Problem 21 - College Algebra with Modeling and Visualization

Question

If possible, find the slope of the line passing through each pair of points. $$ \left(\frac{1}{3},-\frac{3}{5}\right),\left(-\frac{5}{6}, \frac{7}{10}\right) $$

Step-by-Step Solution

Answer

The slope is \( -\frac{39}{35} \).

Step 1: Identify the Points

The given points are \( \left( \frac{1}{3}, -\frac{3}{5} \right) \) and \( \left( -\frac{5}{6}, \frac{7}{10} \right) \). Let's label these points as \( (x_1, y_1) \) and \( (x_2, y_2) \).

Step 2: Recall the Slope Formula

The formula to find 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: Substitute the Points into the Slope Formula

Substitute \( x_1 = \frac{1}{3} \), \( y_1 = -\frac{3}{5} \), \( x_2 = -\frac{5}{6} \), and \( y_2 = \frac{7}{10} \) into the slope formula: \[ m = \frac{\frac{7}{10} - \left(-\frac{3}{5}\right)}{-\frac{5}{6} - \frac{1}{3}} \]

Step 4: Simplify the Numerator

The numerator of the slope formula becomes: \[ \frac{7}{10} + \frac{3}{5} = \frac{7}{10} + \frac{6}{10} = \frac{13}{10} \]

Step 5: Simplify the Denominator

The denominator of the slope formula becomes: \[ -\frac{5}{6} - \frac{1}{3} = -\frac{5}{6} - \frac{2}{6} = -\frac{7}{6} \]

Step 6: Complete the Calculation

Substitute the simplified numerator and denominator back into the slope formula: \[ m = \frac{\frac{13}{10}}{-\frac{7}{6}} = \frac{13}{10} \times -\frac{6}{7} = -\frac{78}{70} \] Then simplify the fraction: \[ -\frac{78}{70} = -\frac{39}{35} \]

Step 7: State the Slope

The slope of the line passing through the points \( \left( \frac{1}{3}, -\frac{3}{5} \right) \) and \( \left( -\frac{5}{6}, \frac{7}{10} \right) \) is \( -\frac{39}{35} \).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, helps us understand geometry through a coordinate system. In this system, each point has a pair of numerical coordinates. These coordinates are expressed as \(x\) and \(y\), representing the point's location on the horizontal and vertical axes, respectively. This approach allows us to study geometric problems using algebra.

For example, to analyze the line passing through the points \( \, \left( \frac{1}{3}, -\frac{3}{5} \right) \, \) and \( \, \left( -\frac{5}{6}, \frac{7}{10} \right) \, \), we plot these points on a set of axes and investigate their relationship. In coordinate geometry, lines, curves, and shapes can be easily represented and manipulated through their equations. Understanding the concepts of coordinate geometry facilitates working with equations of lines, circles, and other shapes.

Fraction Operations

Working with fractions is a core skill in solving problems involving coordinate geometry, especially when the points or slopes are expressed as fractions. Fraction operations involve several key processes: addition, subtraction, multiplication, and division of fractions. Here’s how each of these are handled in the context of calculating the slope of a line:

• Adding/Subtracting Fractions: To combine fractions like \( \frac{7}{10} \) and \( -\frac{3}{5} \), find a common denominator. In this case, \( 10 \) is a common denominator for both fractions, allowing them to be combined as \( \frac{7}{10} + \frac{6}{10} = \frac{13}{10} \).
• Multiplying/Dividing Fractions: When simplifying the slope \( \frac{13}{10} \times -\frac{6}{7} \,\), you multiply the numerators and denominators separately: \(13 \times -6 \) and \(10 \times 7 \,\), then reduce to the simplest form.

Understanding these operations is critical for solving algebraic equations involving fractions, such as finding slopes, intercepts, and angles.

Slope Formula

The slope of a line is a measure that describes its steepness and direction. To find the slope of a line passing through two points, we use the slope formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula calculates how much the \(y\)-coordinate (vertical change) changes per unit change in the \(x\)-coordinate (horizontal change). A positive slope means the line rises as it moves along the horizontal axis, while a negative slope indicates the line falls.

For example, to find the slope between the points \(\frac{1}{3}, -\frac{3}{5}\) and \(-\frac{5}{6}, \frac{7}{10}\), we substitute these coordinates into the formula:\[m = \frac{\frac{7}{10} - (-\frac{3}{5})}{-\frac{5}{6} - \frac{1}{3}} = \frac{\frac{13}{10}}{-\frac{7}{6}} \]After performing the necessary fraction operations, we determine the slope is \(-\frac{39}{35}\). Understanding the slope formula helps in analyzing linear relationships and predicting the behavior of a line on a graph.