrite the expression by dividing by the reciprocal of {-1/9}, which is 9. $$m = \frac{-19}{12} \cdot -9 = \frac{-19}{12} \cdot \frac{-9}{1}$$

Step 5: Multiply and Simplify

Multiply the numerators and denominators and simplify: $$m = \frac{(-19)(-9)}{(12)(1)} = \frac{171}{12}$$ So, the slope of the line going through the given points is $$\frac{171}{12}$$.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses the principles of algebra and geometry to describe the relationships between points, lines, and shapes in a coordinate plane. It deals with locating points on a plane using ordered pairs of numbers (coordinates) like \(x, y\).

In this problem, we have two points given as \(\left(\frac{1}{3}, \frac{3}{4}\right)\) and \(\left(\frac{2}{9},-\frac{5}{6}\right)\).

• The first in the pair corresponds to the x-coordinate.
• The second in the pair corresponds to the y-coordinate.

These coordinates help us define and measure the distance and direction of a line, through crucial calculations like finding the slope, which measures how steep the line is.

Slope Formula

The slope formula helps us determine how inclined a line is by analyzing two points on the line. The slope is expressed as "m," and it signifies the change in \(y\) for a unit change in \(x\).

For two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope formula is:

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

It's a straightforward formula derived from the concept of \'rise over run,\' capturing how much y changes in relation to x. In the provided example, substituting the given points into this formula is the first step in finding the slope between them.
Understanding calculated differences in y-values (rise) and x-values (run) gives insight into the behavior of the line between these points.

Fraction Operations

Handling fractions correctly is a key skill when working with the slope formula, especially for coordinates given as fractions.

Fraction operations include:

• Finding a common denominator, crucial for adding or subtracting fractions. It ensures both fractions have the same base, making them comparable.
• Subtracting fractions involves converting them to have a common denominator and then subtracting the numerators.
• Multiplying and dividing fractions requires multiplying or dividing the numerators and denominators respectively.

In the exercise, subtracting \(\frac{3}{4}\) from \(\frac{-5}{6}\) and \(\frac{1}{3}\) from \(\frac{2}{9}\) needed common denominators.
Fractions were then rearranged for easier subtraction, leading to formula application.

Simplifying Fractions

Once you perform operations like subtraction or finding the slope, the result may need simplification. Cross-check for common factors in both the numerator and denominator.
Simplifying fractions involves:

• Dividing both the numerator and the denominator by their Greatest Common Divisor (GCD).
• Converting improper fractions into mixed numbers when needed.

In our solution, \(\frac{171}{12}\) is obtained as the slope, which simplifies the steps by reducing any unnecessary complexity.
Essential skills such as multiplying the numerators and denominators, and reducing the fractions, help clarify the final outcome, ensuring clarity and correctness in the result.