Problem 22 - College Algebra with Modeling and Visualization

Question

If possible, find the slope of the line passing through each pair of points. $$ \left(-\frac{13}{15},-\frac{7}{8}\right),\left(\frac{1}{10}, \frac{3}{16}\right) $$

Step-by-Step Solution

Answer

The slope is \(\frac{255}{232}\).

Step 1: Identify the Points

The two points given are \((-\frac{13}{15}, -\frac{7}{8})\) and \((\frac{1}{10}, \frac{3}{16})\). Let's label these points as \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\). So, we have \(x_1 = -\frac{13}{15}\), \(y_1 = -\frac{7}{8}\), \(x_2 = \frac{1}{10}\), and \(y_2 = \frac{3}{16}\).

Step 2: Apply the Slope Formula

The formula for the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Step 3: Calculate the Difference in Y-Coordinates

Find the difference in the \(y\)-coordinates: \(y_2 - y_1 = \frac{3}{16} - \left(-\frac{7}{8}\right)\). Convert \(-\frac{7}{8}\) to a fraction with a denominator of 16: \(-\frac{7}{8} = -\frac{14}{16}\). Then, \(y_2 - y_1 = \frac{3}{16} + \frac{14}{16} = \frac{17}{16}\).

Step 4: Calculate the Difference in X-Coordinates

Find the difference in the \(x\)-coordinates: \(x_2 - x_1 = \frac{1}{10} - \left(-\frac{13}{15}\right)\). Convert \(-\frac{13}{15}\) to a fraction with a denominator of 30: \(-\frac{13}{15} = -\frac{26}{30}\). And \(\frac{1}{10} = \frac{3}{30}\). Therefore, \(x_2 - x_1 = \frac{3}{30} + \frac{26}{30} = \frac{29}{30}\).

Step 5: Calculate the Slope

Substitute the differences into the slope formula: \(m = \frac{17/16}{29/30}\). To divide fractions, multiply by the reciprocal: \(m = \frac{17}{16} \times \frac{30}{29} = \frac{510}{464}\). Simplify the fraction \(\frac{510}{464}\) by dividing the numerator and denominator by their greatest common divisor, which is 2: \(\frac{510}{464} = \frac{255}{232}\).

Step 6: Conclusion

The slope of the line passing through the points \((-\frac{13}{15}, -\frac{7}{8})\) and \((\frac{1}{10}, \frac{3}{16})\) is \(\frac{255}{232}\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a fascinating field combining algebra and geometry to discuss geometric figures using coordinates on a plane. Using a pair of numbers, called coordinates, you can specify the exact location of any point on the plane. Each point is given as \(x, y\) where \(x\) is the horizontal coordinate, and \(y\) is the vertical one.
In this exercise, we determine the slope of a line defined by two coordinates. The slope is a value indicating the steepness of a line as it travels from one point to another.

• The greater the slope, the steeper the line.
• Understanding slopes helps predict a line's direction. A positive slope indicates the line rises from left to right, while a negative slope means it descends.
• A slope of zero indicates a horizontal line.

Coordinate geometry teaches us how to compute such slopes through a simple formula, which is immensely helpful in understanding linear relationships and forms a core part of analyzing geometrical structures mathematically.

Fractions

Fractions are a way of representing parts of a whole, playing a significant role in various mathematical calculations. In the context of slope calculation, fractions are vital because both the coordinates and the slope itself may be expressed as fractions. Understanding fractions becomes crucial for accurate computations.
To compute the slope, you must deal with fractions when calculating differences between points and subsequently simplify the results.

• When adding or subtracting fractions, it's important to have a common denominator. For example, adding \(-\frac{7}{8}\) and \(\frac{3}{16}\) requires converting \(-\frac{7}{8}\) to \(-\frac{14}{16}\).
• Multiplying fractions involves multiplying the numerators together and the denominators together. In slope calculation, dividing by fractions often requires taking the reciprocal, changing division into multiplication for simplification.
• Simplifying fractions, as seen in dividing \(\frac{510}{464}\) by its greatest common divisor to become \(\frac{255}{232}\), ensures we express results in the simplest terms possible.

Through handling fractions with care, we ensure our solutions remain accurate and manageable, reinforcing fundamental mathematical skills.

Algebra Problem Solving

Algebra problem solving is an essential skill that applies algebraic principles to find solutions, making sense of mathematical relationships, and solving equations. In this problem, algebra allows us to apply a clear methodology for finding the slope of a line. The process involves several algebraic steps:

• Identify and label the given points, breaking down the problem into manageable pieces.
• Apply the slope formula \((m = \frac{y_2 - y_1}{x_2 - x_1})\), which involves calculating the difference in the \(y\)-coordinates and \(x\)-coordinates using algebraic manipulation.
• Execute operations such as subtraction, multiplication, and simplification, systematically working through each step using algebraic rules.

These steps highlight the practical application of algebra to real-world problems. Algebra teaches us to follow logical sequence of operations and reasoning, enabling clear solutions to problems like this slope problem. It transforms a complex mathematical process into a series of manageable tasks, ultimately yielding a precise result.